resolution of singularities steven dale cutkosky - university of

Resolution of Singularities

Steven Dale Cutkosky

Department of Mathematics, University of Missouri, Columbia,Missouri 65211

To Hema, Ashok and Maya

Contents

Preface vii

Chapter 1. Introduction 1

§1.1. Notation 2

Chapter 2. Non-singularity and Resolution of Singularities 3

§2.1. Newton’s method for determining the branches of a planecurve 3

§2.2. Smoothness and non-singularity 7

§2.3. Resolution of singularities 9

§2.4. Normalization 10

§2.5. Local uniformization and generalized resolution problems 11

Chapter 3. Curve Singularities 17

§3.1. Blowing up a point on A2 17

§3.2. Completion 22

§3.3. Blowing up a point on a non-singular surface 25

§3.4. Resolution of curves embedded in a non-singular surface I 26

§3.5. Resolution of curves embedded in a non-singular surface II 29

Chapter 4. Resolution Type Theorems 37

§4.1. Blow-ups of ideals 37

§4.2. Resolution type theorems and corollaries 40

Chapter 5. Surface Singularities 45

§5.1. Resolution of surface singularities 45

v

vi Contents

§5.2. Embedded resolution of singularities 56

Chapter 6. Resolution of Singularities in Characteristic Zero 61§6.1. The operator 4 and other preliminaries 62§6.2. Hypersurfaces of maximal contact and induction in resolution 66§6.3. Pairs and basic objects 70§6.4. Basic objects and hypersurfaces of maximal contact 75§6.5. General basic objects 81§6.6. Functions on a general basic object 83§6.7. Resolution theorems for a general basic object 89§6.8. Resolution of singularities in characteristic zero 99

Chapter 7. Resolution of Surfaces in Positive Characteristic 105§7.1. Resolution and some invariants 105§7.2. τ(q) = 2 109§7.3. τ(q) = 1 113§7.4. Remarks and further discussion 130

Chapter 8. Local Uniformization and Resolution of Surfaces 133§8.1. Classification of valuations in function fields of dimension 2 133§8.2. Local uniformization of algebraic function fields of surfaces 137§8.3. Resolving systems and the Zariski-Riemann manifold 148

Chapter 9. Ramification of Valuations and Simultaneous Resolution 155

Appendix. Smoothness and Non-singularity II 163§A.1. Proofs of the basic theorems 163§A.2. Non-singularity and uniformizing parameters 169§A.3. Higher derivations 171§A.4. Upper semi-continuity of νq(I) 174

Bibliography 179

Index 185

Preface

The notion of singularity is basic to mathematics. In elementary algebrasingularity appears as a multiple root of a polynomial. In geometry a pointin a space is non-singular if it has a tangent space whose dimension is thesame as that of the space. Both notions of singularity can be detectedthrough the vanishing of derivitives.

Over an algebraically closed field, a variety is non-singular at a pointif there exists a tangent space at the point which has the same dimensionas the variety. More generally, a variety is non-singular at a point if itslocal ring is a regular local ring. A fundamental problem is to remove asingularity by simple algebraic mappings. That is, can a given variety bedesingularized by a proper, birational morphism from a non-singular variety?This is always possible in all dimensions, over fields of characteristic zero.We give a complete proof of this in Chapter 6.

We also treat positive characteristic, developing the basic tools neededfor this study, and giving a proof of resolution of surface singularities inpositive characteristic in Chapter 7.

In Section 2.5 we discuss important open problems, such as resolutionof singularities in positive characteristic and local monomialization of mor-phisms.

Chapter 8 gives a classification of valuations in algebraic function fieldsof surfaces, and a modernization of Zariski’s original proof of local uni-formization for surfaces in characteristic zero.

This book has evolved out of lectures given at the University of Mis-souri and at the Chennai Mathematics Institute, in Chennai, (also knownas Madras), India. It can be used as part of a one year introductory sequence

vii

viii Preface

in algebraic geometry, and would provide an exciting direction after the ba-sic notions of schemes and sheaves have been covered. A core course onresolution is covered in Chapters 2 through 6. The major ideas of resolutionhave been introduced by the end of Section 6.2, and after reading this far,a student will find the resolution theorems of Section 6.8 quite believable,and have a good feel for what goes into their proofs.

Chapters 7 and 8 cover additional topics. These two chapters are inde-pendent, and can be chosen as possible followups to the basic material inthe first 5 chapters. Chapter 7 gives a proof of resolution of singularitiesfor surfaces in positive characteristic, and Chapter 8 gives a proof of localuniformization and resolution of singularities for algebraic surfaces. Thischapter provides an introduction to valuation theory in algebraic geometry,and to the problem of local uniformization.

The appendix proves foundational results on the singular locus that weneed. On a first reading, I recommend that the reader simply look upthe statements as needed in reading the main body of the book. Versionsof almost all of these statements are much easier over algebraically closedfields of characteristic zero, and most of the results can be found in this casein standard textbooks in algebraic geometry.

I assume that the reader has some familiarity with algebraic geometryand commutative algebra, such as can be obtained from an introductorycourse on these subjects. This material is covered in books such as Atiyahand MacDonald [13] or the basic sections of Eisenbud’s book [37], andthe first two chapters of Hartshorne’s book on algebraic geometry [47], orEisenbud and Harris’s book on schemes [38].

I thank Professors Seshadri and Ed Dunne for their encouragement towrite this book, and Laura Ghezzi, Tai Ha, Krishna Hanamanthu, OlgaKashcheyeva and Emanoil Theodorescu for their helpful comments on pre-liminary versions of the manuscript.

For financial support during the preparation of this book I thank theNational Science Foundation, the National Board of Higher Mathematics ofIndia, the Mathematical Sciences Research Insititute and the University ofMissouri.

Steven Dale Cutkosky

Chapter 1

Introduction

An algebraic variety X is defined locally by the vanishing of a system ofpolynomial equations fi ∈ K[x1, . . . , xn],

f1 = · · · = fm = 0.

If K is algebraically closed, points of X in this chart are α = (α1, . . . , αn) ∈AnK which satisfy this system. The tangent space Tα(X) at a point α ∈ X

is the linear subspace of AnK defined by the system of linear equations

L1 = · · · = Lm = 0,

where Li is defined by

Li =n∑j=1

∂fi∂xj

(α)(xj − αj).

We have that dim Tα(X) ≥ dim X, and X is non-singular at the point αif dim Tα(X) = dim X. The locus of points in X which are singular is aproper closed subset of X.

The fundamental problem of resolution of singularities is to performsimple algebraic transformations of X so that the transform Y of X is non-singular everywhere. To be precise, we seek a resolution of singularitiesof X; that is, a proper birational morphism Φ : Y → X such that Y isnon-singular.

The problem of resolution when K has characteristic zero has been stud-ied for some time. In fact we will see (Chapters 2 and 3) that the method ofNewton for determining the analytical branches of a plane curve singularityextends to give a proof of resolution for algebraic curves. The first algebraicproof of resolution of surface singularities is due to Zariski [86]. We give

1

2 1. Introduction

a modern treatment of this proof in Chapter 8. A study of this proof issurely the best introduction to the methods and ideas underlying the recentemphasis on valuation-theoretic methods in resolution problems.

The existence of a resolution of singularities has been completely solvedby Hironaka [52], in all dimensions, when K has characteristic zero. We givea simplified proof of this theorem, based on the proof of canonical resolutionby Encinas and Villamayor ([40], [41]), in Chapter 6.

When K has positive characteristic, resolution is known for curves, sur-faces and 3-folds (with char(K) > 5). The first proof in positive characteris-tic of resolution of surfaces and of resolution for 3-folds is due to Abhyankar[1], [4].

We give several proofs, in Chapters 2, 3 and 4, of resolution of curvesin arbitrary characteristic. In Chapter 5 we give a proof of resolution ofsurfaces in arbitrary characteristic.

1.1. Notation

The notation of Hartshorne [47] will be followed, with the following differ-ences and additions.

By a variety over a field K (or a K-variety), we will mean an open subsetof an equidimensional reduced subscheme of the projective space PnK . Thusan integral variety is a “quasi-projective variety” in the classical sense. Acurve is a one-dimensional variety. A surface is a two-dimensional variety,and a 3-fold is a three-dimensional variety. A subvariety Y of a variety Xis a closed subscheme of X which is a variety.

An affine ring is a reduced ring which is of finite type over a field K.If X is a variety, and I is an ideal sheaf on X, we denote V (I) =

spec(OX/I) ⊂ X. If Y is a subscheme of a variety X, we denote the idealof Y in X by IY . If W1,W2 are subschemes of a variety of X, we will denotethe scheme-theoretic intersection of W1 and W2 by W1 · W2. This is thesubscheme

W1 ·W2 = V (IW1 + IW2) ⊂W.

A hypersurface is a codimension one subvariety of a non-singular variety.

Chapter 2

Non-singularity andResolution ofSingularities

2.1. Newton’s method for determining the branches of aplane curve

Newton’s algorithm for solving f(x, y) = 0 by a fractional power seriesy = y(x

1m ) can be thought of as a generalization of the implicit function

theorem to general analytic functions. We begin with this algorithm becauseof its simplicity and elegance, and because this method contains some ofthe most important ideas in resolution. We will see (in Section 2.5) thatthe algorithm immediately gives a local solution to resolution of analyticplane curve singularities, and that it can be interpreted to give a globalsolution to resolution of plane curve singularities (in Section 3.5). All ofthe proofs of resolution in this book can be viewed as generalizations ofNewton’s algorithm, with the exception of the proof that curve singularitiescan be resolved by normalization (Theorems 2.14 and 4.3).

Suppose K is an algebraically closed field of characteristic 0, K[[x, y]] isa ring of power series in two variables and f ∈ K[[x, y]] is a non-unit, suchthat x - f . Write f =

∑i,j aijx

iyj with aij ∈ K. Let

mult(f) = mini+ j | aij 6= 0,

and

mult(f(0, y)) = minj | a0j 6= 0.

3

4 2. Non-singularity and Resolution of Singularities

Set r0 = mult(f(0, y)) ≥ mult(f). Set

δ0 = min

i

r0 − j: j < r0 and aij 6= 0

.

δ0 =∞ if and only if f = uyr0 , where u is a unit in K[[x, y]]. Suppose thatδ0 <∞. Then we can write

f =∑

i+δ0j≥δ0r0

aijxiyj

with a0r0 6= 0, and the weighted leading form

Lδ0(x, y) =∑

i+δ0j=δ0r0

aijxiyj = a0r0y

r0 + terms of lower degree in y

has at least two non-zero terms. We can thus choose 0 6= c1 ∈ K so that

Lδ0(1, c1) =∑

i+δ0j=δ0r0

aijcj1 = 0.

Write δ0 = p0q0

, where q0, p0 are relatively prime positive integers. We makea transformation

x = xq01 , y = xp01 (y1 + c1).Then

f = xr0p01 f1(x1, y1),where

(2.1) f1(x1, y1) =∑

i+δ0j=δ0r0

aij(c1 + y1)j + x1H(x1, y1).

By our choice of c1, f1(0, 0) = 0. Set r1 = mult(f1(0, y1)). We see thatr1 ≤ r0. We have an expansion

f1 =∑

aij(1)xi1yj1.

Set

δ1 = min

i

r1 − j: j < r1 and aij(1) 6= 0

,

and write δ1 = p1q1

with p1, q1 relatively prime. We can then choose c2 ∈ Kfor f1, in the same way that we chose c1 for f , and iterate this process,obtaining a sequence of transformations

(2.2)x = xq01 , y = xp01 (y1 + c1),x1 = xq12 , y1 = xp12 (y2 + c2),

...

Either this sequence of transformations terminates after a finite number nof steps with δn = ∞, or we can construct an infinite sequence of trans-formations with δn < ∞ for all n. This allows us to write y as a series inascending fractional powers of x.

2.1. Newton’s method for determining the branches of a plane curve 5

As our first approximation, we can use our first transformation to solvefor y in terms of x and y1:

y = c1xδ0 + y1x

δ0 .

Now the second transformation gives us

y = c1xδ0 + c2x

δ0+δ1q0 + y2x

δ0+δ1q0 .

We can iterate this procedure to get the formal fractional series

(2.3) y = c1xδ0 + c2x

δ0+δ1q0 + c3x

δ0+δ1q0

+δ2q0q1 + · · · .

Theorem 2.1. There exists an i0 such that δi ∈ N for i ≥ i0.

Proof. ri = mult(fi(0, yi)) are monotonically decreasing, and positive forall i, so it suffices to show that ri = ri+1 implies δi ∈ N. Without loss ofgenerality, we may assume that i = 0 and r0 = r1. f1(x1, y1) is given by theexpression (2.1). Set

g(t) = f1(0, t) =∑

i+δ0j=δ0r0

aij(c1 + t)j .

g(t) has degree r0. Since r1 = r0, we also have mult(g(t)) = r0. Thusg(t) = a0r0t

r0 , and ∑i+δ0j=δ0r0

aijtj = a0r0(t− c1)r0 .

In particular, since K has characteristic 0, the binomial theorem shows that

(2.4) ai,r0−1 6= 0,

where i is a natural number with i+ δ0(r0 − 1) = δ0r0. Thus δ0 ∈ N.

We can thus find a natural number m, which we can take to be thesmallest possible, and a series

p(t) =∑

biti

such that (2.3) becomes

(2.5) y = p(x1m ).

For n ∈ N, set

pn(t) =n∑i=1

biti.

Using induction, we can show that

mult(f(tm, pn(t))→∞


as n→∞, and thus f(tm, p(t)) = 0. Thus

(2.6) y =∑

bixim

is a branch of the curve f = 0. This expansion is called a Puiseux series(when r0 = mult(f)), in honor of Puiseux, who introduced this theory intoalgebraic geometry.

Remark 2.2. Our proof of Theorem 2.1 is not valid in positive character-istic, since we cannot conclude (2.4). Theorem 2.1 is in fact false over fieldsof positive characteristic. See Exercise 2.4 at the end of this section.

Suppose that f ∈ K[[x, y]] is irreducible, and that we have found asolution y = p(x

1m ) to f(x, y) = 0. We may suppose that m is the smallest

natural number for which it is possible to write such a series. y − p(x1m )

divides f in R1 = K[[x1m , y]]. Let ω be a primitive m-th root of unity in K.

Since f is invariant under the K-algebra automorphism φ of R1 determinedby x

1m → ωx

1m and y → y, it follows that y − p(ωjx

1m )) | f in R1 for all j,

and thus y = p(ωjx1m ) is a solution to f(x, y) = 0 for all j. These solutions

are distinct for 0 ≤ j ≤ m− 1, by our choice of m. The series

g =m−1∏j=0

(y − p(ωjx

1m )

)is invariant under φ, so g ∈ K[[x, y]] and g | f in K[[x, y]], the ring ofinvariants of R1 under the action of the group Zm generated by φ. Since fis irreducible,

f = u

m−1∏j=0

(y − p(ωjx

1m )

) ,

where u is a unit in K[[x, y]].

Remark 2.3. Some letters of Newton developing this idea are translated(from Latin) in [18].

After we have defined non-singularity, we will return to this algorithmin (2.7) of Section 2.5, to see that we have actually constructed a resolutionof singularities of a plane curve singularity.

Exercise 2.4.

1. Construct a Puiseux series solution to

f(x, y) = y4 − 2x3y2 − 4x5y + x6 − x7 = 0

over the complex numbers.

2.2. Smoothness and non-singularity 7

2. Apply the algorithm of this section to the equation

f(x, y) = yp + yp+1 + x = 0

over an algebraically closed field k of characteristic p. What is theresulting fractional series?

2.2. Smoothness and non-singularity

Definition 2.5. Suppose that X is a scheme. X is non-singular at P ∈ Xif OX,P is a regular local ring.

Recall that a local ring R, with maximal ideal m, is regular if the di-mension of m/m2 as an R/m vector space is equal to the Krull dimensionof R.

For varieties over a field, there is a related notion of smoothness.Suppose that K[x1, . . . , xn] is a polynomial ring over a field K,

f1, . . . , fm ∈ K[x1, . . . , xn].

We define the Jacobian matrix

J(f ;x) = J(f1, . . . , fm;x1, . . . , xn) =

∂f1∂x1

· · · ∂f1∂xn

...∂fm∂x1

· · · ∂fm∂xn

.

Let I = (f1, . . . , fm) be the ideal generated by f1, . . . , fm, and define R =K[x1, . . . , xn]/I. Suppose that P ∈ spec(R) has ideal m in K[x1, . . . , xn]and ideal n in R. Let K(P ) = Rn/nn. We will say that J(f ;x) has rankl at P if the image of the l-th Fitting ideal Il(J(f ;x)) of l × l minors ofJ(f, x) in K(P ) is K(P ) and the image of the s-th Fitting ideal Is(J(f ;x))in K(P ) is (0) for s > l.

Definition 2.6. Suppose that X is a variety of dimension s over a field K,and P ∈ X. Suppose that U = spec(R) is an affine neighborhood of P suchthat R ∼= K[x1, . . . , xn]/I with I = (f1, . . . , fm). Then X is smooth over Kif J(f ;x) has rank n− s at P .

This definition depends only on P and X and not on any of the choicesof U , x or f . We verify this is the appendix.

Theorem 2.7. Let K be a field. The set of points in a K-variety X whichare smooth over K is an open set of X.

Theorem 2.8. Suppose that X is a variety over a field K and P ∈ X.

1. Suppose that X is smooth over K at P . Then P is a non-singularpoint of X.


2. Suppose that P is a non-singular point of X and K(P ) is separablygenerated over K. Then X is smooth over K at P .

In the case when K is algebraically closed, Theorems 2.7 and 2.8 areproven in Theorems I.5.3 and I.5.1 of [47]. Recall that an algebraicallyclosed field is perfect. We will give the proofs of Theorems 2.7 and 2.8 forgeneral fields in the appendix.

Corollary 2.9. Suppose that X is a variety over a perfect field K andP ∈ X. Then X is non-singular at P if and only if X is smooth at P overK.

Proof. This is immediate since an algebraic function field over a perfect fieldK is always separably generated over K (Theorem 13, Section 13, ChapterII, [92]).

In the case when X is an affine variety over an algebraically closedfield K, the notion of smoothness is geometrically intuitive. Suppose thatX = V (I) = V (f1, . . . , fm) ⊂ An

K is an s-dimensional affine variety, whereI = (f1, . . . , fm) is a reduced and equidimensional ideal. We interpret theclosed points of X as the set of solutions to f1 = · · · = fm = 0 in Kn. Weidentify a closed point p = (a1, . . . , an) ∈ V (I) ⊂ An

K with the maximalideal m = (x1 − a1, . . . , xn − an) of K[x1, . . . , xn]. For 1 ≤ i ≤ m,

fi ≡ fi(p) + Li mod m2,

where

Li =n∑j=1

∂fi∂xi

(p)(xj − aj).

But fi(p) = 0 for all i since p is a point of X. The tangent space to X inAnK at the point p is

Tp(X) = V (L1, . . . , Lm) ⊂ AnK .

We see that dimTp(X) = n − rank(J(f ;x)(p)). Thus dimTp(X) ≥ s (byRemark A.9), and X is non-singular at p if and only if dimTp(X) = s.

Theorem 2.10. Suppose that X is a variety over a field K. Then the setof non-singular points of X is an open dense set of X.

This theorem is proven when K is algebraically closed in Corollary I.5.3[47]. We will prove Theorem 2.10 when K is perfect in the appendix. Thegeneral case is proven in the corollary to Theorem 11 [85].

Exercise 2.11. Consider the curve y2−x3 = 0 in A2K , over an algebraically

closed field K of characteristic 0 or p > 3.

2.3. Resolution of singularities 9

1. Show that at the point p1 = (1, 1), the tangent space Tp1(X) is theline

−3(x− 1) + 2(y − 1) = 0.2. Show that at the point p2 = (0, 0), the tangent space Tp2(X) is the

entire plane A2K .

3. Show that the curve is singular only at the origin p2.

2.3. Resolution of singularities

Suppose that X is a K-variety, where K is a field.

Definition 2.12. A resolution of singularities of X is a proper birationalmorphism φ : Y → X such that Y is a non-singular variety.

A birational morphism is a morphism φ : Y → X of varieties suchthat there is a dense open subset U of X such that φ−1(U) → U is anisomorphism. IfX and Y are integral andK(X) andK(Y ) are the respectivefunction fields of X and Y , φ is birational if and only if φ∗ : K(X)→ K(Y )is an isomorphism.

A morphism of varieties φ : Y → X is proper if for every valuationring V with morphism α : spec(V ) → X, there is a unique morphism β :spec(V ) → Y such that φ β = α. If X and Y are integral, and K(X) isthe function field of X, then we only need consider valuation rings V suchthat K ⊂ V ⊂ K(X) in the definition of properness.

The geometric idea of properness is that every mapping of a “formal”curve germ into X lifts uniquely to a morphism to Y .

One consequence of properness is that every proper map is surjective.The properness assumption rules out the possibility of “resolving” by takingthe birational resolution map to be the inclusion of the open set of non-singular points into the given variety, or the mapping of the non-singularpoints of a partial resolution to the variety.

Birational proper morphisms of non-singular varieties (over a field ofcharacteristic zero) can be factored by alternating sequences of blow-ups andblow-downs of non-singular subvarieties, as shown by Abramovich, Karu,Matsuki and Wlodarczyk [11].

We can extend our definition of a resolution of singularities to arbitraryschemes. A reasonable category to consider is excellent (or quasi-excellent)schemes (defined in IV.7.8 [45] and on page 260 of [66]). The definition ofexcellence is extremely technical, but the idea is to give minimal conditionsensuring that the the singular locus is preserved by natural base extensionssuch as completion. There are examples of non-excellent schemes whichadmit a resolution of singularities. Rotthaus [74] gives an example of a


regular local ring R of dimension 3 containing a field which is not excellent.In this case, spec(R) is a resolution of singularities of spec(R).

2.4. Normalization

Suppose that R is an affine domain with quotient field L. Let S be theintegral closure of R in L. Since R is affine, S is finite over R, so thatS is affine (cf. Theorem 9, Chapter V [92]). We say that spec(S) is thenormalization of spec(R).

Suppose that V is a valuation ring of L such that R ⊂ V . V is integrallyclosed in L (although V need not be Noetherian). Thus S ⊂ V . Themorphism spec(V ) → spec(R) lifts to a morphism spec(V ) → spec(S), sothat spec(S)→ spec(R) is proper.

If X is an integral variety, we can cover X by open affines spec(Ri)with normalization spec(Si). The spec(Si) patch to a variety Y called thenormalization of X. Y → X is proper, by our local proof.

For a general (reduced but not necessarily integral) variety X, the nor-malization of X is the disjoint union of the normalizations of the irreduciblecomponents of X.

A ring R is normal if Rp is an integrally closed domain for all p ∈spec(R). If R is an affine ring and spec(S) is the normalization of spec(R)we constructed above, then S is a normal ring.

Theorem 2.13 (Serre). A Noetherian ring A is normal if and only if Ais R1 (Ap is regular if ht(p) ≤ 1) and S2 (depthAp ≥ min(ht(p), 2) for allp ∈ spec(A)).

A proof of this theorem is given in Theorem 23.8 [66].

Theorem 2.14. Suppose that X is a 1-dimensional variety over a field K.Then the normalization of X is a resolution of singularities.

Proof. Let X be the normalization of X. All local rings OX,p of pointsp ∈ X are local rings of dimension 1 which are integrally closed, so Theorem2.13 implies they are regular.

As an example, consider the curve singularity y2 − x3 = 0 in A2, withaffine ring R = K[x, y]/(y2 − x3). In the quotient field L of R we have therelation ( yx)2 − x = 0. Thus y

x is integral over R. Since R[ yx ] = K[ yx ] is aregular ring, it must be the integral closure of R in L. Thus spec(R[ yx ]) →spec(R) is a resolution of singularities.

Normalization is in general not enough to resolve singularities in dimen-sion larger than 1. As an example, consider the surface singularity X defined

2.5. Local uniformization and generalized resolution problems 11

by z2−xy = 0 in A3K , where K is a field. The singular locus of X is defined

by the ideal J = (z2 − xy, y, x, 2z).√J = (x, y, z), so the singular locus of

X is the origin in A2. Thus R = K[x, y, z]/(z2 − xy) is R1. Since z2 − xyis a regular element in K[x, y, z], R is Cohen-Macaulay, and hence R is S2

(Theorem 17.3 [66]). We conclude that X is normal, but singular.Kawasaki [58] has proven that under extremely mild assumptions a

Noetherian scheme admits a Macaulayfication (all local rings have maxi-mal depth). In general, Cohen-Macaulay schemes are far from being non-singular, but they do share many good homological properties with non-singular schemes.

A scheme which does not admit a resolution of singularities is givenby the example of Nagata (Example 3 in the appendix to [68]) of a one-dimensional Noetherian domain R whose integral closure R is not a finite R-module. We will show that such a ring cannot have a resolution of singulari-ties. Suppose that there exists a resolution of singularities φ : Y → spec(R).That is, Y is a regular scheme and φ is proper and birational. Let L bethe quotient field of R. Y is a normal scheme by Theorem 2.13. Thus Yhas an open cover by affine open sets U1, . . . , Us such that Ui ∼= spec(Ti),with Ti = R[gi1, . . . , git] where gij ∈ L, and each Ti is integrally closed. Theintegrally closed subring

T = Γ(Y,OY ) =s⋂i=1

Ti

of L is a finite R-module since φ is proper (Theorem III, 3.2.1 [44] or Corol-lary II.5.20 [47] if φ is assumed to be projective).

Exercise 2.15.

1. Let K = Zp(t), where p is a prime and t is an indeterminate. LetR = K[x, y]/(xp + yp − t), X = spec(R). Prove that X is non-singular, but there are no points of X which are smooth over K.

2. Let K = Zp(t), where p > 2 is a prime and t is an indeterminate.Let R = K[x, y]/(x2 + yp − t), X = spec(R). Prove that X isnon-singular, and X is smooth over K at every point except at theprime (yp − t, x).

2.5. Local uniformization and generalized resolutionproblems

Suppose that f is irreducible in K[[x, y]]. Then the Puiseux series (2.5) ofSection 2.1 determines an inclusion

(2.7) R = K[[x, y]]/(f(x, y))→ K[[t]]


with x = tm, y = p(t) the series of (2.5). Since the m chosen in (2.5) ofSection 2.1 is the smallest possible, we can verify that t is in the quotientfield L of R. Since the quotient field of K[[t]] is generated by 1, t, . . . , tm−1

over L, we conclude that R and K[[t]] have the same quotient field. (2.7)is an explicit realization of the normalization of the curve singularity germf(x, y) = 0, and thus spec(K[[t]])→ spec(R) is a resolution of singularities.The quotient field L of K[[t]] has a valuation ν defined for non-zero h ∈ Lby

ν(h(t)) = n ∈ Z

if h(t) = tnu, where u is a unit in K[[t]]. This valuation can be understoodin R by the Puiseux series (2.6).

More generally, we can consider an algebraic function field L over a fieldK. Suppose that V is a valuation ring of L containing K. The problem oflocal uniformization is to find a regular local ring R, essentially of finite typeover K and with quotient field L such that the valuation ring V dominates R(R ⊂ V and the intersection of the maximal ideal of V with R is the maximalideal of R). The exact relationship of local uniformization to resolution ofsingularities is explained in Section 8.3.

If L has dimension 1 over K, then the valuation rings V of L whichcontain K are precisely the local rings of points on the unique non-singularprojective curve C with function field L (cf. Section I.6 [47]). The Newtonmethod of Section 2.1 can be viewed as a solution to the local uniformizationproblem for complex analytic curves.

If L has dimension ≥ 2 over K, there are many valuations rings V of Lwhich are non-Noetherian (see the exercise of Section 8.1).

Zariski proved local uniformization for two-dimensional function fieldsover an algebraically closed field of characteristic zero in [86]. He was ableto patch together local solutions to prove the existence of a resolution ofsingularities for algebraic surfaces over an algebraically closed field of char-acteristic zero. He later was able to prove local uniformization for algebraicfunction fields of characteristic zero in [87]. This method leads to an ex-tremely difficult patching problem in higher dimensions, which Zariski wasable to solve in dimension 3 in [90]. Zariski’s proof of local uniformizationcan be considered as an extension of the Newton method to general val-uations. In Chapter 8, we present Zariski’s proof of resolution of surfacesingularities through local uniformization.

Local uniformization has been proven for two-dimensional function fieldsin positive characteristic by Abhyankar. Abhyankar has proven resolution ofsingularities in positive characteristic for surfaces and for three-dimensionalvarieties, [1],[3],[4].


There has recently been a resurgence of interest in local uniformizationin positive characteristic, and significant progress is being made. Some of therecent papers on this area are: Hauser [50], Heinzer, Rotthaus and Wiegand[51], Kuhlmann [59], [60], Piltant [71], Spivakovsky [77], Teissier [78].

Some related resolution type problems are resolution of vector fields,resolution of differential forms and monomialization of morphisms.

Suppose that X is a variety which is smooth over a perfect field K, andD ∈ Hom(Ω1

X/K ,OX) is a vector field. If p ∈ X is a closed point, andx1, . . . , xn are regular parameters at p, then there is a local expression

D = a1(x)∂

∂x1+ · · ·+ an(x)

∂

∂xn,

where ai(x) ∈ OX,p. We can associate an ideal sheaf ID to D on X by

ID,p = (a1, . . . , an)

for p ∈ X.There are an effective divisor FD and an ideal sheaf JD such that V (JD)

has codimension ≥ 2 in X and ID = OX(−FD)JD. The goal of resolution ofvector fields is to find a proper morphism π : Y → X so that if D′ = π−1(D),then the ideal ID′ ⊂ OY is as simple as possible.

This was accomplished by Seidenberg for vector fields on non-singularsurfaces over an algebraically closed field of characteristic zero in [75]. Thebest statement that can be attained is that the order of JD′ at p (DefinitionA.17) is νp(JD′) ≤ 1 for all p ∈ Y . The basic invariant considered in theproof is the order νp(JD). While this is an upper semi-continuous functionon Y , it can go up under a monoidal transform. However, we have thatνq(JD′) ≤ νp(JD)+1 if π : Y → X is the blow-up of p, and q ∈ π−1(p). Thisshould be compared with the classical resolution problem, where a basicresult is that order cannot go up under a permissible monoidal transform(Lemma 6.4). Resolution of vector fields for smooth surfaces over a perfectfield has been proven by Cano [19]. Cano has also proven a local theoremfor resolution for vector fields over fields of characteristic zero [20], whichimplies local resolution (along a valuation) of a vector field. The statementis that we can achieve νq(JD′) ≤ 1 (at least locally along a valuation) after amorphism π : Y → X. There is an analogous problem for differential forms.A recent paper on this is [21].

We can also consider resolution problems for morphisms f : Y → X ofvarieties. The natural question to ask is if it is possible to perform monoidaltransforms (blow-ups of non-singular subvarieties) over X and Y to producea morphism which is a monomial mapping.


Definition 2.16. Suppose that Φ : X → Y is a dominant morphism of non-singular irreducible K-varieties (where K is a field of characteristic zero).Φ is monomial if for all p ∈ X there exists an etale neighborhood U of p,uniformizing parameters (x1, . . . , xn) on U , regular parameters (y1, . . . , ym)in OY,Φ(p), and a matrix (aij) of non-negative integers (which necessarilyhas rank m) such that

y1 = xa111 · · ·xa1n

n ,...

ym = xam11 · · ·xamnn .

Definition 2.17. Suppose that Φ : X → Y is a dominant morphism ofintegral K-varieties. A morphism Ψ : X1 → Y1 is a monomializationof Φ [27] if there are sequences of blow-ups of non-singular subvarietiesα : X1 → X and β : Y1 → Y , and a morphism Ψ : X1 → Y1 such that thediagram

X1Ψ→ Y1

↓ ↓X

Φ→ Y

commutes, and Ψ is a monomial morphism.

If Φ : X → Y is a dominant morphism from a 3-dimensional variety toa surface (over an algebraically closed field of characteristic 0), then thereis a monomialization of Φ [27]. A generalized multiplicity is defined inthis paper, and it can go up, causing a very high complexity in the proof.An extension of this result to strongly prepared morphisms from n-folds tosurfaces is proven in [33]. It is not known if monomialization is true evenfor birational morphisms of varieties of dimension ≥ 3, although it is truelocally along a valuation, from the following Theorem 2.18. Theorem 2.18is proven when the quotient field of S is finite over the quotient field of Rin [25]. The proof for general field extensions is in [28].

Theorem 2.18 (Theorem 1.1 [26], Theorem 1.1 [28]). Suppose that R ⊂ Sare regular local rings, essentially of finite type over a field K of character-istic zero.

Let V be a valuation ring of K which dominates S. Then there existsequences of monoidal transforms R → R′ and S → S′ such that V domi-nates S′, S′ dominates R′ and there are regular parameters (x1, ...., xm) inR′, (y1, ..., yn) in S′, units δ1, . . . , δm ∈ S′ and an m × n matrix (aij) ofnon-negative integers such that rank(aij) = m is maximal and

(2.8)x1 = ya11

1 .....ya1nn δ1,

...xm = yam1

1 .....yamnn δm.


Thus (since char(K) = 0) there exists an etale extension S′ → S′′,where S′′ has regular parameters y1, . . . , yn such that x1, . . . , xm are puremonomials in y1, . . . , yn.

The standard theorems on resolution of singularities allow one to easilyfind R′ and S′ such that (2.8) holds, but, in general, we will not have theessential condition rank(aij) = m. The difficulty of the problem is to achievethis condition.

This result gives very simple structure theorems for the ramificationof valuations in characteristic zero function fields [35]. We discuss some ofthese results in Chapter 9. A generalization of monomialization in character-istic p function fields of algebraic surfaces is obtained in [34] and especiallyin [35].

We point out that while it seems possible that Theorem 2.18 does hold inpositive characteristic, there are simple examples in positive characteristicwhere a monomialization does not exist. The simplest example is the mapof curves

y = xp + xp+1

in characteristic p.A quasi-complete variety over a field K is an integral finite type K-

scheme which satisfies the existence part of the valuative criterion for proper-ness (Hironaka, Chapter 0, Section 6 of [52] and Chapter 8 of [26]).

The construction of a monomialization by quasi-complete varieties fol-lows from Theorem 2.18. Theorem 2.19 is proven for generically finite mor-phisms in [26] and for arbitrary morphisms in Theorem 1.2 [28].

Theorem 2.19 (Theorem 1.2 [26],Theorem 1.2 [28]). Let K be a field ofcharacteristic zero, Φ : X → Y a dominant morphism of proper K-varieties.Then there exist birational morphisms of non-singular quasi-complete K-varieties α : X1 → X and β : Y1 → Y , and a monomial morphism Ψ :X1 → Y1 such that the diagram

X1Ψ→ Y1

↓ ↓X Φ

→ Y

commutes and α and β are locally products of blow-ups of non-singular sub-varieties. That is, for every z ∈ X1, there exist affine neighborhoods V1 of zand V of x = α(z) such that α : V1 → V is a finite product of monoidal trans-forms, and there exist affine neighborhoods W1 of Ψ(z), W of y = β(Ψ(z))such that β : W1 →W is a finite product of monoidal transforms.


A monoidal transform of a non-singular K-scheme S is the map T → Sinduced by an open subset T of proj(

⊕In), where I is the ideal sheaf of a

non-singular subvariety of S.The proof of Theorem 2.19 follows from Theorem 2.18, by patching a

finite number of local solutions. The resulting schemes may not be separated.It is an extremely interesting question to determine if a monomialization

exists for all morphisms of varieties (over a field of characteristic zero). Thatis, the conclusions of Theorem 2.19 hold, but with the stronger conditionsthat α and β are products of monoidal transforms on proper varieties X1

and Y1.

Chapter 3

Curve Singularities

3.1. Blowing up a point on A2

Suppose that K is an algebraically closed field.Let

U1 = spec(K[s, t]) = A2K , U2 = spec(K[u, v]) = A2

K .

Define a K-algebra isomorphism

λ : K[s, t]s = K[s, t,1s]→ K[u, v]v = K[u, v,

1v]

by λ(s) = 1v , λ(t) = uv. We define a K-variety B0 by patching U1 to U2 on

the open sets

U2 − V (v) = spec(K[u, v]v) and U1 − V (s) = spec(K[s, t]s)

by the isomorphism λ. The K-algebra homomorphisms

K[x, y]→ K[s, t]

defined by x 7→ st, y 7→ t, and

K[x, y]→ K[u, v]

defined by x 7→ u, y 7→ uv, are compatible with the isomorphism λ, so weget a morphism

π : B(p) = B0 → U0 = spec(K[x, y]) = A2K ,

where p denotes the origin of U0. Upon localization of the above maps, weget isomorphisms

K[x, y]y ∼= K[s, t]t and K[x, y]x ∼= K[u, v]u.

17

18 3. Curve Singularities

Thus π : U1 − V (t) ∼= U0 − V (y), π : U2 − V (u) ∼= U0 − V (x), and π is anisomorphism over U0 − V (x, y) = U0 − p. Moreover,

π−1(p) ∩ U1 = V (st, t) = V (t) ⊂ spec(K[s, t]) = spec(K[s]),

π−1(p) ∩ U2 = V (u, uv) = V (u) ⊂ spec(K[u, v]) = spec(K[1s]),

by the identification s = 1v . Thus π−1(p) ∼= P1. Set E = π−1(p). We have

“blown up p” into a codimension 1 subvariety of B(p), isomorphic to P1.Set R = K[x, y], m = (x, y). Then

U1 = spec(K[s, t]) = spec(K[x

y, y]) = spec(R[

x

y]),

U2 = spec(K[x,y

x]) = spec(R[

y

x]).

We see that

B(p) = spec(R[x

y]) ∪ spec(R[

y

x]) = proj(

⊕n≥0

mn).

Suppose that q ∈ π−1(p) is a closed point. If q ∈ U1, then its associatedideal mq is a maximal ideal of R1 = K[xy , y] which contains (x, y)R1 = yR1.Thus mq = (y, xy − α) for some α ∈ K. If we set y1 = y, x1 = x

y − α, we seethat there are regular parameters (x1, y1) in OB(p),q such that

x = y1(x1 + α), y = y1.

By a similar calculation, if q ∈ π−1(p) and q ∈ U2, there are regular param-eters (x1, y1) in OB(p),q such that

x = x1, y = x1(y1 + β)

for some β ∈ K. If the constant α or β is non-zero, then q is in both U1

and U2. Thus the points in π−1(p) can be expressed (uniquely) in one of theforms

x = x1, y = x1(y1 + α) with α ∈ K, or x = x1y1, y = y1.

Since B(p) is projective over spec(R), it certainly is proper over spec(R).However, it is illuminating to give a direct proof.

Lemma 3.1. B(p)→ spec(R) is proper.

Proof. Suppose that V is a valuation ring containing R. Then yx or x

y ∈ V .Say y

x ∈ V . Then R[ yx ] ⊂ V , and we have a morphism

spec(V )→ spec(R[y

x]) ⊂ B(p)

which lifts the morphism spec(V )→ spec(R).

3.1. Blowing up a point on A2 19

More generally, suppose that S = spec(R) is an affine surface over a fieldL, and p ∈ S is a non-singular closed point. After possibly replacing S withan open subset spec(Rf ), we may assume that the maximal ideal of p in Ris mp = (x, y). We can then define the blow up of p in S by

(3.1) π : B(p) = proj(⊕n≥0

mnp )→ S.

We can write B(p) as the union of two affine open subsets:

B(p) = spec(R[x

y]) ∪ spec(R[

y

x]).

π is an isomorphism over S − p, and π−1(p) ∼= P1.Suppose that S is a surface, and p ∈ S is a non-singular point, with ideal

sheaf mp ⊂ OS . The blow-up of p ∈ S is

π : B(p) = proj(⊕n≥0

mnp )→ S.

π is an isomorphism away from p, and if U = spec(R) ⊂ S is an affine openneighborhood of p in S such that

(x, y) = Γ(U,mp) ⊂ R

is the maximal ideal of p in R, then the map π : π−1(U)→ U is defined bythe construction (3.1).

Suppose that C ⊂ A2K is a curve. Since K[x, y] is a unique factorization

domain, there exists f ∈ K[x, y] such that V (f) = C.If q ∈ C is a closed point, we will denote the corresponding maximal

ideal of K[x, y] by mq. We set

νq(C) = maxr | f ∈ mrq

(more generally, see Definition A.17). νq(C) is both the multiplicity and theorder of C at q.

Lemma 3.2. q ∈ A2K is a non-singular point of C if and only if νq(C) = 1.

Proof. Let R = K[x, y]mq , and suppose that mq = (x, y). Let T = R/fR.

If νq(C) ≥ 2, then f ∈ m2q , and mqT/m

2qT∼= mq/m

2q has dimension

2 > 1, so that T is not regular and q is singular on C. However, if νq(C) = 1,then we have

f ≡ αx+ βy mod m2q ,

where α, β ∈ K and at least one of α and β is non-zero. Without loss ofgenerality, we may suppose that β 6= 0. Thus

y ≡ −αβx mod m2

q + (f),


and x is a K generator of mqT/m2qT∼= mq/m

2q + (f). We see that

dimK mqT/m2qT = 1

and q is non-singular on C.

Remark 3.3. This lemma is also true in the situation where q is a non-singular point on a non-singular surface S (over a field K) and C is a curvecontained in S. We need only modify the proof by replacing R with theregular local ring R = OS,q, which has regular parameters (x, y) which area K(q) basis of mq/m

2q , and since R is a unique factorization domain, there

is f ∈ R such that f = 0 is a local equation of C at q.

Let p be the origin in A2K , and suppose that νp(C) = r > 0. Let

π : B(p)→ A2K be the blow-up of p. The strict transform C of C in B(p) is

the Zariski closure of π−1(C − p) ∼= C − p in B(p). Let E = π−1(p) be theexceptional divisor . Set-theoretically, π−1(C) = C ∪ E.

For some aij ∈ K, we have a finite sum

f =∑i+j≥r

aijxiyj

with aij 6= 0 for some i, j with i + j = r. In the open subset U2 =spec(K[x1, y1]) of B(p), where

x = x1, y = x1y1,

x1 = 0 is a local equation for E. Also,

f = xr1f1,

wheref1 =

∑i+j≥r

aijxi+j−r1 yj1.

x1 - f1 since νp(C) = r, so that f1 = 0 is a local equation of the stricttransform C of C in U2.

In the open subset U1 = spec(K[x1, y1]) of B(p), where

x = x1y1, y = y1,

y1 = 0 is a local equation for E and

f = yr1f1,

wheref1 =

∑i+j≥r

aijxi1yi+j−r1

is a local equation of C in U1.

3.1. Blowing up a point on A2 21

As a scheme, we see that π−1(C) = C + rE. The scheme-theoreticpreimage of C is called the total transform of C. Define the leading form off to be

L =∑i+j=r

aijxiyj .

Suppose that q ∈ π−1(p). There are regular parameters (x1, y1) at q ofone of the forms

x = x1, y = x1(y1 + α) or x = x1y1, y = y1.

In the first case f1 = fxr1

= 0 is a local equation of C at q, where

f1 =∑i+j=r

aij(y1 + α)j + x1Ω

for some polynomial Ω. Thus νq(C) ≤ r, and νq(C) = r implies∑i+j=r

aij(y1 + α)j = a0ryr1.

We then see that ∑i+j=r

aijyj1 = a0r(y1 − α)r

andL = a0r(y − αx)r.

In the second case f1 = fyr1

= 0 is a local equation of C at q, where

f1 =∑i+j=r

aijxi1 + y1Ω

for some series Ω. Thus νq(C) ≤ r, and νq(C) = r implies∑i+j=r

aijxi1 = ar0x

r1

andL = ar0x

r.

We then see that there exists a point q ∈ π−1(p) such that νq(C) = ronly when L = (ax+ by)r for some constants a, b ∈ K. Since r > 0, there isat most one point q ∈ π1(p) where the multiplicity does not drop.

Exercise 3.4. Suppose that K[x1, . . . , xn] is a polynomial ring over a per-fect field K and f ∈ K[x1, . . . , xn] is such that f(0, . . . , 0) = 0. Letm = (x1, . . . , xn), a maximal ideal of K[x1, . . . , xn].

Define ord f(0, . . . , 0, xn) = r if

xrn | f(0, . . . , 0, xn) and xr+1n - f(0, . . . , 0, xn).


Show that R = (K[x1, . . . , xn]/(f))m is a regular local ring iford f(0, . . . , 0, xn) = 1.

3.2. Completion

Suppose that A is a local ring with maximal ideal m. A coefficient field ofA is a subfield L of A which is mapped onto A/m by the quotient mappingA→ A/m.

A basic theorem of Cohen is that an equicharacteristic complete localring contains a coefficient field (Theorem 27 Section 12, Chapter VIII [92]).This leads to Cohen’s structure theorem (Corollary, loc. cit.), which showsthat an equicharacteristic complete regular local ring A is isomorphic to aformal power series ring over a field. In fact, if L is a coefficient field ofA, and if (x1, . . . , xn) is a regular system of parameters of A, then A is thepower series ring

A = L[[x1, . . . , xn]].

We further remark that the completion of a local ring R is a regular localring if and only if R is regular (cf. Section 11, Chapter VIII [92]).

Lemma 3.5. Suppose that S is a non-singular algebraic surface defined overa field K, p ∈ S is a closed point, π : B = B(p)→ S is the blow-up of p, andsuppose that q ∈ π−1(p) is a closed point such that K(q) is separable overK(p). Let R1 = OS,p and R2 = OB,q, and suppose that K1 is a coefficientfield of R1, (x, y) are regular parameters in R1. Then there exist a coefficientfield K2 = K1(α) of R2 and regular parameters (x1, y1) of R2 such that

π∗ : R1 → R2

is the map given by ∑i,j≥0

aijxiyj →

∑i,j≥0

aijxi+j1 (y1 + α)j ,

where aij ∈ K1, or K2 = K1 and

π∗ : R1 → R2

is the map given by ∑i,j≥0

aijxiyj →

∑i,j≥0

aijxi1yi+j1 .

Proof. We have R1 = K1[[x, y]], and a natural homomorphism

OS → OS,p → R1

3.2. Completion 23

which induces

q ∈ B(p)×S spec(R1) = spec(R1[y

x]) ∪ spec(R1[

x

y]).

Let mq be the ideal of q in B(p)×S spec(R1). If q ∈ spec(R1[ yx ]), then

K(q) ∼= R1[y

x]/mq

∼= K1[y

x]/(f(

y

x))

for some irreducible polynomial f( yx) in the polynomial ring K1[ yx ]. SinceK(q) is separable overK1

∼= K(p), f is separable. We havemq = (x, f( yx)) ⊂R1[ yx ].

Let α ∈ R1[ yx ]/mq be the class of yx . We have the residue map φ : R2 → Lwhere L = R1[ yx ]/mq.

There is a natural embedding of K1 in R2. We have a factorization off(t) = (t − α)γ(t) in L[t], where t − α and γ(t) are relatively prime. ByHensel’s Lemma (Theorem 17, Section 7, Chapter VIII [92]) there is α ∈ R2

such that φ(α) = α and f(t) = (t − α)γ(t) in R2[t], where φ(γ(t)) = γ(t).The subfield K2 = K1[α] of R2 is thus a coefficient field of R2, and mqR2 =(x, yx −α). Thus R2 = K2[[x, yx −α]]. Set x1 = x, y1 = y

x −α. The inclusion

R1 = K1[[x, y]]→ R2 = K2[[x1, y1]]

is natural. A series ∑aijx

iyj

with coefficients aij ∈ K1 maps to the series∑aijx

i+j1 (y1 + α)j .

We now give an example to show that even if a regular local ring containsa field, there may not be a coefficient field of the completion of the ringcontaining that field. Thus the above lemma does not extend to non-perfectfields.

Let K = Zp(t), where t is an indeterminate. Let R = K[x](xp−t). Sup-pose that R has a coefficient field L containing K. Let φ : R→ R/m be theresidue map. Since φ | L is an isomorphism, there exists λ ∈ L such thatλp = t. Thus (xp − t) = (x− λ)p in R. But this is impossible, since (xp − t)is a generator of mR, and is thus irreducible.

Theorem 3.6. Suppose that R is a reduced affine ring over a field K, andA = Rp, where p is a prime ideal of R. Then the completion A = Rp of Awith respect to its maximal ideal is reduced.


When K is a perfect field, this is a theorem of Chevalley (Theorem 31,Section 13, Chapter VIII [92]). The general case follows from Scholie IV7.8.3 (vii) [45].

However, the property of being a domain is not preserved under com-pletion. A simple example is f = y2 − x2 + x3. f is irreducible in C[x, y],but is reducible in the completion C[[x, y]]:

f = y2 − x2 − x3 = (y − x√

1 + x)(y + x√

1 + x).

The first parts of the expansions of the two factors are

y − x− 12x2 + · · ·

andy + x+

12x2 + · · · .

Lemma 3.7 (Weierstrass Preparation Theorem). Let K be a field, andsuppose that f ∈ K[[x1, . . . , xn, y]] is such that

0 < r = ν(f(0, . . . , 0, y)) = maxn | yn divides f(0, . . . , 0, y) <∞.Then there exist a unit series u in K[[x1, . . . , xn, y]] and non-unit seriesai ∈ K[[x1, . . . , xn]] such that

f = u(yr + a1yr−1 + · · ·+ ar).

A proof is given in Theorem 5, Section 1, Chapter VII [92].A concept which will be important in this book is the Tschirnhausen

transformation, which generalizes the ancient notion of “completion of thesquare” in the solution of quadratic equations.

Definition 3.8. Suppose that K is a field of characteristic p ≥ 0 andf ∈ K[[x1, . . . , xn, y]] has an expression

f = yr + a1yr−1 + · · ·+ ar

with ai ∈ K[[x1, . . . , xn]] and p = 0 or p - r. The Tschirnhausen transfor-mation of f is the change of variables replacing y with

y′ = y +a1

r.

f then has an expression

f = (y′)r + b2(y′)r−2 + · · ·+ br

with bi ∈ K[[x1, . . . , xn]] for all i.

Exercise 3.9. Suppose that (R,m) is a complete local ring and L1 ⊂ R isa field such that R/m is finite and separable over L1. Use Hensel’s Lemma(Theorem 17, Section 7, Chapter VIII [92]) to prove that there exists acoefficient field L2 of R containing L1.

3.3. Blowing up a point on a non-singular surface 25

3.3. Blowing up a point on a non-singular surface

We define the strict transform, the total transform, and the multiplicity (ororder) νp(C) analogously to the definition of Section 3.1 (More generally,see Section 4.1 and Definition A.17).

Lemma 3.10. Suppose that X is a non-singular surface over an alge-braically closed field K, and C is a curve on X. Suppose that p ∈ X andνp(C) = r. Let π : B(p)→ X be the blow-up of p, C the strict transform ofC, and suppose that q ∈ π−1(p). Then νq(C) ≤ r, and if r > 0, there is atmost one point q ∈ π−1(p) such that νq(C) = r.

Proof. Let f = 0 be a local equation of C at p. If (x, y) are regularparameters in OX,p, we can write

f =∑

aijxiyj

as a series of order r in OX,p ∼= K[[x, y]]. Let

L =∑i+j=r

aijxiyj

be the leading form of f .If q ∈ π−1(p), then OB(p),q has regular parameters (x1, y1) such that

x = x1, y = x1(y1 + α), or x = x1y1, y = y1.

This substitution induces the inclusion

K[[x, y]] = OX,p → OB(p),q = K[[x1, y1]].

First suppose that (x1, y1) are defined by

x = x1, y = x1(y1 + α).

Then a local equation for C at q is f1 = fxr1

, which has the expansion

f1 =∑


for some series Ω. We finish the proof as in Section 3.1.

Suppose that C is the curve y2−x3 = 0 in A2K . The only singular point

of C is the origin p. Let π : B(p) → A2K be the blow-up of p, C the strict

transform of C, and E the exceptional divisor E = π−1(p).On U1 = spec(K[xy , y]) ⊂ B(p) we have coordinates x1, y1 with x = x1y1,

y = y1. A local equation for E on U1 is y1 = 0. Also,

y2 − x3 = y21(1− x3

1y1).

A local equation for C on U1 is 1− x31y1 = 0, which is a unit on E ∩ U1, so

C ∩ E ∩ U1 = ∅.


On U2 = spec(K[x, yx ]) ⊂ B(p) we have coordinates x1, y1 with x = x1,y = x1y1. A local equation for E on U2 is x1 = 0. Also,

y2 − x3 = x21(y

21 − x1).

A local equation for C on U2 is y21−x1 = 0, which has order ≤ 1 everywhere.

Thus C is non-singular, andC → C

is a resolution of singularities. Finally,

Γ(C,OC) = K[x1, y1]/(y21 − x1) = K[y1] = K[

y

x].

This is the normalization of K[x, y]/(y2 − x3) that we computed in Section2.4.

As a further example, consider the curve C with equation y2 − x5 inA2. The only singular point of C is the origin p. Let π : B(p) → A2 bethe blow-up of p, E be the exceptional divisor, C be the strict transform ofC. There is only one singular point q on C. Regular parameters at q are(x1, y1), where x = x1, y = x1y1. C has the local equation

y21 − x3

1 = 0.

In this example νq(C) = νp(C) = 2, so the multiplicity has not dropped.However, this singularity is resolved after blowing up q, as we calculated

in the previous example.These examples suggest an algorithm to resolve curve singularities. First

blow up all singular points. If the resulting curve is not resolved, blow upthe new singular points, and repeat as long as the curve is singular.

This algorithm ends after a finite number of iterations in a non-singularcurve. In Sections 3.4 and 3.5 we will prove this for curves embedded in anon-singular surface, first over an algebraically closed field K of character-istic zero, and then over an arbitrary field.

3.4. Resolution of curves embedded in a non-singularsurface I

In this section we consider curves embedded in a non-singular surface overan algebraically closed field K of characteristic zero, and prove that theirsingularities can be resolved.

The theorem is proved by passing to the completion of the local ring ofthe surface at a singular point of the curve. This allows us to view a localequation of the curve as a power series in two variables. Our main invariantis the multiplicity of the curve at a point. We know that this multiplicity

3.4. Resolution of curves embedded in a non-singular surface I 27

cannot go up after blowing up, so we must show that it will eventually godown, after enough blowing up.

Theorem 3.11. Suppose that C is a curve on a non-singular surface Xover an algebraically closed field K of characteristic zero. Then there existsa sequence of blow ups of points λ : Y → X such that the strict transformC of C on Y is non-singular.

Proof. Let r = max(νp(C) | p ∈ C. If r = 1, C is non-singular, so we mayassume that r > 1. The set p ∈ C | ν(p) = r is a subset of the singularlocus of C, which is a proper closed subset of the 1-dimensional variety C,so it is a finite set.

The proof is by induction on r.We can construct a sequence of projective morphisms

(3.2) · · · → Xnπn→ · · · π1→ X0 = X,

where each πn+1 : Xn+1 → Xn is the blow-up of all points on the stricttransform Cn of C which have multiplicity r on Cn. If this sequence is finite,then there is an integer n such that all points on the strict transform Cn ofC have multiplicity ≤ r − 1. By induction on r, we can repeat this processto construct the desired morphism Y → X which induces a resolution of C.

We will assume that the sequence (3.2) is infinite, and derive a contra-diction. If it is infinite, then for all n ∈ N there are closed points pn ∈ Cnwhich have multiplicity r on Cn and such that πn+1(pn+1) = pn for all n.Let Rn = OXn,pn for all n. We then have an infinite sequence of completionsof quadratic transforms of local rings

R0 → R1 → · · · → Rn → · · · .

Suppose that (x, y) are regular parameters in R0 = OX,p, and f = 0 is alocal equation of C in R0 = K[[x, y]]. f is reduced by Theorem 3.6. After alinear change of variables in (x, y), we may assume that ν(f(0, y)) = r. Bythe Weierstrass preparation theorem,

f = u(yr + a1(x)yr−1 + · · ·+ ar(x)),

where u is a unit series. We now “complete the r-th power of f”. Set

(3.3) y = y +a1(x)r

.

Then

(3.4) yr + a1(x)yr−1 + · · ·+ ar(x) = yr + b2(x)yr−2 + · · ·+ br(x)

for some series bi(x). We may thus assume that

(3.5) f = yr + a2(x)yr−2 + · · ·+ ar(x).


Since f is reduced, we must have some ai(x) 6= 0. Set

(3.6) n = min

mult(ai(x))i

.

We have n ≥ 1, since ν(f) = r.

OX1,p1 has regular parameters x1, y1 such that

1. x = x1, y = x1(y1 + α) with α ∈ K, or2. x = x1y1, y = y1

In case 2,f = yr1f1,

where

f1 = 1 +a2(x1y1)

y21

+ · · ·+ ar(x1y1)yr1

is a local equation of the strict transform C1 of C at p1. f1 is a unit, so thatνp1(C1) = 0. Thus case 2 cannot occur. In case 1,

f = xr1f1

where

f1 = (y1 + α)r +a2(x1)x2

1

(y1 + α)r−2 + · · ·+ ar(x1)xr1

.

f1 = 0 is a local equation at p1 for C1. If α 6= 0, then νp1(C1) ≤ r − 1, sothis case cannot occur.

If α = 0, and νp1(C1) = r, then f1 has the form of (3.5), with n decreasedby 1.

Thus for some index i with i ≤ n we have that pi must have multiplicity< r on Ci, a contradiction to the assumption that (3.2) has infinite length.

The transformation of (3.3) is called a Tschirnhausen transformation(Definition 3.8). The Tschirnhausen transformation finds a (formal) curve ofmaximal contact H = V (y) for C at p. The Tschirnhausen transformationwas introduced as a fundamental method in resolution of singularities byAbhyankar.

Definition 3.12. Suppose that C is a curve on a non-singular surface Sover a field K, and p ∈ C. A non-singular curve H = V (g) ⊂ spec(OS,p) isa formal curve of maximal contact for C at p if whenever

Sn → Sn−1 → · · · → S1 → spec(OS,p)

is a sequence of blow-ups of points pi ∈ Si, such that the strict transformCi of C in Si contains pi with νpi(Ci) = νp(C) for i ≤ n, then the stricttransform Hi of H in Si contains pi.

3.5. Resolution of curves embedded in a non-singular surface II 29

Exercise 3.13.

1. Resolve the singularities by blowing up:a. x2 = x4 + y4,b. xy = x6 + y6,c. x3 = y2 + x4 + y4,d. x2y + xy3 = x4 + y4,e. y2 = xn,f. y4 − 2x3y2 − 4x5y + x6 − x7.

2. Suppose that D is an effective divisor on a non-singular surface S.That is, there exist curves Ci on S and numbers ri ∈ N such thatID = Ir1C1

· · · IrnCn . D has simple normal crossings (SNCs) on S iffor every p ∈ S there exist regular parameters (x, y) in OS,p suchthat if f = 0 is a local equation for D at p, then f = unit xayb inOS,p.

Find a sequence of blow-ups of points making the total trans-form of y2 − x3 = 0 a SNCs divisor.

3. Suppose that D is an effective divisor on a non-singular surface S.Show that there exists a sequence of blow-ups of points

Sn → · · · → S

such that the total transform π∗(D) is a SNCs divisor.

3.5. Resolution of curves embedded in a non-singularsurface II

In this section we consider curves embedded in a non-singular surface, de-fined over an arbitrary field K.

Lemma 3.14. Suppose that K is a field, S is a non-singular surface overK, C is a curve on S and p ∈ C is a closed point. Suppose that π : B =B(p) → S is the blow-up of p, C is the strict transform of C on B andq ∈ π−1(p) ∩ C. Then

νq(C) ≤ νp(C),

and ifνq(C) = νp(C),

then K(p) = K(q).

Proof. Let R = OS,p, (x, y) be regular parameters in R. Since there is anatural embedding of π−1(p) in B ×S spec(R), it follows that

q ∈ B ×S spec(R) = spec(R[x

y]) ∪ spec(R[

y

x]).


Without loss of generality, we may assume that q ∈ spec(R[ yx ]). Let K1∼=

K(p) be a coefficient field of R, m ⊂ R[ yx ] the ideal of q. Since R[ yx ]/(x, y)∼= K1[ yx ], there exists an irreducible monic polynomial h(t) ∈ K1[t] such that

mR[y

x] = (x, h(

y

x)).

Let f ∈ R be such that f = 0 is a local equation of C, and let r = νp(C).We have an expansion in R,

f =∑i+j≥r

aijxiyj

with aij ∈ K1. Thus f = xrf1, where f1 = 0 is a local equation of the stricttransform C of C in spec(R[ yx ]) and

f1 =∑i+j=r

aij(y

x)j + xΩ

in R[ yx ]. Let n = νq(C). Then

h(y

x)n divides

∑i+j=r

aij(y

x)j

in K1[ yx ]. Also,

deg(∑i+j=r

aij(y

x)j) ≤ r

in K1[ yx ] implies n ≤ r and νq(C) = r implies deg(h( yx)) = 1, so thath = y

x − α with α ∈ K1 and (by Lemma 3.5) there exist regular parameters(x1, y1) in mq ⊂ OB(p),q such that

OS,p = K1[[x, y]]→ K1[[x1, y1]] = OB(p),q

is the natural K1∼= K(p) algebra homomorphism

x = x1, y = x1(y1 + α).

Theorem 3.15. Suppose that C is a curve which is a subvariety of a non-singular surface X over an infinite field K. Then there exists a sequence ofblow-ups of points λ : Y → X such that the strict transform C of C on Y isnon-singular.

Proof. Let r = maxνp(C) | p ∈ C. If r = 1, C is non-singular, so we mayassume that r > 1. The set p ∈ C | νp(C) = r is a subset of the singularlocus of C, which is a proper closed subset of the 1-dimensional variety C,so it is a finite set.

The proof is by induction on r.


We can construct a sequence of projective morphisms

(3.7) · · · → Xnπn→ · · · → X1

π1→ X0 = X,

where each πn+1 : Xn+1 → Xn is the blow-up of all points on the stricttransform Cn of C which have multiplicity r on Cn. If this sequence isfinite, then there is an integer n such that all points on the strict transformCn of C have multiplicity ≤ r − 1. By induction on r, we can then repeatthis process to construct the desired morphism Y → X which induces aresolution of C.

We will assume that the sequence (3.7) is infinite, and derive a contra-diction. If it is infinite, then for all n ∈ N there are closed points pn ∈ Cnwhich have multiplicity r on Cn and are such that πn+1(pn+1) = pn. LetRn = OXn,pn for all n. We then have an infinite sequence of completions ofquadratic transforms (blow-ups of maximal ideals) of local rings

R = R0 → R1 → · · · → Rn → · · · .

We will define

δpi ∈1r!

N

such that

(3.8) δpi = δpi−1 − 1

for all i ≥ 1. We can thus conclude that pi has multiplicity < r on thestrict transform Ci of C for some natural number i ≤ δp + 1. From thiscontradiction it will follow that (3.7) is a sequence of finite length.

Suppose that f ∈ R0 = OX,p is such that f = 0 is a local equation of Cand (x, y) are regular parameters in R = OX,p such that r = mult(f(0, y)).We will call such (x, y) good parameters for f . Let K ′ be a coefficient fieldof R. There is an expansion

f =∑i+j≥r

aijxiyj

with aij ∈ K ′ for all i, j and a0r 6= 0. Define

(3.9) δ(f ;x, y) = min

i

r − j| j < r and aij 6= 0

.

We have δ(f ;x, y) ≥ 1 since (x, y) are good parameters We thus have anexpression (with δ = δ(f ;x, y))

f =∑

i+jδ≥rδaijx

iyj = Lδ +∑

i+jδ>rδ

aijxiyj ,


where

(3.10) Lδ =∑

i+jδ=rδ

aijxiyj = a0ry

r + Λ

is such that a0r 6= 0 and Λ is not zero.Suppose that (x, y) are fixed good parameters of f . Define

(3.11)

δp = supδ(f ;x, y1) | y = y1+n∑i=1

bixi with n ∈ N and bi ∈ K ′ ∈ 1

r!N∪∞.

We cannot have δp =∞, since then there would exist a series

y = y1 +∞∑i=1

bixi

such that δ(f ;x, y1) =∞, and thus there would be a unit series γ in R suchthat

f = γyr1.

But then r = 1 since f is reduced in R, a contradiction. We see then thatδp ∈ 1

r!N.After possibly making a substitution

y = y1 +n∑i=1

bixi

with bi ∈ K ′, we may assume that δp = δ(f ;x, y).Let δ = δp.Since νp1(C1) = r, by Lemma 3.14, we have an expression

R1 = K ′[[x1, y1]],

where R→ R1 is the natural K ′-algebra homomorphism such that either

x = x1, y = x1(y1 + α)

for some α ∈ K ′, orx = x1y1, y = y1.

We first consider the case where x = x1y1, y = y1. A local equation ofC1 (in spec(R1)) is f1 = 0, where

f1 =f

yr1=

∑i+j=r

aijxi1 + y1Ω = a0r + x1g + y1h

for some Ω, g, h ∈ R1. Thus f1 is a unit in R1, a contradiction.


Now consider the case where x = x1, y = x1(y1 + α) with 0 6= α ∈ K ′.A local equation of C1 is

f1 =f

xr1=

∑i+j=r


for some Ω ∈ R1. Since νp1(C1) = r,∑i+j=r

aij(y1 + α)j = a0ryr1.

Substituting t = y1 + α, we have∑i+j=r

aijtj = a0r(t− α)r.

If we now substitute t = yx and multiply the series by xr, we obtain the

leading form L of f ,

L =∑i+j=r

aijxiyj = a0r(y − αx)r = a0ry

r + · · ·+ (−1)ra0rαrxr.

Comparing with (3.10), we see that r = rδ and δ = 1, so that Lδ = L. Butwe can replace y with y−αx to increase δ, a contradiction to the maximalityof δ. Thus νp1(C1) < r, a contradiction.

Finally, consider the case x = x1, y = x1y1. Set δ′ = δ − 1. A localequation of C1 is

f1 = fxr1

=∑

i+jδ≥rδaijx

i+j−r1 yj1

=∑

i+jδ′=rδ′

ai−j+r,jxi1yj1 +

∑i+jδ′>rδ′

ai−j+r,jxi1yj1,

where i = i+ j − r. Since

Lδ′(x1, y1) =1xr1Lδ(x1, x1y1)

has at least two non-zero terms, we see that δ(f1;x1, y1) = δ′ = δ − 1.We will now show that δp1 = δ(f1;x1, y1). Suppose not. Then we can

make a substitution

y1 = y′1 −∑

bixi1 = y′1 − bxd1 + higher order terms in x1

with 0 6= b ∈ K ′ such that

δ(f1;x1, y′1) > δ(f1;x1, y1).

Then we have an expression∑i+jδ′=rδ′

ai−j+r,jxi1(y

′1 − bxd1)j = a0r(y′1)

r +∑

i+jδ′>rδ′

bijxi1(y

′1)j ,


so that δ′ = d ∈ N, and∑i+jδ′=rδ′

ai−j+r,jxi1(y

′1 − bxd1)j = a0r(y′1)

r.

Thus ∑i+jδ=rδ

aijxi+j−r1 yj1 =

∑i+jδ′=rδ′

ai−j+r,jxi1yj1 = a0r(y1 + bxδ

′1 )r.

If we now multiply these series by xr1, we obtain

Lδ =∑

i+jδ=rδ

aijxiyj = a0r(y + bxδ)r.

But we can now make the substitution y′ = y − bxδ and see that

δ(f ;x, y′) > δ(f ;x, y) = δp,

a contradiction, from which we conclude that δp1 = δ′ = δp − 1. We canthen inductively define δpi for i ≥ 0 by this procedure so that (3.8) holds.The conclusions of the theorem now follow.

Remark 3.16.

1. This proof is a generalization of the algorithm of Section 2.1. Amore general version of this, valid in arbitrary two-dimensional reg-ular local rings, can be found in [5] or [73].

2. Good parameters (x, y) for f which achieve δp = δ(f ;x, y) are suchthat y = 0 is a (formal) curve of maximal contact for C at p (Defi-nition 3.12).

Exercise 3.17.

1. Prove that the δp defined in formula (3.11) is equal to

δp = supδ(f ;x, y) | (x, y) are good parameters for f.

Thus δp does not depend on the initial choice of good parameters,and δp is an invariant of p.

2. Suppose that νp(C) > 1. Let π : B(p)→ X be the blow-up of p, Cbe the strict transform of C. Show that there is at most one pointq ∈ π−1(p) such that νq(C) = νp(C).

3. The Newton polygon N(f ;x, y) is defined as follows. Let I be theideal in R = K[[x, y]] generated by the monomials xαyβ such thatthe coefficient aαβ of xαyβ in f(x, y) =

∑aαβx

αyβ is not zero. Set

P (f ;x, y) = (α, β) ∈ Z2|xαyβ ∈ I.

Now define N(f ;x, y) to be the smallest convex subset of R2 suchthat N(f ;x, y) contains P (f ;x, y) and if (α, β) ∈ N(f ;x, y), (s, t) ∈


R2+, then (α + s, β + t) ∈ N(f ;x, y). Now suppose that (x, y)

are good parameters for f (so that ν(f(0, y)) = ν(f) = r). Then(0, r) ∈ N(f ;x, y). Let the slope of the steepest segment ofN(f ;x, y) be s(f ;x, y). We have 0 ≥ s(f ;x, y) ≥ −1, since aαβ = 0if α+ β < r. Show that

δ(f ;x, y) = − 1s(f ;x, y)

.

Chapter 4

Resolution TypeTheorems

4.1. Blow-ups of ideals

Suppose that X is a variety, and J ⊂ OX is an ideal sheaf. The blow-up ofJ is

π : B = B(J ) = proj(⊕n≥0

J n)→ X.

B is a variety and π is proper. If X is projective then B is projective. π isan isomorphism over X − V (J ), and JOB is a locally principal ideal sheaf.

If U ⊂ X is an open affine subset, and R = Γ(U,OX), I = Γ(U,J ) =(f1, . . . , fm) ⊂ R, then

π−1(U) = B(I) = proj(⊕n≥0

In).

If X is an integral scheme, we have

proj(⊕n≥0

In) =m⋃i=1

spec(R[f1

fi, · · · , fm

fi]).

Suppose that W ⊂ X is a subscheme with ideal sheaf IW on X.The total transform of W , π∗(W ), is the subscheme of B with the ideal

sheaf

Iπ∗(W ) = IWOB.

Set-theoretically, π∗(W ) is the preimage of W , π−1(W ).

37

38 4. Resolution Type Theorems

Let U = X − V (J ). The strict transform W of W is the Zariski closureof π−1(W ∩ U) in B(J ).

Lemma 4.1. Suppose that R is a Noetherian ring, I, J ⊂ R are ideals andI = q1 ∩ · · · ∩ qm is a primary decomposition, with primes pi =

√qi. Then

∞⋃n=0

(I : Jn) =⋂

i|J 6⊂pi

qi.

The proof of Lemma 4.1 follows easily from the definition of a primaryideal. Geometrically, Lemma 4.1 says that

⋃∞n=0(I : Jn) removes the pri-

mary components qi of I such that V (qi) ⊂ V (J).

Thus we see that the strict transform W of W has the ideal sheaf

IW =∞⋃n=0

(IWOB : J nOB).

For q ∈ B,

IW ,q = f ∈ OB,q | fJ nq ⊂ IWOB,q for some n ≥ 0.The strict transform has the property that

W = B(JOW ) = proj(⊕n≥0

(JOW )n)

This is shown in Corollary II.7.15 [47].

Theorem 4.2 (Universal Property of Blowing Up). Suppose that I is anideal sheaf on a variety V and f : W → V is a morphism of varieties suchthat IOW is locally principal. Then there is a unique morphism g : W →B(I) such that f = π g.

This is proved in Proposition II.7.14 [47].

Theorem 4.3. Suppose that C is an integral curve over a field K. Considerthe sequence

(4.1) · · · → Cnπn→ · · · → C1

π1→ C,

where Cn+1 → Cn is obtained by blowing up the (finitely many) singularpoints on Cn. Then this sequence is finite. That is, there exists n such thatCn is non-singular.

Proof. Without loss of generality, C = spec(R) is affine. Let R be theintegral closure of R in the function field K(C) of C. R is a regular ring.Let C = spec(R). All ideals in R are locally principal (Proposition 9.2, page94 [13]). By Theorem 4.2, we have a factorization

C → Cn → · · · → C1 → C

4.1. Blow-ups of ideals 39

for all n. Since C → C is finite, Ci+1 → Ci is finite for all i, and thereexist affine rings Ri such that Ci = spec(Ri). For all n we have sequencesof inclusions

R→ R1 → · · · → Rn → R.

Here Ri 6= Ri+1 for all i, since a maximal ideal m in Ri is locally principalif and only if (Ri)m is a regular local ring. Since R is finite over R, we havethat (4.1) is finite.

Corollary 4.4. Suppose that X is a variety over a field K and C is anintegral curve on X. Consider the sequence

· · · → Xnπn→ Xn−1 → · · · → X1

π1→ X,

where πi+1 is the blow-up of all points of X which are singular on the stricttransform Ci of C on Xi. Then this sequence is finite. That is, there existsan n such that the strict transform Cn of C is non-singular.

Proof. The induced sequence

· · · → Cn → · · · → C1 → C

of blow-ups of points on the strict transform Ci of C is finite by Theorem4.3.

Theorem 4.5. Suppose that V and W are varieties and V → W is aprojective birational morphism. Then V ∼= B(I) for some ideal sheaf I ⊂OW .

This is proved in Proposition II.7.17 [47].Suppose that Y is a non-singular subvariety of a varietyX. The monoidal

transform of X with center Y is π : B(IY )→ X. We define the exceptionaldivisor of the monoidal transform π (with center Y ) to be the reduced divisorwith the same support as π∗(Y ).

In general, we will define the exceptional locus of a proper birationalmorpism φ : V → W to be the (reduced) closed subset of V on which φ isnot an isomorphism.

Remark 4.6. If Y has codimension greater than or equal to 2 in X, thenthe exceptional divisor of the monoidal transform π : B(IY ) → X withcenter Y , is the exceptional locus of π. However, if X is singular and notlocally factorial, it may be possible to blow up a non-singular codimension 1subvariety Y of X to get a monoidal transform π such that the exceptionallocus of π is a proper closed subset of the exceptional divisor of π. For anexample of this kind, see Exercise 4.7 at the end of this section. If X isnon-singular and Y has codimension 1 in X, then π is an isomorphism and


we have defined the exceptional divisor to be Y . This property will be usefulin our general proof of resolution in Chapter 6.

Exercise 4.7.

1. Let K be an algebraically closed field, and X be the affine surface

X = spec(K[x, y, z]/(xy − z2)).

Let Y = V (x, z) ⊂ X, and let π : B → X be the monoidal trans-form centered at the non-singular curve Y . Show that π is a res-olution of singularities. Compute the exceptional locus of π, andcompute the exceptional divisor of the monoidal transform π of Y .

2. Let K be an algebraically closed field, and X be the affine 3-fold

X = spec(K[x, y, z, w]/(xy − zw).

Show that the monodial transform π : B → X with center Y isa resolution of singularities in each of the following cases, and de-scribe the exceptional locus and exceptional divisor (of the monoidaltransform):

a. Y = V ((x, y, z, w)),b. Y = V ((x, z)),c. Y = V ((y, z)).

Show that the monoidal transforms of cases b and c factor themonoidal transform of case a.

4.2. Resolution type theorems and corollaries

Resolution of singularities. Suppose that V is a variety. A resolution ofsingularities of V is a proper birational morphism φ : W → V such that Wis non-singular.

Principalization of ideals. Suppose that V is a non-singular variety, andI ⊂ OV is an ideal sheaf. A principalization of I is a proper birationalmorphism φ : W → V such that W is non-singular and IOW is locallyprincipal.

Suppose that X is a non-singular variety, and I ⊂ OX is a locallyprincipal ideal. We say that I has simple normal crossings (SNCs) at p ∈X if there exist regular parameters (x1, . . . , xn) in OX,p such that Ip =xa1

1 · · ·xann OX,p for some natural numbers a1, . . . , an.

4.2. Resolution type theorems and corollaries 41

Suppose that D is an effective divisor on X. That is, D = r1E1 + · · ·+rnEn, where Ei are irreducible codimension 1 subvarities of X, and ri arenatural numbers. D has SNCs if ID = Ir1E1

· · · IrnEn has SNCs.Suppose that W ⊂ X is a subscheme, and D is a divisor on X. We say

that W has SNCs with D at p if

1. W is non-singular at p,

2. D is a SNC divisor at p,

3. there exist regular parameters (x1, . . . , xn) in OX,p and r ≤ n suchthat IW,p = (x1, . . . , xr) (or IW,p = OX,p) and

ID,p = xa11 · · ·x

ann OX,p

for some ai ∈ N.

Embedded resolution. Suppose that W is a subvariety of a non-singularvariety X. An embedded resolution of W is a proper morphism π : Y → Xwhich is a product of monoidal transforms, such that π is an isomorphism onan open set which intersects every component of W properly, the exceptionaldivisor of π is a SNC divisor D, and the strict transform W of W has SNCswith D.

Resolution of indeterminacy. Suppose that φ : W → V is a rationalmap of proper K-varieties such that W is non-singular. A resolution ofindeterminacy of φ is a proper non-singular K-variety X with a birationalmorphism ψ : X →W and a morphism λ : X → V such that λ = φ ψ.

Lemma 4.8. Suppose that resolution of singularities is true for K-varietiesof dimension n. Then resolution of indeterminacy is true for rational mapsfrom K-varieties of dimension n.

Proof. Let φ : W → V be a rational map of proper K-varieties, where Wis non-singular. Let U be a dense open set of W on which φ is a morphism.Let Γφ be the Zariski closure of the image in W ×V of the map U →W ×Vdefined by p 7→ (p, φ(p)). By resolution of singularities, there is a properbirational morphism X → Γφ such that X is non-singular.

Theorem 4.9. Suppose that K is a perfect field, resolution of singularities istrue for projective hypersurfaces over K of dimension n, and principalizationof ideals is true for non-singular varieties of dimension n over K. Thenresolution of singularities is true for projective K-varieties of dimension n.


Proof. Suppose V is an n-dimensional projective K-variety. If V1, . . . , Vrare the irreducible components of V , we have a projective birational mor-phism from the disjoint union of the Vi to V . Thus it suffices to assume thatV is irreducible. The function field K(V ) of V is a finite separable extensionof a rational function field K(x1, . . . , xn) (Chapter II, Theorem 30, page 104[92]). By the theorem of the primitive element,

K(V ) ∼= K(x1, . . . , xn)[xn+1]/(f).

We can clear the denominator of f so that

f =∑

ai1...in+1xi11 · · ·x

in+1

n+1

is irreducible in K[x1, . . . , xn+1]. Let

d = maxi1 + · · · in+1 | ai1...in+1 6= 0.

SetF =

∑ai1...in+1X

d−(i1+···+in+1)0 Xi1

1 · · ·Xin+1

n+1 ,

the homogenization of f . Let V be the variety defined by F = 0 in Pn+1.Then K(V ) ∼= K(V ) implies there is a birational rational map from V toV . That is, there is a birational morphism φ : V → V , where V is a denseopen subset of V . Let Γφ be the Zariski closure of (a, φ(a)) | a ∈ V in V × V . We have birational projection morphisms π1 : Γφ → V andπ2 : Γφ → V . Γφ is the blow-up of an ideal sheaf J on V (Theorem 4.5).By resolution of singularities for n-dimensional hypersurfaces, we have aresolution of singularities f : W ′ → V . By principalization of ideals in non-singular varieties of dimension n, we have a principalization g : W → W ′

for JOW ′ . By the universal property of blowing up (Theorem 4.2), wehave a morphism h : W → Γφ. Hence π1 h : W → V is a resolution ofsingularities.

Corollary 4.10. Suppose that C is a projective curve over an infinite perfectfield K. Then C has a resolution of singularities.

Proof. By Theorem 3.15, resolution of singularities is true for projectiveplane curves over K. All ideal sheaves on a non-singular curve are locallyprincipal, since the local rings of points are Dedekind local rings.

Note that Theorem 4.3 and Corollary 4.4 are stronger results than Corol-lary 4.10. However, the ideas of the proofs of resolution of plane curves inTheorems 3.11 and 3.15 extend to higher dimensions, while Theorem 4.3does not. As a consequence of embedded resolution of curve singularities ona surface, we will prove principalization of ideals in non-singular varieties ofdimension two. This simple proof is by Abhyankar (Proposition 6, [73]).

4.2. Resolution type theorems and corollaries 43

Theorem 4.11. Suppose that S is a non-singular surface over a field K,and J ⊂ OS is an ideal sheaf. Then there exists a finite sequence of blow-upsof points T → S such that JOT is locally principal.

Proof. Without loss of generality, S is affine. Let

J = Γ(J ,OS) = (f1, . . . , fm).

By embedded resolution of curve singularities on a surface (Exercise 3.13 ofSection 3.4) applied to

√f1 · · · fm ∈ S, there exists a sequence of blow-ups

of points π1 : S1 → S such that f1 · · · fm = 0 is a SNC divisor everywhereon S1.

Let p1, . . . , pr be the finitely many points on S1 where JOS1 is notlocally principal (they are contained in the singular points of V (

√JOS1)).

Suppose that p ∈ p1, . . . , pr. By induction on the number of generators ofJ , we may assume that JOS1,p = (f, g). After possibly multiplying f and gby units in OS1,p, there are regular parameters (x, y) in OS1,p such that

f = xayb, g = xcyd.

Set tp = (a − c)(b − d). (f, g) is principal if and only if tp ≥ 0. By ourassumption, tp < 0. Let π2 : S2 → S1 be the blow-up of p, and suppose thatq ∈ π−1(p). We will show that tq > tp.

The only two points q where (f, g) may not be principal have regularparameters (x1, y1) such that

x = x1, y = x1y1 or x = x1y1, y = y1.

If x = x1, y = x1y1, then

f = xa+b1 yb1, g = xc+d1 yd1 ,

tq = (a+ b− (c+ d))(b− d) = (b− d)2 + (a− c)(b− d) > tp.

The case when x = x1y1, y = y1 is similar.By induction on mintp | JOS1,p is not principal we can principalize J

after a finite number of blow-ups of points.

Chapter 5

Surface Singularities

5.1. Resolution of surface singularities

In this section, we give a simple proof of resolution of surface singularities incharacteristic zero. The proof is by the good point algorithm of Abhyankar([7], [62], [73]).

Theorem 5.1. Suppose that S is a projective surface over an algebraicallyclosed field K of characteristic 0. Then there exists a resolution of singu-larities

Λ : T → S.

Theorem 5.1 is a consequence of Theorems 5.2, 4.9 and 4.11.

Theorem 5.2. Suppose that S is a hypersurface of dimension 2 in a non-singular variety V of dimension 3, over an algebraically closed field K ofcharacteristic 0. Then there exists a sequence of blow-ups of points andnon-singular curves contained in the strict transform Si of S,

Vn → Vn−1 → · · · → V1 → V,

such that the strict transform Sn of S on Vn is non-singular.

The remainder of this section will be devoted to the proof of Theorem5.2. Suppose that V is a non-singular three-dimensional variety over analgebraically closed field K of characteristic 0, and S ⊂ V is a surface.

For a natural number t, define (Definitions A.17 and A.20)

Singt(S) = p ∈ V | νp(S) ≥ t.

By Theorem A.19, Singt(S) is Zariski closed in V .

45

46 5. Surface Singularities

Letr = maxt | Singt(S) 6= ∅

be the maximal multiplicity of points of S. There are two types of blow-upsof non-singular subvarieties on a non-singular three-dimensional variety, ablow-up of a point, and a blow-up of a non-singular curve.

We will first consider the blow-up π : B(p)→ V of a closed point p ∈ V .Suppose that U = spec(R) ⊂ V is an affine open neighborhood of p, and phas ideal mp = (x, y, z) ⊂ R. Then

π−1(U) = proj(⊕

mnp ) = spec(R[

y

x,z

x]) ∪ spec(R[

x

y,z

y]) ∪ spec(R[

x

z,y

z]).

The exceptional divisor is E = π−1(p) ∼= P2.At each closed point q ∈ π−1(p), we have regular parameters (x1, y1, z1)

of the following forms:

x = x1, y = x1(y1 + α), z = x1(z1 + β),

with α, β ∈ K, x1 = 0 a local equation of E, or

x = x1y1, y = y1, z = y1(z1 + α)

with α ∈ K, y1 = 0 a local equation of E, or

x = x1z1, y = y1z1, z = z1,

z1 = 0 a local equation of E.We will now consider the blow-up π : B(C)→ V of a non-singular curve

C ⊂ V . If p ∈ V and U = spec(R) ⊂ V is an open affine neighborhood of pin V such that mp = (x, y, z) and the ideal of C is I = (x, y) in R, then

π−1(U) = proj(⊕

In) = spec(R[x

y]) ∪ spec(R[

y

x]).

Also, π−1(p) ∼= P1 and π−1(C ∩ U) ∼= (C ∩ U) × P1. Let E = π−1(C) bethe exceptional divisor. E is a projective bundle over C. At each pointq ∈ π−1(p), we have regular parameters (x1, y1, z1) such that

x = x1, y = x1(y1 + α), z = z1,

where α ∈ K, x1 = 0 is a local equation of E, or

x = x1y1, y = y1, z = z1,

where y1 = 0 is a local equation of E.In this section, we will analyze the blow-up

π : B(W ) = B(IW )→ V

of a non-singular subvariety W of V above a closed point p ∈ V , bypassing to a formal neighborhood spec(OV,p) of p and analyzing the mapπ : B(IW,p) → spec(OV,p). We have a natural isomorphism B(IW,p) ∼=

5.1. Resolution of surface singularities 47

B(IW ) ×V spec(OV,p). Observe that we have a natural identification ofπ−1(p) with π−1(p).

We say that an ideal I ⊂ OV,p is algebraic if there exists an ideal J ⊂OV,p such that I = J . This is equivalent to the statement that there existsan ideal sheaf I ⊂ OV such that Ip = I.

If I ⊂ OV,p is algebraic, so that there exists an ideal sheaf I ⊂ OV suchthat Ip = I, then we can extend the blow-up π : B(I) → spec(OV,p) to ablow-up π : B(I)→ V .

The maximal ideal mpOV is always algebraic. However, the ideal sheafI of a non-singular (formal) curve in spec(OV,p) may not be algebraic.

One example of a formal, non-algebraic curve is

I = (y − ex) ⊂ C[[x, y]] = OA2C,0.

A more subtle example, which could occur in the course of this section, isthe ideal sheaf of the irreducible curve

J = (y2 − x2 − x3, z) ⊂ R = K[x, y, z],

which we studied after Theorem 3.6. In R = K[[x, y, z]] we have regularparameters

x = y − x√

1 + x, y = y + x√

1 + x, z = z.

ThusJR = (xy, z) = (x, z) ∩ (y, z) ⊂ R = K[[x, y, z]].

In this example, we may be tempted to blow-up one of the two formalbranches x = 0, z = 0 or y = 0, z = 0, but the resulting blown-up schemewill not extend to a blow-up of an ideal sheaf in A3

K .A situation which will arise in this section when we will blow up a formal

curve which will actually be algebraic is given in the following lemma.

Lemma 5.3. Suppose that V is a non-singular three-dimensional variety,p ∈ V is a closed point and π : V1 = B(p)→ V is the blow-up of p with ex-ceptional divisor E = π−1(p) ∼= P2. Let R = OV,p. We have a commutativediagram of morphisms of schemes

B = B(mpR) → V1 = B(p)π ↓ ↓ π

spec(R) → V

such that π−1(mp) → π−1(p) = E is an isomorphism of schemes. Supposethat I ⊂ R is any ideal, and I ⊂ OB is the strict transform of I. Then thereexists an ideal sheaf J on V1 such that JOB = IEOB + I.


Proof. IOE is an ideal sheaf on E, so there exists an ideal sheaf J ⊂ OV1

such that IE ⊂ J and J /IE ∼= IOE . Thus J has the desired property.

Lemma 5.4. Suppose that V is a non-singular three-dimensional variety,S ⊂ V is a surface, C ⊂ Singr(S) is a non-singular curve, π : B(C) → V

is the blow-up of C, and S is the strict transform of S in B(C). Supposethat p ∈ C. Then νq(S) ≤ r for all q ∈ π−1(p), and there exists at most onepoint q ∈ π−1(p) such that νq(S) = r. In particular, if E = π−1(C), theneither Singr(S)∩E is a non-singular curve which maps isomorphically ontoC, or Singr(S) ∩ E is a finite union of points.

Proof. By the Weierstrass preparation theorem and after a Tschirnhausentransformation (Definition 3.8), a local equation f = 0 of S in OS,p =K[[x, y, z]] has the form

(5.1) f = zr + a2(x, y)zr−2 + · · ·+ ar(x, y).

But f ∈ (I(r)C,p) = IrC,p implies ∂f

∂z ∈ Ir−1C,p , and r!z = ∂r−1f

∂zr−1 ∈ IC,p. Thusz ∈ IC,p. After a change of variables in x and y, we may assume thatIC,p = (x, z). Then f ∈ IrC,p implies xi | ai for all i. If q ∈ π−1(p), thenOB(C),q has regular parameters (x1, y, z1) such that

x = x1z1, z = z1

orx = x1, z = x1(z1 + α)

for some α ∈ K.In the first case, a local equation of the strict transform of S is a unit.

In the second case, the strict transform of S has a local equation

f1 = (z1 + α)r +a2

x21

(z1 + α)r−2 + · · ·+ arxr1.

We have ν(f1) ≤ r, and ν(f1) < r if α 6= 0.

Lemma 5.5. Suppose that p ∈ Singr(S) is a point, π : B(p) → V is theblow-up of p, S is the strict transform of S in B(p), and E = π−1(p). Thenνq(S) ≤ r for all q ∈ π−1(p), and either Singr(S)∩E is a non-singular curveor Singr(S) ∩ E is a finite union of points.

Proof. By the Weierstrass preparation theorem and after a Tschirnhausentransformation, a local equation f = 0 of S in OS,p = K[[x, y, z]] has theform

(5.2) f = zr + a2(x, y)zr−2 + · · ·+ ar(x, y).


If q ∈ π−1(p) and νq(S) ≥ r, then OB(C),q has regular parameters (x1, y, z1)such that

x = x1y1, y = y1, z = y1z1.

orx = x1, y = x1(y1 + α), z = x1z1

for some α ∈ K, and ν1(S) = r. Thus Singr(S) ∩ E is contained in theline which is the intersection of E with the strict transform of z = 0 inB(C)×V spec(OV,p).

Definition 5.6. Singr(S) has simple normal crossings (SNCs) if

1. all irreducible components of Singr(S) (which could be points orcurves) are non-singular, and

2. if p is a singular point of Singr(S), then there exist regular param-eters (x, y, z) in OV,p such that ISingr(S),p = (xy, z).

Lemma 5.7. Suppose that Singr(S) has simple normal crossings, W is apoint or an irreducible curve in Singr(S), π : V ′ = B(W )→ V is the blow-up of W and S′ is the strict transform of S on V ′. Then Singr(S′) hassimple normal crossings.

Proof. This follows from Lemmas 5.4 and 5.5 and a simple local calculationon V ′.

Definition 5.8. A closed point p ∈ S is a pregood point if, in a neighborhoodof p, Singr(S) is either empty, a non-singular curve through p, or a a unionof two non-singular curves intersecting transversally at p (satisfying 2 ofDefinition 5.6).

Definition 5.9. p ∈ S is a good point if p is pregood and for any sequence

Xn → Xn−1 → · · · → X1 → spec(OV,p)

of blow-ups of non-singular curves in Singr(Si), where Si is the strict trans-form of S ∩ spec(OV,p) on Xi, then q is pregood for all closed points q ∈Singr(Sn). In particular, Singr(Sn) contains no isolated points.

A point which is not good is called bad.

Lemma 5.10. Suppose that all points of Singr(S) are good. Then thereexists a sequence of blow ups

V ′ = Vn → · · · → V1 → V

of non-singular curves contained in Singr(Si), where Si is the strict trans-form of S in Vi, such that Singr(S′) = ∅ if S′ is the strict transform of Son V ′.


Proof. Suppose that C is a non-singular curve in Singr(S). Let π1 : V1 =B(C) → V be the blow-up of C, and let S1 be the strict transform of S.If Singr(S1) 6= ∅, we can choose another non-singular curve C1 in Singr(S1)and blow-up by π2 : V2 = B(C1) → V1. Let S2 be the strict transform ofS1. We either reach a surface Sn such that Singr(Sn) = ∅, or we obtain aninfinite sequence of blow-ups

· · · → Vn → Vn−1 → · · · → V

such that each Vi+1 → Vi is the blow-up of a curve Ci in Singr(Si), whereSi is the strict transform of S on Vi. Each curve Ci which is blown up mustmap onto a curve in S by Lemma 5.5. Thus there exists a curve γ ⊂ Ssuch that there are infinitely many blow-ups of curves mapping onto γ inthe above sequence. Let R = OV,γ , a two-dimensional regular local ring.IS,γ is a height 1 prime ideal in this ring, and P = Iγ,γ is the maximal idealof R. We have

dimR+ trdegK R/P = 3

by the dimension formula (Theorem 15.6 [66]). ThusR/P has transcendencedegree 1 over K. Let t ∈ R be the lift of a transcendence basis of R/P overK. Then K[t] ∩ P = (0), so the field K(t) ⊂ R. We can write R = AQ,where A is a finitely generated K(t)-algebra (which is a domain) and Q isa maximal ideal in A. Thus R is the local ring of a non-singular point qon the K(t) surface spec(A). q is a point of multiplicity r on the curve inspec(A) with ideal sheaf IS,γ in R.

The sequence

· · · → Vn ×V spec(R)→ Vn−1 ×V spec(R)→ · · · → spec(R)

consists of infinitely many blow-ups of points on a K(t)-surface, which areof multiplicity r on the strict transform of the curve spec(R/IS,γ). But thisis impossible by Theorem 3.15.

Lemma 5.11. The number of bad points on S is finite.

Proof. Let

B0 = isolated points of Singr(S) ∪ singular points of Singr(S).

Singr(S)−B0 is a non-singular 1-dimensional subscheme of V . Let

π1 : V1 = B(Singr(S)−B0)→ V −B0

be its blow-up. Let S1 be the strict transform of S, and let

B1 = isolated points of Singr(S1) ∪ singular points of Singr(S1).

We can iterate to construct a sequence

· · · → (Vn −Bn)→ · · · → V,


where πn : Vn −Bn → V − Tn are the induced surjective maps, with

Tn = B0 ∪ π1(B1) ∪ · · · ∪ πn(Bn).Let Sn be the strict transform of S on Vn.

let C ⊂ Singr(S) be a curve. spec(OS,C) is a curve singularity of mul-tiplicity r, embedded in a non-singular surface over a field of transcendencedegree 1 over K (as in the proof of Lemma 5.10). πn induces by base change

Sn ×S spec(OS,C)→ spec(OS,C),

which corresponds to an open subset of a sequence of blow-ups of pointsover spec(OS,C). So for n >> 0, Sn ×S spec(OS,C) is non-singular. Thusthere are no curves in Singr(Sn) which dominate C. For n >> 0 thereare thus no curves in Singr(Sn) which dominate curves of Singr(S), so thatSingr(Sn) ∩ (Vn − Bn) is empty for large n, and all bad points of S arecontained in a finite set Tn.

Theorem 5.12. Let

(5.3) · · · → Vn → Vn−1 → · · · → V1 → V

be the sequence where πn : Vn → Vn−1 is the blow-up of all bad points on thestrict transform Sn−1 of S. Then this sequence is finite, so that it terminatesafter a finite number of steps with a Vm such that all points of Singr(Sm)are good.

We now give the proof of Theorem 5.12.Suppose there is an infinite sequence of the form of (5.3). Then there is

an infinite sequence of points pn ∈ Singr(Sn) such that πn(pn) = pn−1 forall n. We then have an infinite sequence of homomorphisms of local rings

R0 = OV,p → R1 = OV1,p1 → · · · → Rn = OVn,pn → · · · .By the Weierstrass preparation theorem and after a Tshirnhausen transfor-mation (Definition 3.8) there exist regular parameters (x, y, z) in R0 suchthat there is a local equation f = 0 for S in R0 of the form

f = zr + a2(x, y)zr−2 + · · ·+ ar(x, y)

with ν(ai(x, y)) ≥ i for all i. Since νpi(Si) = r for all i, we have regularparameters (xi, yi, zi) in Ri for all i and a local equation fi = 0 for Si suchthat

xi−1 = xi, yi−1 = xi(yi + αi), zi−1 = xi−1zi−1

with αi ∈ K, or

xi−1 = xiyi, yi−1 = yi, zi−1 = yi−1zi−1,

andfi = zri + a2,i(xi, yi)zr−2

i + · · · ar,i(xi, yi),


where

aji(xi, yi) =

aj,i−1(xi,xi(yi+αi))

xji

or aj,i−1(xiyi,yi)

yji

for all j. Thus the sequence

K[[x, y]→ K[[x1, y1]]→ · · ·is a sequence of blow-ups of a maximal ideal, followed by completion. InK[[xi, yi]] we have relations

aj = γj,ixcj,ii y

dj,ii aj,i−1

for all i and 2 ≤ j ≤ r, where γji is a unit. By embedded resolution of curvesingularities (Exercise 3.13), there exists m0 such that

∏rj=2 aj(x, y) = 0 is

a SNC divisor in Ri for all i ≥ m0. Thus for i ≥ m0 we have an expression

(5.4) fi = zri + a2i(xi, yi)xa2ii yb2ii zr−2

i + · · ·+ ari(xi, yi)xarii ybrii ,

where the aji are units (or zero) in Ri and aji + bji ≥ j if aji 6= 0.We observe that, by the proof of Theorem A.19 and Remark A.21,

ISingr(Si),pi=√J,

where J is the ideal in Ri generated by

∂j+k+lfi

∂xji∂yki ∂z

li

| 0 ≤ j + k + l < r.

Thus ISingr(Si),pimust be one of (xi, yi, zi), (xi, zi), (yi, zi) or (xiyi, zi).

Singr(Si) has SNCs at pi unless√J = (xiyi, zi) is the completion of an

ideal of an irreducible (singular) curve on Vi. Let E = π−1i (pi−1) ∼= P2.

By construction, we have that xi = 0 or yi = 0 is a local equation of Eat pi−1. Without loss of generality, we may suppose that xi = 0 is a localequation of E. zi = 0 is a local equation of the strict transform of z = 0 inVi ×V spec(R), so that (xi, zi) is the completion at qi of the ideal sheaf ofa subscheme of E. Thus there is a curve C in Vi such that IC,pi = (xi, zi)by Lemma 5.3. The ideal sheaf J = (ISingr(Si)

: IC) in OVi is such that(J ∩ IC)pi ∼= ISingr(Si),pi

, so that (xiyi, zi) is not the completion of an idealof an irreducible curve on Vi, and Singr(Si) has simple normal crossings atpi.

We may thus assume that m0 is sufficently large that Singr(Si) has SNCseverywhere on Vi for i ≥ m0.

We now make the observation that we must have bji 6= 0 and aki 6= 0for some j and k if i ≥ m0. If we did have bji = 0 for all j (with a similaranalysis if aji = 0 for all j) in (5.4), then, since

ISingr(Si),pi=√J,


where J is the ideal in Ri generated by

∂j+k+lfi

∂xji∂ykj ∂z

li

| 0 ≤ j + k + l < r,

we have (with the condition that bji = 0 for all j) that√J = (xi, zi). Since

Singr(Si) has SNCs in Vi, there exists a non-singular curve C ⊂ Singr(Si)such that xi = zi = 0 are local equations of C in Ri. Let λ : W → Vi be theblow-up of C. If q ∈ λ−1(pi), we have regular parameters (u, v, y) in OW,qsuch that

xi = u, zi = u(v + β)

orxi = uv, zi = v.

If f ′ = 0 is a local equation of the strict transform S′ of Si in W , we havethat νq(f ′) < r except possibly if xi = u, zi = uv. Then

f ′ = vr + a2iua2i−2vr−2 + · · ·+ ariu

ari−r.

We see that f ′ is of the form of (5.4), with bji = 0 for all j, but

minajij| 2 ≤ j ≤ r

has decreased by 1. We see, using Lemma 5.4, that after a finite numberof blow-ups of non-singular curves in Singr of the strict transform of S, themultiplicity must be < r at all points above pi. Thus pi is a good point, acontradiction to our assumption. Thus bji 6= 0 and aki 6= 0 in (5.4) for somej and k if i ≥ m0.

If follows that, for i ≥ m0, we must have

xi = xi+1, yi = xi+1yi+1, zi = xi+1zi+1

orxi = xi+1yi+1, yi = yi+1, zi = yi+1zi+1.

Thus in (5.4), we must have for 2 ≤ j ≤ r,

(5.5)aj,i+1 = aji + bji − j, bj,i+1 = bji oraj,i+1 = aji, bj,i+1 = aj,i + bji − j

respectively, for all i ≥ m0.Suppose that ζ ∈ R. Then the fractional part of ζ is

ζ = t

if ζ = a+ t with a ∈ Z and 0 ≤ t < 1.

Lemma 5.13. For i ≥ m0, pi is a good point on Vi if there exists j suchthat aji 6= 0 and


1.ajij≤ aki

k,

bjij≤ bki

k

for all k with 2 ≤ k ≤ r and aki 6= 0, while

2. ajij

+

bjij

< 1.

Proof. Since νpi(Si) = r, aji+ bji ≥ j. Condition 2 thus implies that eitheraji ≥ j or bji ≥ j. Without loss of generality, aji ≥ j. Then condition 1implies that aki ≥ k for all k, and

ISingr(Si),pi⊂ (xi, zi)Ri.

Since Singr(Si) has SNCs, there exists a non-singular curve C ⊂ Singr(Si)such that IC,pi = (xi, zi). Let π : V ′ = B(C) → Vi be the blow-up of C,and let S′ be the strict transform of S on V ′. By Lemma 5.7, Singr(S′)has SNCs. A direct local calculation shows that there is at most one pointq ∈ π−1(pi) such that νq(S′) = r, and if such a point q exists, then OV ′,qhas regular parameters (x′, y′, z′) such that

xi = x′, yi = y′, zi = x′z′.

A local equation f ′ = 0 of S′ at q is

f ′ = (z′)r + a2i(x′)a2i−2(y′)b2i(z′)r−2 + · · ·+ ari(x′)ari−r(y′)bri .

Thus we have an expression of the form (5.4), and conditions 1 and 2 of thestatement of this lemma hold for f ′. After repeating this procedure a finitenumber of times we will construct V → Vi such that νa(S) < r for all pointsa on the strict transform S of S such that a maps to pi, since aji

j + bjij must

drop by 1 every time we blow up.

Lemma 5.14. Let

δj,k,i =(ajij− aki

k

) (bjij− bki

k

).

Thenδj,k,i+1 ≥ δjki,

and if δjki < 0 then

δjk,i+1 − δjki ≥1r4.

Proof. We may assume that the first case of (5.5) holds. Then

δj,k,i+1 = δj,k,i +(bjij− bki

k

)2

.


If δj,k,i < 0 we must have bjij −

bkik 6= 0. Thus(

bjij− bki

k

)2

≥ 1j2k2

≥ 1r4.

Corollary 5.15. There exists m1 such that i ≥ m1 implies condition 1 ofLemma 5.13 holds.

Proof. There exists m1 such that i ≥ m1 implies δjki ≥ 0 for all j, k. Thenthe pairs (akik ,

bkik ) with 2 ≤ k ≤ r are totally ordered by ≤, so there exists

a minimal element (ajij ,bjij ).

Lemma 5.16. There exists an i′ ≥ m1 such that we have condition 2 ofLemma 5.13,

aji′

j+

bji′

j < 1.

Proof. A calculation shows thatajij

+

bjij

≥ 1

implies aj,i+1

j

+

bj,i+1

j

≤

ajij

+

bjij

− 1r.

With the above i′, pi′ is a good point, since pi′ satisfies the conditionsof Lemma 5.13. Thus the conclusions of Theorem 5.12 must hold.

Now we give a the proof of Theorem 5.2. Let r be the maximal mul-tiplicity of points of S. By Theorem 5.12, there exists a finite sequenceof blow-ups of points Vm → V such that all points of Singr(Sm) are goodpoints, where Sm is the strict transform of S on Vm. By Lemma 5.10, thereexists a sequence of blow-ups of non-singular curves Vn → Vm such thatSingr(Sn) = ∅, where Sn is the strict transform of S on Vn. By descendinginduction on r we can reach the case where Sing2(Sn) = ∅, so that Sn isnon-singular.

Remark 5.17. Zariski discusses early approaches to resolution in [91].Some early proofs of resolution of surface singularities are by Albanese [12],Beppo Levi [61], Jung [57] and Walker [82]. Zariski gave several proofs ofresolution of algebraic surfaces over fields of characteristic zero, includingthe first algebraic proof [86], which we present later in Chapter 8.


Exercise 5.18.

1. Suppose that S is a surface which is a subvariety of a non-singularthree-dimensional variety V over an algebraically closed field K ofcharacteristic 0, and p ∈ S is a closed point.

a. Show that a Tschirnhausen transformation of a suitable localequation of S at p gives a formal hypersurface of maximalcontact for S at p.

b. Show that if f = 0 is a local equation of S at p, and (x, y, z)are regular parameters at p such that ν(f) = ν(f(0, 0, z)),then ∂r−1f

∂zr−1 = 0 is a hypersurface of maximal contact (A.20)for f = 0 in a neighborhood of the origin.

2. Follow the algorithm to reduce the multiplicity of f = zr+xayb = 0,where a+ b ≥ r.

5.2. Embedded resolution of singularities

In this section we will prove the following theorem, which is an extension ofthe main resolution theorem, Theorem 5.2 of the previous section.

Theorem 5.19. Suppose that S is a surface which is a subvariety of a non-singular three-dimensional variety V over an algebraically closed field K ofcharacteristic 0. Then there exists a sequence of monoidal transforms overthe singular locus of S such that the strict transform of S is non-singular,and the total transform π∗(S) is a SNC divisor.

Definition 5.20. A resolution datum R = (E0, E1, S, V ) is a 4-tuple whereE0, E1, S are reduced effective divisors on the non-singular three-dimensionalvariety V such that E = E0 ∪ E1 is a SNC divisor.

R is resolved at p ∈ S if S is non-singular at p and E ∪ S has SNCs atp. For r > 0, let

Singr(R) = p ∈ S | νp(S) ≥ r and R is not resolved at p.

Observe that Singr(R) = Singr(S) if r > 1, and Singr(R) is a proper Zariskiclosed subset of S for r ≥ 1.

For p ∈ S, let

η(p) = the number of components of E1 containing p.

We have 0 ≤ η(p) ≤ 3. For r, t ∈ N, let

Singr,t(R) = p ∈ Singr(R) | η(p) ≥ t.

5.2. Embedded resolution of singularities 57

Singr,t(R) is a Zariski closed subset of Singr(R) (and of S). For the rest ofthis section we will fix

r = ν(R) = ν(S,E) = maxνp(S) | p ∈ S,R is not resolved at p,

t = η(R) = the maximum number of components of E1

containing a point of Singr(R).

Definition 5.21. A permissible transform of R is a monoidal transformπ : V ′ → V whose center is either a point of Singr,t(R) or a non-singularcurve C ⊂ Singr,t(R) such that C makes SNCs with E.

Let F be the exceptional divisor of π. Then π∗(E)∪F is a SNC divisor.We define the strict transform R′ of R to be R′ = (E′

0, E′1, S

′), where S′ isthe strict transform of S, and we also define

E′0 = π∗(E0)red + F,

E′1 = strict transform of E1 if ν(S′, π∗((E0 + E1)red + F ) = ν(R),

E′0 = ∅, E′

1 = π∗(E0 + E1)red + F if ν(S′, π∗((E0 + E1)red + F ) < ν(R).

Lemma 5.22. With the notation of the previous definition,

ν(R′) ≤ ν(R)

andν(R′) = ν(R) implies η(R′) ≤ η(R).

The proof of Lemma 5.22 is a generalization of Lemmas 5.3 and 5.4.

Definition 5.23. p ∈ Singr,t(R) is a pregood point if the following twoconditions hold:

1. In a neighborhood of p, Singr,t(R) is either a non-singular curvethrough p, or two non-singular curves through p intersecting trans-versally at p (satisfying 2 of Definition 5.6).

2. Singr,t(R) makes SNCs with E at p. That is, there exist regularparameters x, y, z in OV,p such that the ideal sheaf of each com-ponent of E and each curve in Singr,t(R) containing p is generatedby a subset of x, y, z at p.

p ∈ S is a good point if p is a pregood point and for any sequence

π : Xn → Xn−1 → · · · → X1 → spec(OV,p)

of permissible monodial transforms centered at curves in Singr,t(Ri), whereRi is the transform of R on Xi, then q is a pregood point for all q ∈ π−1(p).

A point p ∈ Singr,t(R) is called bad if p is not good.

Lemma 5.24. The number of bad points in Singr,t(R) is finite.


Lemma 5.25. Suppose that all points of Singr,t(R) are good. Then thereexists a sequence of permissible monoidal transforms, π : V ′ → V , cen-tered at non-singular curves contained in Singr,t(Si), where Si is the stricttransform of S on the i-th monoidal transform, such that if R′ is the stricttransform of R on V ′, then either

ν(R′) < ν(R)

orν(R′) = ν(R) and η(R′) < η(R).

The proofs of the above two lemmas are similar to the proofs of Lemmas5.11 and 5.10 respectively, with the use of embedded resolution of planecurves.

Theorem 5.26. Suppose that R is a resolution datum such that E0 = ∅.Let

(5.6) · · · → Vnπn→ Vn−1 → · · ·

π1→ V0 = V

be the sequence where πn is the monoidal transform centered at the unionof bad points in Singr,t(Ri), where Ri is the transform of R on Vi. Thenthis sequence is finite, so that it terminates in a Vm such that all points ofSingr,t(Rm) are good.

Proof. Suppose that (5.6) is an infinite sequence. Then there exists aninfinite sequence of points pn ∈ Vn such that πn+1(pn+1) = pn for all n andpn is a bad point. Let Rn = OVn,pn for n ≥ 0. Since the sequence (5.6) isinfinite, its length is larger than 1, so we may assume that t ≤ 2.

Let f = 0 be a local equation of S at p0, and let gk, 0 ≤ k ≤ t, be localequations of the components of E1 containing p0. In R0 = OV,p there areregular parameters (x, y, z) such that ν(f(0, 0, z)) = r and ν(gk(0, 0, z)) = 1for all k. This follows since for each equation this is a non-trivial Zariski opencondition on linear changes of variables in a set of regular parameters. Bythe Weierstrass preparation theorem, after multiplying f and the gk by unitsand performing a Tschirnhausen transformation on f , we have expressions

(5.7)f = zr +

∑rj=2 aj(x, y)z

r−j ,

gk = z − φk(x, y), (or gk is a unit)

in R0. z = 0 is a local equation of a formal hypersurface of maximal contactfor f = 0. Thus we have regular parameters (x1, y1, z1) in R1 = OV1,p1

defined byx = x1, y = x1(y1 + α), z = x1z1,

with α ∈ K, orx = x1y1, y = y1, z = y1z1.

5.2. Embedded resolution of singularities 59

The strict transforms f1 of f and gk1 of gk have the form of (5.7) withaj replaced by aj

xj1(or aj

yj1), φk with φk

x1(or φk

y1). After a finite number of

monoidal transforms centered at points pj with ν(fj) = r, we have forms inRi = OVi,pi for the strict transforms of f and gk:

(5.8) fi = zri +∑r

j=2 aj(xi, yi)xajii y

bjii zr−ji ,

gki = zrk + φki(xi, yi)xckii ydkii ,

such that aj , φki are either units or zero (as in the proof of Theorem 5.12).We further have that local equations of the exceptional divisor Ei0 of Vi → Vare one of xi = 0, yi = 0 or xiyi = 0. This last case can only occur if t < 2.Note that our assumption η(pi) = t implies gki is not a unit in Ri for1 ≤ k ≤ t.

We can further assume, by an extension of the argument in the proof ofTheorem 5.12, that all irreducible curves in Singr,t(Ri) are non-singular.

By Lemma 5.14, we can assume that the set

T =(

ajij,bjij

), | aj 6= 0

∪ (cki, dki) | φki 6= 0

is totally ordered. Now by Lemma 5.16 we can further assume thataj0ij0

+

bj0ij0

< 1,

where (aj0ij0 ,bj0ij0

) is the minimum of the (ajij ,bjij ). Recall that, if t > 0,∏t

i=1 gi = 0 makes SNCs with the exceptional divisor Ei0 of Vi → V . Theseconditions imply that, after possibly interchanging the gki, interchangingxi, yi and multiplying xi, yi by units in Ri, that the gki can be described byone of the following cases at pi:

1. t = 0.

2. t = 1,a. g1i = zi + xλi , λ ≥ 1.b. g1i = zi + xλi y

µ, λ ≥ 1, µ ≥ 1.c. g1i = zi

3. t = 2,a. g1i = zi, g2i = zi + xi.b. gi1 = zi + xi, gi2 = zi + εxi, where ε is a unit in Ri such that

the residue of ε in the residue field of Ri is not 1.c. gi1 = zi + xi, gi2 = zi + εxλi y

µi , where ε is a unit in Ri, λ ≥ 1

and λ+ µ ≥ 2. If µ > 0 we can take ε = 1.

We can check explicitly by blowing up permissible curves that pi is agood point. In this calculation, if t = 0, this follows from the proof of


Theorem 5.12, recalling that xi = 0, yi = 0 or xiyi = 0 are local equationsof E0 at pi. If t = 1, we must remember that we can only blow up a curveif it lies on g1 = 0, and if t = 2, we can only blow up the curve g1 = g2 = 0.Although we are constructing a sequence of permissible monoidal transformsover spec(OVi,pi), this is equivalent to constructing such a sequence overspec(OVi,pi), as all curves blown up in this calculation are actually algebraic,since all irreducible curves in Singr,t(Ri) are non-singular.

We have thus found a contradiction, by which we conclude that thesequence (5.6) has finite length.

The proof of Theorem 5.19 now follows easily from Theorem 5.26, Lemma5.25 and descending induction on (r, t) ∈ N×N, with the lexicographic order.

Remark 5.27. This algorithm is outlined in [73] and [62].

Exercise 5.28. Resolve the following (or at least make (ν(R), η(R)) dropin the lexicographic order):

1. R = (∅, E1, S), where S has local equation f = z = 0, and E1 isthe union of two components with respective local equations g1 =z + xy, g2 = z + x;

2. R = (∅, E1, S), where S has local equation f = z3 + x7y7 = 0 andE1 is the union of two components with respective local equationsg1 = z + xy, g2 = z + y(x2 + y3).

Chapter 6

Resolution ofSingularities inCharacteristic Zero

The first proof of resolution in arbitrary dimension (over a field of charac-teristic zero) was by Hironaka [52]. The proof consists of a local proof, anda complex web of inductions to produce a global proof. The local proof usessophisticated methods in commutative algebra to reduce the problem of re-ducing the multiplicity of the strict transform of an ideal after a permissiblesequence of monoidal transforms to the problem of reducing the multiplic-ity of the strict transform of a hypersurface. The use of the Tschirnhausentransformation to find a hypersurface of maximal contact lies at the heart ofthis proof. The order ν and the invariant τ (of Chapter 7) are the principalinvariants considered.

De Jong’s theorem [36], referred to at the end of Chapter 7, can berefined to give a new proof of resolution of singularities of varieties of char-acteristic zero ([9], [16], [70]). The resulting theorem is not as strong as theclassical results on resolution of singularities in characteristic zero (Section6.8), as the resulting resolution X ′ → X may not be an isomorphism overthe non-singular locus of X.

In recent years several different proofs of canonical resolution of singu-larities have been constructed (Bierstone and Milman [14], Bravo, Encinasand Villamayor [17], Encinas and Villamayor [40], [41], Encinas and Hauser[39], Moh [67], Villamayor [79], [80], and Wlodarczyk [84]). These papers

61

62 6. Resolution of Singularities in Characteristic Zero

have significantly simplified Hironaka’s original proofs in [47] and [55], andrepresent a great advancement of the subject.

In this chapter we prove the basic theorems of resolution of singularitiesin characteristic 0. Our proof is based on the algorithms of [40] and [41].The final statements of resolution are proved in Section 6.8, and in Exercise6.42 at the end of that section.

Throughout this chapter, unless explicitely stated otherwise, we willassume that all varieties are over a field K of characteristic zero. Supposethat X1 and X2 are subschemes of a variety W . We will denote the reducedset-theoretic intersection of X1 and X2 by X1∩X2. However, we will denotethe scheme-theoretic intersection of X1 and X2 by X1 ·X2. X1 ·X2 is thesubscheme of W with ideal sheaf IX1 + IX2 .

6.1. The operator 4 and other preliminaries

Definition 6.1. Suppose that K is a field of characteristic zero, R =K[[x1, . . . , xn]] is a ring of formal power series overK, and J = (f1, . . . , fr) ⊂R is an ideal. Define

4(J) = 4R(J) = (f1, . . . , fr) +(∂fi∂xj| 1 ≤ i ≤ r, 1 ≤ j ≤ n

).

4(J) is independent of the choice of generators fi of J and regular param-eters xj in R.

Lemma 6.2. Suppose that W is a non-singular variety and J ⊂ OW is anideal sheaf. Then there exists an ideal 4(J) = 4W (J) ⊂ OW such that

4(J)OW,q = 4(J)

for all closed points q ∈W .

Proof. We define 4(J) as follows. We can cover W by affine open subsetsU = spec(R) which satisfy the conclusions of Lemma A.11, so that Ω1

R/K =dy1R⊕ · · · ⊕ dynR. If Γ(U, J) = (f1, . . . , fr), we define

Γ(U,4(J)) = (fi,∂fi∂yj| 1 ≤ i ≤ r, 1 ≤ j ≤ n).

This definition is independent of all choices made in this expression.Suppose that q ∈W is a closed point, and x1, . . . , xn are regular param-

eters in OW,q. Let U = spec(R) be an affine neighborhood of q as above.Let A = OW,q. Then dy1, . . . , dyn and dx1, . . . , dxn are A-bases of Ω1

A/k

by Lemma A.11 and Theorem A.10. Thus the determinant

|( ∂yi∂xj

)|(q) 6= 0

6.1. The operator 4 and other preliminaries 63

from which the conclusions of the lemma follow.

Given J ⊂ OW and q ∈W , we define νJ(q) = νq(J) (Definition A.17). IfX is a subvariety of W , we denote νX = νIX . We can define 4r(J) for r ∈ Ninductively, by the formula 4r(J) = 4(4r−1(J)). We define 40(J) = J .

Lemma 6.3. Suppose that W is a non-singular variety, (0) 6= J ⊂ OW isan ideal sheaf, and q ∈W (which is not necessarily a closed point). Then:

1. νq(J) = b > 0 if and only if νq(4(J)) = b− 1.

2. νq(J) = b > 0 if and only if νq(4b−1(J)) = 1.

3. νq(J) ≥ b > 0 if and only if q ∈ V (4b−1(J)).4. νJ : W → N is an upper semi-continuous function.

Proof. The lemma follows from Theorem A.19.

If f : X → I is an upper semi-continuous function, we define max f tobe the largest value assumed by f on X, and we define a closed subset ofX,

Max f = q ∈ X | f(q) = max f.We have Max νJ = V (4b−1(J)) if b = max νJ .

Suppose that X is a closed subset of a n-dimensional variety W . Weshall denote by

(6.1) R(1)(X) ⊂ Xthe union of irreducible components of X which have dimension n− 1.

Lemma 6.4. Suppose that W is a non-singlar variety, J ⊂ OW is anideal sheaf, b = max νJ and Y ⊂ Max νJ is a non-singular subvariety. Letπ : W1 → W be the monoidal transform with center Y . Let D be theexceptional divisor of π. Then:

1. There is an ideal sheaf J1 ⊂ OW1 such that

J1 = I−bD J

and ID - J1.2. If q ∈W1 and p = π(q), then

νJ1(q) ≤ νJ(p).

3. max νJ ≥ max νJ1.

Proof. Suppose that q ∈W1 and p = π(q). Let A be the Zariski closure ofq in W1, and B the Zariski closure of p in W . Then A → B is proper.By upper semi-continuity of νJ , there exists a closed point x ∈ B such thatνJ(x) = νJ(p). Let y ∈ π−1(x) ∩ A be a closed point. By semi-continuity


of νJ1, we have νJ1

(q) ≤ νJ1(y), so it suffices to prove 2 when q and p are

closed points.By Corollary A.5, there is a regular system of parameters y1, . . . , yn in

OW,p such that y1 = y2 = · · · = yr = 0 are local equations of Y at p. ByRemark A.18, we may assume that K = K(p) = K(q). Then we can makea linear change of variables in y1, . . . , yn to assume that there are regularparameters y1, . . . , yn in OW1,q such that

y1 = y1, y2 = y1y2, . . . , yr = y1yr, yr+1 = yr+1, . . . , yn = yn.

y1 = 0 is a local equation of D. By Remark A.18, if suffices to verify theconclusions of the theorem in the complete local rings

R = OW,p = K[[y1, . . . , yn]

andS = OW1,q = K[[y1, . . . , yn]].

Y ⊂ Max νJ implies νη(J) = b, where η is the general point of the componentof Y whose closure contains p. Suppose that f ∈ JR. Let t = νR(f) ≥ b.Then there is an expansion

f =∑

ai1,...,inyi11 · · · y

inn

in R, with ai1,...,in ∈ K and ai1,...,in = 0 if i1 + · · · + ir < t. Further, thereexist i1, . . . , in with i1 + · · ·+ in = t such that ai1,...,in 6= 0.

In S, we havef = yt1f1,

wheref1 =

∑ai1,...,iny

i1+···+ir−t1 · · · yinn

is such that y1 does not divide f1. Thus we have νS(f1) ≤ νR(f). Inparticular, we have yb1 | JS; and since there exists f ∈ R with νR(f) = b,yb+1

1 - JS. ThusJ1 = I−bD J

satisfies 1 andνJ1

(q) ≤ νJ(p) = b.

Suppose that W is a non-singular variety and J ⊂ OW is an ideal sheaf.Suppose that Y ⊂ W is a non-singular subvariety. Let Y1, . . . , Ym be thedistinct irreducible (connected) components of Y . Let π : W1 → W bethe monoidal transform with center Y and exceptional divisor D. Let D =D1+· · ·+Dm, where Di are the distinct irreducible (connected) components

6.1. The operator 4 and other preliminaries 65

of D, indexed so that π(Di) = Yi. Then by an argument as in the proof ofLemma 6.4, there exists an ideal sheaf J1 ⊂ OW1 such that

(6.2) JOW1 = Ic1D1· · · Icm

DmJ1,

where J1 is such that for 1 ≤ i ≤ m we have IDi - J1 and ci = νηi(J), withηi the generic point of Yi. J1 is called the weak transform of J .

There is thus a function c on W1, defined by c(q) = 0 if q 6∈ D andc(q) = ci if q ∈ Di, such that

(6.3) JOW1 = IcDJ1.

In the situation of Lemma 6.4, J1 has properties 1 and 2 of that lemma.With the notation of Lemma 6.4, suppose that J = IX is the ideal sheaf

of a subvariety X of W . The weak transform X of X is the subscheme ofW1 with ideal sheaf J1. We see that the weak transform X of X is mucheasier to calculate than the strict transform X of X. We have inclusions

X ⊂ X ⊂ π∗(X).

Example 6.5. Suppose that X is a hypersurface on a variety W , r =max νX and Y ⊂ Max νX is a non-singular subvariety. Let π : W1 → Wbe the monoidal transform of W with center Y and exceptional divisor E.In this case the strict transform X and the weak transform X are the samescheme, and π∗(X) = X + rE.

Example 6.6. Let X ⊂ W = A3 be the nodal plane curve with idealIX = (z, y2−x3) ⊂ k[x, y, z]. Let m = (x, y, z), and let π : B = B(m)→ A3

be the blow-up of the point m. The strict transform X of X is non-singular.The most interesting point in X is the point q ∈ π−1(m) with regular

parameters (x1, y1, z1) such that

x = x1, y = x1y1, z = x1z1.

Then

Iπ∗(X),q = (x1z1, x21y

21 − x3

1),

IX,q = (z1, x1y21 − x2

1),

IX,q = (z1, y21 − x1).

In this example, X, X and π∗(X) are all distinct.


6.2. Hypersurfaces of maximal contact and induction inresolution

Suppose that W is a non-singular variety, J ⊂ OW is an ideal sheaf andb ∈ N. Define

Sing(J, b) = q ∈W | νJ(q) ≥ b.

Sing(J, b) is Zariski closed in W by Lemma 6.3. Suppose that Y ⊂ Sing(J, b)is a non-singular (but not necessarily connected) subvariety of W . Let π1 :W1 → W be the monodial transform with center Y , and let D1 = π−1(Y )be the exceptional divisor.

Recall (6.3) that the weak transform J1 of J is defined by

JOW1 = J cD1J1,

where c is locally constant on connected components of D1. We have c ≥ bby upper semi-continuity of νJ .

We can now define J1 by

(6.4) JOW1 = J bD1J1,

so that

J1 = J c−bD1J1.

Suppose that

(6.5) Wnπn→Wn−1 → · · · →W1

π1→W

is a sequence of monoidal transforms such that each πi is centered at anon-singular subvariety Yi ⊂ Sing(Ji, b), where Ji is defined inductively by

Ji−1OWi = IbDiJi

and Di is the exceptional divisor of πi. We will say that (6.5) is a resolutionof (W,J, b) if Sing(Jn, b) = ∅.

Definition 6.7. Suppose that r = max νJ , q ∈ Sing(J, r). A non-singularcodimension 1 subvariety H of an affine neighborhood U of q in W is calleda hypersurface of maximal contact for J at q if

1. Sing(J, r) ∩ U ⊂ H, and

2. if

Wn → · · · →W1 →W

is a sequence of monoidal transforms of the form of (6.5) (withb = r), then the strict transform Hn of H on Un = Wn ×W U issuch that Sing(Jn, r) ∩ Un ⊂ Hn.

6.2. Hypersurfaces of maximal contact and induction in resolution 67

With the assumptions of Definition 6.7, a non-singular codimension 1subvariety H of U = spec(OW,q) is called a formal hypersurface of maximalcontact for J at q if 1 and 2 of Definition 6.7 hold, with U = spec(OW,q).

If X ⊂W is a codimension 1 subvariety, with r = max νX , q ∈ max νX ,then our definition of hypersurface of maximal contact for J = IX coincideswith the definition of a hypersurface of maximal contact for X given inDefinition A.20. In this case, the ideal sheaf Jn in (6.5) is the ideal sheaf ofthe strict transform Xn of X on Wn.

Now suppose that X ⊂ W is a singular hypersurface, and K is alge-braically closed. Let

r = maxνX > 1

be the set of points of maximal multiplicity on X. Let q ∈ Sing(IX , r) be aclosed point of maximal multiplicity r.

The basic strategy of resolution is to construct a sequence

Wn → · · · →W1 →W

of monoidal transforms centered at non-singular subvarieties of the locus ofpoints of multiplicity r on the strict transform of X so that we eventuallyreach a situation where all points of the strict transform of X have multiplic-ity < r. We know that under such a sequence the multiplicity can never goup, and always remains ≤ r (Lemma 6.4). However, getting the multiplicityto drop to less that r everywhere is much more difficult.

Notice that the desired sequence Wn → W1 will be a resolution of theform of (6.5), with (Ji, b) = (IXi , r), where Xi is the strict transform of Xon Wi.

Let q ∈ Sing(IX , r) be a closed point. After Weierstrass preparation andperforming a Tschirnhausen transformation, there exist regular parameters(x1, . . . , xn, y) in R = OW,q such that there exists f ∈ R such that f = 0 isa local equation of X at q, and

(6.6) f = yr +r∑i=2

ar(x1, . . . , xn)yr−i.

Set H = V (y) ⊂ spec(R). Then H = spec(S), where S = K[[x1, . . . , xn]].We observe that H is a (formal) hypersurface of maximal contact for X

at q. The verification is as follows.We will first show that H contains a formal neighborhood of the locus

of points of maximal multiplicity on X at q. Suppose that Z ⊂ Sing(IX , r)is a subvariety containing q. Let J = IZ,q.

Write J = P1∩· · ·∩Pr, where the Pi are prime ideals of the same heightin R. Let Ri = RPi . By semi-continuity of νX , we have f ∈ P ri RPi for all


i. Since ∂∂y : RPi → RPi , we have (r − 1)!y = ∂r−1f

∂yr−1 ∈ JRPi for all i. Thusy ∈

⋂JRPi = J and Z ∩ spec(R) ⊂ H.

Now we will verify that if Y ⊂ Sing(IX , r) is a non-singular subvariety,π : W1 → W is the monoidal transform with center Y , X1 is the stricttransform of X and q1 is a closed point of X1 such that π(q1) = q whichhas maximal multiplicity r, then q1 is on the strict transform H1 of H (overthe formal neighborhood of q). Suppose that q1 is such a point. That is,q1 ∈ π−1(q) ∩ Sing(IX1 , r). We have shown above that y ∈ IY . Withoutchanging the form of (6.6), we may then assume that there is s ≤ n such thatx1 = . . . = xs = y = 0 are (formal) local equations of Y at q, and there isa formal regular system of parameters x1(1), . . . , xn(1), y(1) in OW1,q1 suchthat

(6.7)

x1 = x1(1),xi = x1(1)xi(1) for 2 ≤ i ≤ s,y = x1(1)y(1),xi = xi(1) for s < i ≤ n.

Since Y ⊂ Sing(IX , r), we have qi ∈ (x1, . . . , xs)i for all i. Then we have a(formal) local equation f1 = 0 of X1 at q1, where

(6.8)

f1 =f

xr1= y(1)r +

r∑i=2

aixi1y(1)r−i

= y(1)r +r∑i=2

ai(1)(x1(1), . . . , xn(1))y(1)r−i

and y1(1) = 0 is a local equation of H1 at q1, while x1(1) = 0 is a localequation of the exceptional divisor of π. With our assumption that f1 alsohas multiplicity r, f1 has an expression of the same form as (6.6), so byinduction, H is a (formal) hypersurface of maximal contact.

We see that (at least over a formal neighborhood of q) we have reducedthe problem of resolution of X to some sort of resolution problem on the(formal) hypersurface H.

We will now describe this resolution problem. Set

I = (ar!22 , . . . , a

r!rr ) ⊂ S = K[[x1, . . . , xn]].

Observe that the scheme of points of multiplicity r in the formal neighbor-hood of q on X coincides with the scheme of points of multiplicity ≥ r! ofI,

Sing(fR, r) = Sing(I, r!).However, we may have νR(I) > r!.

Let us now consider the effect of the monodial transform π of (6.7)on I. Since H is a hypersurface of maximal contact, π certainly induces

6.2. Hypersurfaces of maximal contact and induction in resolution 69

a morphism H1 → H, where H1 is the strict transform of H. By directverification, we see that the transform I1 of I (this transform is defined by(6.4)) satisfies

I1S1 =1xr!1

IS1,

where S1 = K[[x1(1), . . . , xn(1)]] = OH1,q1 , and this ideal is thus, by (6.8),the ideal

I1S1 = (a2(1)r!2 , . . . , ar(1)

r!r ),

which is obtained from the coefficients of f1, our local equation of X1, in theexact same way that we obtained the ideal I from our local equation f ofX. We further observe that νS1(I1S1) ≥ r! if and only if νq1(X1) ≥ r. Wesee then that to reduce the multiplicity of the strict transform of X over q,we are reduced to resolving a sequence of the form (6.5), but over a formalscheme.

The above is exactly the procedure which is followed in the proof ofresolution for curves given in Section 3.4, and the proof of resolution forsurfaces in Section 5.1.

When X is a curve (dim W = 2) the induced resolution problem on theformal curve H is essentially trivial, as the ideal I is in fact just the idealgenerated by a monomial, and we only need blow up points to divide outpowers of the terms in the monomial.

When X is a surface (dim W = 3), we were able to make use of anotherinduction, by realizing that resolution over a general point of a curve in thelocus of points of maximal multiplicity reduces to the problem of resolutionof curves (over a non-closed field). In this way, we were able to reduceto a resolution problem over finitely many points in the locus of points ofmaximal multiplicity on X. The potentially worrisome problem of extendinga resolution of an object over a formal germ of a hypersurface to a globalproper morphism over W was not a great difficulty, since the blowing upof a point on the formal surface H always extends trivially, and the onlyother kind of blow-ups we had to consider were the blow-ups of non-singularcurves contained in the locus of maximal multiplicity. This required a littleattention, but we were able to fairly easily reduce to a situation were theseformal curves were always algebraic. In this case we first blew up points toreduce to the situation where the ideal I was locally a monomial ideal. Thenwe blew up more points to make it a principal monomial ideal. At this point,we finished the resolution process by blowing up non-singular curves in thelocus of maximal multiplicity on the strict transform of the surface X, whichamounts to dividing out powers of the terms in the principal monomial ideal.

In Section 5.2, we extended this algorithm to obtain embedded resolutionof the surface X. This required keeping track of the exceptional divisors


which occur under the resolution process, and requiring that the centers ofall monoidal transforms make SNCs with the previous exceptional divisors.We introduced the η invariant, which counts the number of components ofthe exceptional locus which have existed since the multiplicity last dropped.

In this chapter we give a proof of resolution in arbitrary dimension (overfields of characteristic zero) which incorporates all of these ideas into ageneral induction statement. The fact that we must consider some kindof covering by local hypersurfaces which are not related in any obvious wayis incorporated in the notion of General Basic Object.

In the algorithm of this chapter Tschirnhausen is replaced by a methodof Giraud for finding algebraic hypersurfaces of maximal contact for an idealJ of order b, by finding a hypersurface in 4b−1(J).

6.3. Pairs and basic objects

Definition 6.8. A pair is (W0, E0), where W0 is a non-singular varietyover a field K of characteristic zero, and E0 = D1, . . . , Dr is an orderedcollection of reduced non-singular divisors on W0 such that D1 + · · ·+Dr isa reduced simple normal crossings divisor.

Suppose that (W0, E0) is a pair, and Y0 is a non-singular closed subva-riety of W0. Y0 is a permissible center for (W0, E0) if Y0 has SNCs with E0.Let π : W1 →W be the monoidal transform of W with center Y0 and excep-tional divisor D. By abuse of notation we identify Di (1 ≤ i ≤ r) with itsstrict transform on W1 (which could become the empty set if Di is a unionof components of Y0). By a local analysis, using the regular parameters ofthe proof of Lemma 6.4, we see that D1 + · · ·+Dr +D is a SNC divisor onW1.

Now set E1 = D1, . . . , Dr, Dr+1 = D. We have thus defined a newpair (W1, E1), called the transform of (W0, E0) by the map π. The diagram

(W1, E1)→ (W0, E0)

will be called a transformation. Observe that some of the strict transformsDi may be the empty set.

Definition 6.9. Suppose that (W0, E0) is a pair with E0 = D1, . . . , Dr,and W0 is a non-singular subvariety of W0. Define scheme-theoretic intersec-tions Di = Di · W0 for all i. Suppose that Di are non-singular for 1 ≤ i ≤ rand that D1 + · · · + Dr is a SNC divisor. Set E0 = D1, . . . , Dr. Then(W0, E0) is a pair, and

(W0, E0)→ (W0, E0)

is called an immersion of pairs.

6.3. Pairs and basic objects 71

Definition 6.10. A basic object is (W0, (J0, b), E0), where (W0, E0) is a pair,J0 is an ideal sheaf of W0 which is non-zero on all connected components ofW0, and b is a natural number. The singular locus of (W0, (J0, b), E0) is

Sing(J0, b) = q ∈W0 | νq(J0) ≥ b = V (4b−1(J0)) ⊂W0.

Definition 6.11. A basic object (W0, (J0, b), E0) is a simple basic object if

νq(J) = b if q ∈ Sing(J, b).

Suppose that (W0, (J0, b), E0) is a basic object and Y0 is a non-singularclosed subvariety of W0. Y0 is a permissible center for (W0, (J0, b), E0) ifY0 is permissible for (W0, E0) and Y0 ⊂ Sing(J0, b). Suppose that Y0 is apermissible center for (W0, (J0, b), E0). Let π : W1 → W0 be the monoidaltransform of W0 with center Y0 and exceptional divisor D. Thus we have atransformation of pairs (W1, E1)→ (W0, E0). Let J1 be the weak transformof J0 on W1, so that there is a locally constant function c1 on D (constanton each connected component of D) such that the total transform of J0 is

J0OW1 = Ic1D J1.

We necessarily have c1 ≥ b, so we can define

J1 = I−bD J0OW1 = Ic1−bD J1.

We have thus defined a new basic object (W1, (J1, b), E1), called the trans-form of (W0, (J0, b), E0) by the map π. The diagram

(W1, (J1, b), E1)→ (W0, (J0, b), E0)

will be called a transformation.Observe that a transform of a basic object is a basic object, and (by

Lemma 6.4) the transform of a simple basic object is a simple basic object.Suppose that

(6.9) (Wk, (Jk, b), Ek)→ · · · → (W0, (J0, b), E0)

is a sequence of transformations.

Definition 6.12. (6.9) is a resolution of (W0, (J0, b), E0) if Sing(Jk, b) = ∅.

Definition 6.13. Suppose that (6.9) is a sequence of transformations. SetJ0 = J0, and let J i+1 be the weak transform of J i for 0 ≤ i ≤ k − 1. For0 ≤ i ≤ k define

w-ordi : Sing(Ji, b)→1bZ

by

w-ordi(q) =νq(J i)b


and defineordi : Sing(Ji, b)→

1bZ

by

ordi(q) =νq(Ji)b

.

If we assume that Yi ⊂ Maxw-ordi ⊂ Sing(Ji, b) for 1 ≤ i ≤ k in (6.9),then

max w-ord0 ≥ · · · ≥ max w-ordkby Lemma 6.4.

Definition 6.14. Suppose that (6.9) is a sequence of transformations suchthat Yi ⊂ Maxw-ordi ⊂ Sing(Ji, b) for 0 ≤ i < k. Let k0 be the smallestindex such that max w-ordk0−1 > max w-ordk0 = · · · = max w-ordk. Inparticular, k0 is defined to be 0 if max w-ord0 = · · · = max w-ordk. SetEk = E+

k ∪ E−k , where E−

k is the ordered set of divisors D in Ek which arestrict transforms of divisors of Ek0 and E+

k = Ek − E−k . Define

tk : Sing(Jk, b)→ Q× Z,where Q× Z has the lexicographic order by tk(q) = (w-ordk(q), η(q)),

η(q) =

CardD ∈ Ek | q ∈ D if w-ordk(q) < max w-ordk0 ,CardD ∈ E−

k | q ∈ D if w-ordk(q) = maxw-ordk0 .

This definition allows us to inductively define upper semi-continuousfunctions t0, t1, . . . , tk on a sequence of transformations (6.9) such that Yi ⊂Maxw-ordi ⊂ Sing(Ji, b) for 1 ≤ i ≤ k.

Lemma 6.15. Suppose that (6.9) is a sequence of transformations such thatYi ⊂ Max ti ⊂ Maxw-ordi for 0 ≤ i ≤ k. Then

max tk ≤ · · · ≤ max tk.

Lemma 6.15 follows from Lemma 6.4 and a local analysis, using theregular parameters of the proof of Lemma 6.4.

Definition 6.16. Suppose that (W0, (J0, b), E0) is a basic object, and

(6.10) (Wk, (Jk, b), Ek)→ · · · → (W0, (J0, b), E0)

is a sequence of transformations such that maxw-ordk = 0. Then we saythat (Wk, (Jk, b), Ek) is a monomial basic object (relative to (6.10)).

Suppose that (Wk, (Jk, b), Ek) is a monomial basic object (relative toa sequence (6.10)). Let E0 = D1, . . . , Dr and let Di be the exceptionaldivisor of πi : Wi → Wi−1, where i = i+ r. Then we have an expansion (ina neighborhood of Sing(Jk, b)) of the weak transform Jk of J :

Jk = Id1D1· · · IdkDk ,

6.3. Pairs and basic objects 73

where the di are functions on Wk which are zero on Wk − Di and di is alocally constant function on Di for 1 ≤ i ≤ k.

Suppose that B = (Wk, (Jk, b), Ek) is a monomial basic object (relativeto (6.10)). We define

Γ(B) : Sing(Jk, b)→ IM = Z×Q× ZN

byΓ(B)(q) = (−Γ1(q),Γ2(q),Γ3(q)),

where

Γ1(q) = min

p such that there exist i1, . . . , ip,with r < i1, . . . , ip ≤ r + k, such thatdi1(q) + · · ·+ dip(q) ≥ b for some q ∈ Di1 ∩ · · · ∩Dip

,

Γ2(q) = max

di1 (q)+···+dip (q)

b , with r < i1, . . . , ip ≤ r + k,such that p = Γ1(q),di1(q) + · · ·+ dip(q) ≥ b, q ∈ Di1 ∩ · · · ∩Dip

,

Γ3(q) = max

(i1, . . . , ip, 0, . . .) such that Γ2(q) =

di1 (q)+···+dip (q)

b ,with r < i1, . . . , ip ≤ r + k, q ∈ Di1 ∩ · · · ∩Dip

.

Observe that Γ(B) is upper semi-continuous, and MaxΓ(B) is non-singular.

Lemma 6.17. Suppose that B = (Wk, (Jk, b), Ek) is a monomial basic ob-ject (relative to a sequence (6.10)). Then Yk = Max Γ(B) is a permissiblecenter for B, and if (Wk+1, (Jk+1, b), Ek+1)→ (Wk, (Jk, b), Ek) is the trans-formation induced by the monoidal transform πk+1 : Wk+1 →Wk with centerYk, then B1 = (Wk+1, (Jk+1, b), Ek+1) is a monomial basic object and

max Γ(B) > max Γ(B1)

in the lexicographic order.

Proof. MaxΓ(B) is certainly a permissible center. We will show that

max Γ(B) > max Γ(B1).

We will first assume that max(−Γ1) < −1. Let i = i + r. There existlocally constant functions di on W such that di(q) = 0 if q 6∈ Di and di islocally constant on Di such that

Jk = Id1D1· · · IdkDk .

LetDi be the strict transform ofDi onWk+1 and letDk+1 be the exceptionaldivisor of the blow-up πk+1 : Wk+1 →Wk of Yk = Max Γ(B). Then

Jk+1 = Id1D1

· · · Idk+1

Dk+1

,


where for 1 ≤ i ≤ k we have di(q) = di(πk+1(q)) if q ∈ Di and di(q) = 0 ifq 6∈ Di. We have

dk+1(q) = di1(πk+1(q)) + · · ·+ dip(πk+1(q))− b

if q ∈ Dk+1, and dk+1(q) = 0 if q 6∈ Dk+1.For q ∈ Sing(Jk+1, b) we have Γ(B1)(q) = Γ(B)(πk+1(q)) if πk+1(q) 6∈

MaxΓ(B), so it suffices to prove that

Γ(B1)(q) < Γ(B)(πk+1(q))

if q ∈ Sing(Jk+1, b) and πk+1(q) ∈ MaxΓ(B). Suppose that

Γ(B1)(q) = (−p1, w1, (j1, . . . , jp1 , 0, . . .))

andΓ(B)(πk+1(q)) = max Γ(B) = (−p, w, (i1, . . . , ip, 0, . . .)).

We will first verify that

(6.11) p1 = Γ1(q) ≥ Γ1(πk+1(q)) = p.

If k + 1 6∈ j1, . . . , jp1, then the inequality

b ≤ dj1(q) + · · ·+ djp1 (q) ≤ dj1(πk+1(q)) + · · ·+ djp1 (πk+1(q))

implies p1 ≥ p.Suppose that k + 1 ∈ j1, . . . , jp1, so that j1 = k + 1. If p1 < p, there

exists ik ∈ i1, . . . , ip such that ik 6∈ j1, . . . , jp1. Let

d = dj1(q) + · · ·+ djp1 (q).

Thend = dj2(q) + · · ·+ djp1 (q) + di1(πk+1(q)) + · · ·+ dip(πk+1(q))− b≤ (di1(πk+1(q)) + · · ·+ dik−1

(πk+1(q)) + dik+1(πk+1(q))

+ · · ·+ dip(πk+1(q))− b)+dik(πk+1(q)) + dj2(πk+1(q)) + · · ·+ djp1 (πk+1(q))

< dik(πk+1(q)) + dj2(πk+1(q)) + · · ·+ djp1 (πk+1(q))< b,

a contradiction. Thus p1 ≥ p.Now assume that p1 = p. We will verify that w1 ≤ w. If k + 1 6∈

j1, . . . , jp1, we have

w1 =dj1 (q)+···+djp1 (q)

b

≤ dj1 (πk+1(q))+···+djp1 (πk+1(q))

b≤ w.

If k + 1 ∈ j1, . . . , jp1, then j1 = k + 1. We must have

dj2(πk+1(q)) + · · ·+ djp1 (πk+1(q)) < b,

6.4. Basic objects and hypersurfaces of maximal contact 75

so thatw1b =

(di1(πk+1(q)) + · · ·+ dip(πk+1(q))− b

)+ dj2(q) + · · ·+ djp1 (q)

≤ di1(πk+1(q)) + · · ·+ dip(πk+1(q))+

(dj2(πk+1(q)) + · · ·+ djp1 (πk+1(q))− b

)< wb.

Thus w1 < w.Assume that p = p1 and w = w1. We have shown that k + 1 6∈

j1, . . . , jp1 in this case. Since

Di1 ∩ · · · ∩Dip = ∅,we must have (j1, . . . , jp, 0, . . .) < (i1, . . . , ip, 0, . . .).

It remains to verify the lemma in the case when max(−Γ1(B)) = −1.This case is not difficult, and is left to the reader.

6.4. Basic objects and hypersurfaces of maximal contact

Suppose that B = (W, (J, b), E) is a basic object and W λλ∈Λ is a (finite)open cover of W . We then have associated basic objects

Bλ = (W λ, (Jλ, b), Eλ)

for λ ∈ Λ obtained by restriction to W λ. Observe that

Sing(Jλ, b) = Sing(J, b) ∩W λ.

If B1 = (W1, (J1, b), E1) → (W, (J, b), E) is a transformation induced by amonoidal transform π : W1 →W , then there are associated transformations

Bλ1 = (W λ

1 , (Jλ1 , b), E

λ1 )→ (W λ, (Jλ, b), Eλ),

where W λ1 = π−1(W λ) and Jλ1 = J1 | W λ

1 . In particular, W λ1 is an open

cover of W1.

Definition 6.18. Let W1 = W × A1K with projection π : W1 → W . Sup-

pose that (W, (J, b), E) is a basic object. Then there is a basic object(W1, (J1, b), E1), where J1 = JOW1 . If E = D1, . . . , Dr and D′

i = π−1(Di)for 1 ≤ i ≤ r, then E1 = D′

1, . . . , D′r. The diagram

(W1, (J1, b), E1)→ (W, (J, b), E)

is called a restriction.

Remark 6.19. Suppose that (W, (J, b), E) is a basic object and Wh ⊂W is a non-singular hypersurface. Suppose that IWh

⊂ 4b−1(J). ThenSing(J, b) ⊂ V (4b−1(J)) ⊂Wh, and νq(4b−1(J) = 1 for any q ∈ Sing(J, b).

Conversely, if (W, (J, b), E) is a simple basic object and q ∈ Sing(J, b),then νq(4b−1(J)) = 1, so there exist an affine neighborhood U of q in Wand a non-singular hypersurface Uh ⊂ U such that U ∩ Sing(J, b) ⊂ Uh.


Lemma 6.20. Suppose that

(W1, (J1, b), E1)→ (W, (J, b), E)

is a transformation of basic objects. Let D be the exceptional divisor ofW1 →W . Then for all integers i with 1 ≤ i ≤ b we have

4b−iW (J)OW1 ⊂ IiD

and1IiD4b−iW (J)OW1 ⊂ 4b−i

W1(J1).

Proof. Let Y ⊂ Sing(J, b) be the center of π : W1 →W .

The first inclusion follows since νq(4b−iW (J)) ≥ i if q ∈ Y , so that IiD

divides 4b−iW (J)OW1 .

Now we will prove the second inclusion. The second inclusion is trivial ifi = b, since JOW1 = IbDJ1 by definition. We will prove the second inclusionby descending induction on i. Suppose that the inclusion is true for somei > 1. Let q ∈ D be a closed point, p = π(q). Let K ′ = K(q). By LemmaA.16, it suffices to prove the inclusion on W ×K K ′, so we may assume thatK(q) = K. Then there exist regular parameters x0, x1, . . . , xn in R = OW,psuch that IY,p = (x0, x1, . . . , xm), with m ≤ n; x0,

x1x0, . . . , xmx0

, xm+1, . . . , xn,is a regular system of parameters in R1 = OW1,q, and ID,q = (x0). It sufficesto show that for f ∈ 4b−(i−1)

W (J)p,

(6.12)f

xi−10

∈ 4b−(i−1)W1

(J1)q.

If f ∈ 4b−iW (J)p ⊂ 4b−(i−1)

W (J)p, then the formula follows by induction, sowe may assume that f = D(g) with g ∈ 4b−i

W (J)p, D an R-derivation. SetD′ = x0D. D′ is an R1-derivation on R1 = OW1,q since D′( xix0

) ∈ OW1,q for1 ≤ i ≤ m. By induction,

g

xi0∈ 1IiD4b−iW (J)OW1,q ⊂ 4b−i

W1(J1)q ⊂ 4b−(i−1)

W1(J1)q.

ThusD′(

g

xi0) ∈ 4b−(i−1)

W1(J1)q,

D′(g

xi0) =

D(g)xi−1

0

− iD(x0)g

xi0,

which impliesf

xi−10

=D(g)xi−1

0

= D′(g

xi0) + iD(x0)

g

xi0∈ 4b−(i−1)

W1(J1)q,

and (6.12) follows.


Lemma 6.21. Suppose that (W, (J, b), E) is a simple basic object, and Wh ⊂W is a non-singular hypersurface such that IWh

⊂ 4b−1(J). Suppose that

(W1, (J1, b), E1)→ (W, (J, b), E)

is a transformation with center Y . Let (Wh)1 be the strict transform of Wh

on W1. Then Y ⊂ Wh, (Wh)1 is a non-singular hypersurface in W1, andI(Wh)1 ⊂ 4b−1(J1).

Proof. Let D be the exceptional divisor of π : W1 → W . Suppose thatq ∈ D is a closed point, and

p = π(q) ∈ Y ⊂ Sing(J, b) = V (4b−1(J)) ⊂Wh.

Recall that Y ⊂ Sing(J, b) by the definition of transformation. Let f = 0be a local equation for Wh at p, and g = 0 a local equation of D at q. Thenfg = 0 is a local equation of (Wh)1 at q, and f

g ∈ 4b−1(J1)q by Lemma 6.20.

Thus

Sing(J1, b) = V (4b−1(J1)) ⊂ V (f

g).

Definition 6.22. Suppose that B = (W, (J, b), E) is a simple basic objectand W is a non-singular closed subvariety of W such that IW ⊂ 4

b−1(J).Then the coefficient ideal of B on W is

C(J) =b−1∑i=0

4i(J)b!b−iOW .

Lemma 6.23. With the notation of Lemma 6.21 and Definition 6.22,

Sing(J, b) = Sing(C(J), b!) ⊂Wh ⊂W.

Proof. It suffices to check this at closed points q ∈Wh. By Lemma A.16 wemay assume that K(q) = K. Then there exist regular parameters x1, . . . , xnat q in W such that x1 = 0 is a local equation for Wh at q. Let R = OW,q =K[[x1, . . . , xn]], S = OWh,q = K[[x2, . . . , xn]] with natural projectionR→ S.It suffices to show that

(6.13) νR(J) ≥ b if and only if νS(4j(JS)) ≥ b− j

for j = 0, 1, . . . , b− 1. Suppose that f ∈ J ⊂ R. Write

f =∑i≥0

aixi1

with ai ∈ S for all i. We have νR(f) ≥ b if and only if νS(aj) ≥ b − j forj = 0, 1, . . . , b− 1.


We also have aj ∈ 4j(J)S, since aj is the projection in S of 1j!∂jf

∂xj1. Thus

we have the if direction of (6.13).

The ideal 4j(J) is spanned by elements of the form

ci1,...,in =∂i1+···+inf

∂xi11 · · · ∂xinn

with f ∈ J and i1 + · · ·+ in ≤ j, and the projection of ci1,...,in in S is

i1!∂i2+···+inai1∂xi22 · · · ∂x

inn

.

Thus we have the only if direction of (6.13).

Theorem 6.24. Suppose that (W, (J, b), E) is a simple basic object, andW λλ∈Λ is an open cover of W such that for each λ there is a non-singularhypersurface W λ

h ⊂ W λ such that IWλh⊂ 4b−1(Jλ) and W λ

h has simplenormal crossings with Eλ. Let Eλh be the set of non-singular reduced divisorson W λ

h obtained by intersection of E with W λh . Recall that R(1)(Z) denotes

the codimension 1 part of a closed reduced subscheme Z, as defined in (6.1).

1. If R = R(1)(Sing(J, b)) 6= ∅, then R is non-singular, open andclosed in Sing(J, b), and has simple normal crossings with E. Fur-thermore, the monoidal transform of W with center R induces atransformation

(W1, (J1, b), E1)→ (W, (J, b), E)

such that R(1)(Sing(J1, b)) = ∅.2. If R(1)(Sing(J, b)) = ∅, for each λ ∈ Λ, the basic object

(W λh , (C(Jλ), b!), Eλh)

has the following properties:a. Sing(Jλ, b) = Sing(C(Jλ), b!).b. Suppose that

(6.14) (Ws, (Js, b), Es)→ · · · → (W, (J, b), E)

is a sequence of transformations and restrictions, with inducedsequences for each λ

(W λs , (J

λs , b), E

λs )→ · · · → (W λ, (Jλ, b), Eλ).

Then for each λ there is an induced sequence of transforma-tions and restrictions

((Wh)λs , (C(Jλ)s, b!), (Eh)λs )→ (W λh , (C(Jλ), b!), Eλh)


and a diagram where the vertical arrows are closed immersionsof pairs

(6.15)(W λ

s , Eλs ) → · · · → (W λ, Eλ)

↑ ↑((Wh)λs , (Eh)

λs ) → · · · → (W λ

h , Eλh)

such that

Sing(C(Jλ)i, b!) = Sing(Jλi , b)

for all i.

Proof. We may assume that W λ = W .First suppose that T = R(1)(Sing(J, b)) 6= ∅. Then T ⊂ Sing(J, b) ⊂Wh

implies T is a union of connected components of Wh, so T is non-singular,open and closed in Sing(J, b), and has SNCs with E.

Suppose that q ∈ T . Let f = 0 be a local equation of T at q in W . Since(W, (J, b), E) is a simple basic object, Jq = f bOW,q. If π1 : W1 → W is themonoidal transform with center T , then (J1)q1 = OW,q1 for all q1 ∈W1 suchthat π1(q1) ∈ T , and necessarily, R(1)(Sing(J1, b)) = ∅.

Now suppose that R(1)(Sing(J, b)) = ∅, and (6.14) is a sequence oftransformations and restrictions of simple basic objects.

Suppose that (W1, (J1, b), E1) → (W, (J, b), E) is a transformation withcenter Y0. By Lemma 6.21 we have Y0 ⊂Wh, the strict transform (Wh)1 ofWh is a non-singular hypersurface in W1, and

I(Wh)1 ⊂ 4b−1(J1).

Since Wh has SNCs with E, Y ⊂ Wh and Y has SNCs with E, it followsthat Y has SNCs with E ∪ Wh. Thus (Wh)1 has SNCs with E1, and if(Eh)1 = E1 · (Wh)1 is the scheme-theoretic intersection, we have a closedimmersion of pairs

((Wh)1, (Eh)1)→ (W1, E1).

Since the case of a restriction is trivial, and each (Wi, (Ji, b), Ei) is a sim-ple basic object, we thus can conclude by induction that we have a diagramof closed immersions of pairs (6.15) such that

(6.16) I(Wh)j ⊂ 4b−1(Jj)

for all j.We may assume that the sequence (6.14) consists only of transforma-

tions. Let Dk be the exceptional divisor of Wk →Wk−1. It remains to showthat

(6.17) Sing(C(J)j , b!) = Sing(Jj , b)


for all j. By Lemma A.16 and Remark A.18 we may assume that K isalgebraically closed. For k > 0 and 0 ≤ i ≤ b− 1, set[

4b−i(J)]k

=1IiDk

[4b−i(J)

]k−1OWk

,

so that

C(J)k =b∑i=1

[4b−i(J)

] b!i

kO(Wh)k .

We will establish the following formulas (6.18) and (6.19) for 0 ≤ k ≤ sand 0 ≤ i ≤ b− 1:

(6.18)[4b−i(J)

]k⊂ 4b−i(Jk).

Suppose that q ∈ Sing(C(J0)k, b!) is a closed point. Set Rk = OWk,q, Sk =O(Wh)k,q. Then there are regular parameters (xk,1, . . . , xk,n, zk) in Rk suchthat I(Wh)k,q = (zk), and there are generators fk of JkRk such that

(6.19) fk =∑α

ak,αzαk

with ak,α ∈ K[[xk1, . . . , xk,n]] = Sk such that for 0 ≤ α ≤ b− 1

(ak,α)b!b−α ∈ C(J)kSk.

Assume that the formulas (6.18) and (6.19) hold. (6.18) implies C(J)k ⊂C(Jk), so by Lemma 6.23, Sing(Jk, b) ⊂ Sing(C(J)k, b!). Now (6.19) showsthat Sing(C(J)k, b!) ⊂ Sing(Jk, b), so that Sing(C(J)k, b!) = Sing(Jk, b), asdesired.

We will now verify formulas (6.18) and (6.19). (6.18) is trivial if k = 0,and (6.19) with k = 0 follows from the proof of Lemma 6.23.

We now assume that (6.18) and (6.19) are true for k = t, and provethem for k = t+ 1. Since [

4b−i(J)]t⊂ 4b−i(Jt),

we have[4b−i(J)

]t+1

=1

(IDt+1)i

[4b−i(J)

]t⊂ 1

(IDt+1)i4b−i(Jt) ⊂ 4b−i(Jt+1),

where the last inclusion follows from the second formula of Lemma 6.20. Wehave thus verified formula (6.18) for k = t+ 1.

Let qt+1 ∈ Sing(C(J0)t+1, b!) be a closed point, and qt the image of qt+1

in Sing(C(J0)t, b!).By assumption, there exist regular parameters (xt1, . . . , xtn, zt) in Rt

which satisfy (6.19). Recall that we have reduced to the assumption that

6.5. General basic objects 81

K is algebraically closed. Let Yt be the center of Wt+1 → Wt. By (6.16),there exist regular parameters (x1, . . . , xn) in St = K[[xt1, . . . , xtn]] ⊂ Rtsuch that (x1, . . . , xn, zt) are regular parameters in Rt, there exists r withr ≤ n such that x1 = · · · = xr = zt = 0 are local equations of Yt in OWt,qt ,and Rt+1 = OWt+1,qt+1 has regular parameters (xt+1,1, . . . , xt+1,n, zt+1) suchthat

x1 = xt+1,1,xi = xt+1,1xt+1,i for 2 ≤ i ≤ r,zt = xt+1,1zt+1,xi = xt+1,i for r < i ≤ n.

In particular, xt+1,1 = 0 is a local equation of the exceptional divisor Dt+1

and zt+1 = 0 is a local equation of (Wh)t+1. We have generators

ft+1 =ft

xbt+1,1

=∑α

at+1,αzαt+1

of Jt+1Rt+1 = 1xbt+1,1

JtRt+1 such that

at+1,α =at,α

xb−αt,1

if α < b, and

(at+1,α)b!b−α =

1(xt,1)b!

(at,α)b!b−α ∈ 1

(IDt+1)b!C(J)tSt+1 = C(J)t+1St+1.

6.5. General basic objects

Definition 6.25. Suppose that (W0, E0) is a pair. We define a general basicobject (GBO) (F0,W0, E0) on (W0, E0) to be a collection of pairs (Wi, Ei)with closed sets Fi ⊂Wi which have been constructed inductively to satisfythe following three properties:

1. F0 ⊂W0.

2. Suppose that Fi ⊂ Wi has been defined for the pair (Wi, Ei).If πi+1 : (Wi+1, Ei+1) → (Wi, Ei) is a restriction, then Fi+1 =π−1i+1(Fi).

3. Suppose that Fi ⊂ Wi has been defined for the pair (Wi, Ei). Ifπi+1 : (Wi+1, Ei+1) → (Wi, Ei) is a transformation centered atYi ⊂ Fi, then Fi+1 ⊂ Wi+1 is a closed set such that Fi − Yi ∼=Fi+1 − πi+1(Yi) by the map πi+1.

The closed sets Fi will be called the closed sets associated to (F0,W0, E0).


If(W1, E1)→ (W0, E0)

is a transformation with center Y0 ⊂ F0, then we can use the pairs (Wi, Ei)and closed subsets Fi ⊂ Wi in Definition 6.25 which map to (W1, E1) by asequence of restrictions and transformations satisfying 2 and 3 to define ageneral basic object (F1,W1, E1) over (W1, E1). We will say that Y0 is apermissible center for (F0,W0, E0) and call

(F1,W1, E1)→ (F0,W0, E0)

a transformation of general basic objects.If

(W1, E1)→ (W0, E0)

is a restriction, then we can use the pairs (Wi, Ei) and closed subsets Fidefined in Definition 6.25 above which map to (W1, E1) by a sequence ofrestrictions and transformations satisfying (2) and (3) to define a generalbasic object (F1,W1, E1) over (W1, E1). We will call

(F1,W1, E1)→ (F0,W0, E0)

a restriction of general basic objects.A composition of restrictions and transformations of general basic ob-

jects

(6.20) (Fr,Wr, Er)→ · · · → (F0,W0, E0)

will be called a permissible sequence for the GBO (F0,W0, E0). (6.20) willbe called a resolution if Fr = ∅.

Definition 6.26. Suppose that (W0, E0) is a pair, and (F0,W0, E0) is aGBO over (W0, E0), with associated closed sets Fi. Further, suppose thatW λ

0 λ∈Λ is a finite open covering of W0 and d ∈ N.Suppose that we have a collection of basic objects Bλ

0 λ∈Λ with

Bλ0 = (W λ

0 , (aλ0 , b

λ), Eλ0 ).

We say that Bλ0 λ∈Λ is a d-dimensional structure on the GBO (F0,W0, E0)

if for each λ ∈ Λ

1. we have an immersion of pairs

jλ0 : (W λ0 , E

λ0 )→ (W λ

0 , Eλ0 ),

where W λ0 has dimension d, and

2. if(Fr,Wr, Er)→ · · · → (F0,W0, E0)

6.6. Functions on a general basic object 83

is a permissible sequence for the GBO (F0,W0, E0) (with associatedclosed sets Fi ⊂Wi for 0 ≤ i ≤ r), then the sequence of morphismsof pairs

(W λr , E

λr )→ · · · → (W λ

0 , Eλ0 )

induces diagrams of immersions of pairs

(W λr , E

λr ) → · · · → (W λ

0 , Eλ0 )

↑ ↑(W λ

r , Eλr ) → · · · → (W λ

0 , Eλ0 )

such that

(W λr , (a

λr , b

λ), Eλr )→ · · · → (W λ0 , (a

λ0 , b

λ), Eλ0 )

is a sequence of transformations and restrictions of basic objects,and

F λi = Fi ∩W λi = Sing(aλi , b

λ)for 0 ≤ i ≤ r.

6.6. Functions on a general basic object

Proposition 6.27. Let (F0,W0, E0) be a GBO which admits a d-dimen-sional structure, and fix notation as in Definition 6.26. Let

F λ0 = Sing(aλ0 , bλ).

There is a function ordd : F0 → Q such that for all λ ∈ Λ, the composi-tion

F λ0jλ0→ F0

ordd→ Qis the function ordλ : F λ0 → Q defined by Definition 6.13 for

F λ0 = Sing(aλ0 , bλ).

Proof. Let Bλ0 λ∈Λ, where Bλ

0 = (W λ0 , (a

λ0 , b

λ), Eλ0 ), be the given d-dimen-sional structure on (F0,W0, E0). Let E0 = D1, . . . , Dr. We have closedimmersions jλ0 : W λ

0 →W λ0 for all λ ∈ Λ.

Suppose that q0 ∈ F0 is a closed point, and λ, β ∈ Λ are such that thereare qλ0 ∈ W λ

0 and qβ0 ∈ Wβ0 such that jλ0 (qλ0 ) = jβ0 (qβ0 ) = q0. We must show

thatνqλ0

(aλ0)

bλ=νqβ0

(aβ0 )

bβ.

We will prove this by expressing these numbers intrinsically, in terms of theGBO (F0,W0, E0), so that they are independent of λ. We will carry out thecalculation for Bλ

0 .

Let (F1,W1, E1)π1→ (F0,W0, E0) be the restriction whereW1 = W0×A1

K .Let q1 = (q0, 0) ∈W1. Let L1 = q0 × A1

K ; then L1 ⊂ F1 = π−11 (F0).


For any integer N > 0 we can construct a permissible sequence of pairs

(6.21) (WN , EN ) πN→ · · · π2→ (W1, E1)π1→ (W0, E0)

where for i ≥ 1, Wi+1 → Wi is the blow-up of a point qi. qi is definedas follows. Let Li be the strict transform of L1 on Wi, and let Di be theexceptional divisor of πi, where i = i+ r. Then qi = Li ∩Di.

We have q1 ∈ L1 ⊂ F1. We must have L2 ⊂ F2 by 3 of Definition 6.25.Thus π2 induces a transformation of GBOs (F2,W2, E2) → (F1,W1, E1),and q2 ∈ F2. By induction, we have for any N > 0 a permissible sequence

(6.22) (FN ,WN , EN ) πN→ · · · π2→ (F1,W1, E1)π1→ (F0,W0, E0).

(6.22) induces a permissible sequence of basic objects with qλ1 = (qλ0 , 0)and centers qλi for i ≥ 1:

(6.23) (W λN , (a

λN , b

λ), EλN ) πN→ · · · π1→ (W λ0 , (a

λ0 , b

λ), Eλ0 ).

Observe that Li ⊂ W λi for all i ≥ 1, so that qλi ∈ W λ

i for all i ≥ 1. Setb′ = νqλ0

(aλ0).

Suppose that (x1, . . . , xd) are regular parameters at qλ0 in W λ0 . Then

(x1, . . . , xd, t) are regular parameters at qλ1 in W λ1 (where A1

K = spec(K[t])),and thus there are regular parameters (x1(N), x2(N), . . . , xd(N), t) in W λ

N

at qλN which are defined by xi = xi(N)tN−1 for 1 ≤ i ≤ d. t = 0 is a localequation of the exceptional divisor Dλ

Nof W λ

N → W λN−1.

Suppose that f ∈ OWλ0 ,q

λ0

and νqλ0 (f) = r. Let K ′ = K(qλ0 ). We have anexpression

f =∑

i1+···id≥rai1,...,idx

i11 · · ·x

idd ∈ K

′[[x1, . . . , xd]] = OWλ0 ,q

λ0,

where ai1,...,id ∈ K ′. Thus in K ′[[x1(N), . . . , xd(N), t]] = OWλN ,q

λN

we havean expansion

f =∑

i1+···+id≥rai1,...,idx1(N)i1 · · ·xd(N)idt(i1+···+id)(N−1)

= t(N−1)r

∑i1+···+id=r

ai1,...,idx1(N)i1 · · ·xd(N)id + tN−1ΩN

.

Thus we have an expression f = t(N−1)rfN in OWλN ,qN

, where t - fN .

Since νqλ0 (aλ0) = b′, we have

aλ0OWλN

= I(N−1)b′

DλN

KλN ,


where IDλN

- KλN . We thus have an expression

(6.24) aλN = I(N−1)(b′−bλ)

DλN

KλN ,

and νqλN (aλN ) ≥ bλ since qλN ∈ FN ∩ W λN = Sing(aλN , b

λ).

Under the closed immersion W λN ⊂ W λ

N we have (for N ≥ 1) DλN

=DN · W λ

N , where DλN

has dimension d and is irreducible. Since F λN ⊂ W λN ,

it follows that

dim(FN ·DN ) = dim(F λN ·DN ) ≤ dim DλN

= d,

and the following three conditions are equivalent:

1. dim(FN ·DN ) = dim(F λN ·DN ) = d.

2. DλN ⊂ F λN .

3. (N − 1)(b′ − bλ) ≥ bλ.

Thus b′

bλ= 1 holds if and only if b′ − bλ = 0, which holds if and only if,

for any N , dim(FN ·DN ) < d. Thus the condition

νqλ0(aλ0)

bλ=b′

bλ= 1

is independent of λ.Now assume that b′ − bλ > 0. Then, for N sufficently large,

(N − 1)(b− bλ) ≥ bλ,

and FN ·DN = DλN

is a permissible center of dimension d for (FN ,Wn, EN ).

We now define an extension of (6.22) by a sequence of transformations(6.25)(FN+S ,WN+S , EN+S)

πN+S→ · · · πN+1→ (FN ,WN , EN ) πN→ · · · π1→ (F0,W0, E0),

where πN+i is the transformation with center YN+i = FN+i ·DN+i, and weassume that dimYN+i = d for i = 0, 1, . . . , S − 1. Observe that

(6.26) dimYN+i = d if and only if DλN+i

= DN+i · WλN+i ⊂ FN+i.

We thus have YN+i = DλN+i

. (6.26) induces an extension of (6.23) by asequence of transformations

(6.27)(W λ

N+S , (aλN+S , b

λ), EN+S)πN+S→ · · ·

πN+1→ (W λN , (a

λN , b

λ), EN ) πN→ · · · π1→ (W λ0 , (a

λ0 , b

λ), E0).

The centers of πi in (6.26) are Dλj

for j = N, . . . , N+S−1. Thus WN+i+1 →WN+i is the identity for i = 0, . . . , S − 1.


We see then that

aλN+i = I(N−1)(b′−bλ)−ibλ

DλN+i

KλN

for i = 1, . . . , S. The statement that the extension (6.27) is permissible isprecisely the statment that (N − 1)(b′ − bλ) ≥ Sbλ, which is equivalent tothe statement that dimFN+i · DN+i = d for i = 0, 1, . . . , S − 1. This laststatement is equivalent to

S ≤ `N = b(N − 1)(b′ − bλ)bλ

c,

where bxc denotes the greatest integer in x. Thus `N is determined both byN and by the GBO (F0,W0, E0). The limit

limN→∞

`NN − 1

=b′

bλ− 1

is defined in terms of the GBO, hence is independent of λ.

Theorem 6.28. Let (F0,W0, E0) be a GBO which admits a d-dimensionalstructure, and fix notation as in Definition 6.26. Let Fi be the closed setsassociated to the GBO. Let F λi = Sing(aλi , b

λ), and let w-ordλi , tλi be the

functions of Definition 6.13 defined for F λi .Consider any permissible sequence of transformations

(Fr,Wr, Er)πr→ · · · π1→ (F0,W0, E0).

1. Then for 1 ≤ i ≤ r, there are functions w-orddi : Fi → Q such thatthe composition

F λijλi→ Fi

w-orddi→ Q

is the function w-ordλi : F λi → Q.

2. Suppose that the centers Yi of πi+1 are such that Yi ⊂ Maxw-orddi .Then there are functions tdi : Fi → Q×Z such that the composition

F λijλi→ Fi

tdi→ Q× Z

is the function tλi : F λi → Q× Z.

Proof. Let E0 = D1, . . . , Ds. The functions ti are completely determinedby w-orddi , with the constraint that Yi ⊂ Maxw-orddi for all i, so we need onlyshow that w-orddi can be defined for GBOs with d-dimensional structure.

The functions w-ordλ0 are equal to ordλ0 , which extend to ord0 on F0

by Proposition 6.27. We thus have defined w-ord0 = ord0. Let Di be theexceptional divisor of πi, where i = i+ s, and let Dλ

i= W λ

i ·Di.


For each λ and i we have expressions

(6.28) aλi = Icλ1

Dλ1

· · · Icλi

Dλi

aλi ,

where aλi is the weak transform of aλ0 on W λi , and the cj are locally constant

functions on Dλj

We further have that

aλi =1

IbλDλi

aλi−1OWλi.

Suppose that ηi−1 is a generic point of an irreducible component Y ′i of

Yi such that ηi−1 is contained in W λi (and thus necessarily in W λ

i ), and thatqi ∈ Di is a point which we do not require to be closed, but is contained inDλi

= W λi ·Di, and πi(qi) is contained in the Zariski closure of ηi−1.

From (6.28), we deduce that(6.29)

(aλi )qi =

1

IbλDλi

Icλ1 (qi)

Dλ1

· · · Icλi−1(qi)

Dλi−1

I

Pi−1j=1 c

λj (ηi−1)νηi−1 (I

Dλj

)

!+νηi−1 (aλi−1)

Dλi

aλi

qi

.

Thus

(6.30)cλi (qi)bλ

=

w-ordλi−1(ηi−1) +i−1∑j=1

cλj (ηi−1)bλ

νηi−1(IDλj

)

− 1.

We further have

ordλi (qi) = w-ordλi (qi) +1bλ

i∑j=1

cλj (qj).

Suppose that q′i ∈ Fi, and define q′j for 0 ≤ j ≤ i− 1 by q′j = πj+1(q′j+1).To simplify notation, we can assume that q′j ∈ Yj for all j.

Let Y ′j be the irreducible component of Yj which contains q′j , and let η′j

be the generic point of Y ′j for 0 ≤ j ≤ i− 1.

Suppose that λ ∈ Λ is such that q′i ∈W λi . We have associated points qj

and ηj in W λj for 0 ≤ j ≤ i.

We thus have sequences of points

q0, q1, . . . , qi and η0, η1, . . . , ηi−1 with qj , ηj ∈ W λj for 0 ≤ j ≤ i

and

q′0, q′1, . . . , q′i and η′0, η′1, . . . , η′i−1 with q′j , η′j ∈W λ

j for 0 ≤ j ≤ i.


From equation (6.30) and induction we see that the values

(6.31)cλj (qj)bλ

for j = 1, . . . , i

and w-ordλi (qi) are determined by:

1. The rational numbers

(6.32) ordd0(q′0), ordd1(q

′1), . . . , orddi (q

′i),

(6.33) ordd0(η′0), ordd1(η

′1), . . . , orddi−1(η

′i−1).

2. The functions

(6.34) Y H(q′s, j) =

1 if q′s ∈ Ys and Ys ⊂ Dj locally at qs,0 if q′s 6∈ Ys or Ys 6⊂ Dj locally at qs.

Our theorem now follows from Proposition 6.27, since w-ordλi (qi) is com-pletely determined by the data (6.32), (6.33) and (6.34).

Proposition 6.29. Let B0 = (F0,W0, E0) be a GBO which admits a d-dimensional structure. Fix notation as in Definition 6.26. Let

F λi = Sing(aλi , bλ).

Consider a permissible sequence of transformations

Br = (Fr,Wr, Er)πr→ · · · π1→ (F0,W0, E0)

such that max w-ordr = 0. Then there are functions

Γr = Γr(Br) : Fr → IM = Z×Q× ZN

such that the composition

F λrjλr→ Fr

Γr→ IM

is the function

Γ = Γ(W λr , (a

λr , b

λ), Eλr ) : F λr → IM

of Definition 6.16.

Proof. Suppose that E0 = D1, . . . , Ds. The function Γ on the basicobjects (W λ

r , (aλr , b

λ), Eλr ) depends only on the values of cibλ

for 1 ≤ i ≤ r(with the notation of the proof of Theorem 6.28). Thus the conclusionsfollow from (6.31) of the proof of Theorem 6.28.

6.7. Resolution theorems for a general basic object 89

6.7. Resolution theorems for a general basic object

Theorem 6.30. Let (Fd0 ,W0, Ed0) be a GBO with associated closed sets

F di which admits a d-dimensional structure. Fix notation as in Definition6.26. Let F λi = Sing(aλi , b

λ) = F di ∩ W λi .

Consider a permissible sequence of transformations

(6.35) (Fdr ,Wr, Edr )

πr→ · · · π1→ (Fd0 ,W0, Ed0)

such that the center Yi of πi+1 satisfies

Yi ⊂ Max ti ⊂ Maxw-ordi ⊂ Fifor i = 0, 1 . . . , r − 1 and max w-ordr > 0.

This condition implies w-ordi(q) ≤ w-ordi−1(πi(q)), ti(q) ≤ ti−1(πi(q))for q ∈ Fi and 1 ≤ i ≤ r. Thus

max w-ord0 ≥ max w-ord1 ≥ · · · ≥ max w-ordr,

max t0 ≥ max t1 ≥ · · · ≥ max tr.

Let R(1)(Max tr) be the set of points where Max tr has dimension d − 1.Then R(1) is open and closed in Max tr and non-singular in Wr.

1. If R(1)(Max tr) 6= ∅, then R(1) is a permissible center for (Fdr ,Wr,Edr ) with associated transformation

(Fdr+1,Wr+1, Edr+1)

πr+1→ (Fdr ,Wr, Edr ),

and either R(1)(Max tr+1) = ∅ or max tr > max tr+1.

2. If R(1)(Max tr) = ∅, there is a GBO (Fd−1r ,Wr, E

d−1r ) with (d−1)-

dimensional structure and associated closed sets F d−1i such that

if

(6.36) (Fd−1N ,WN , E

d−1N )→ · · · → (Fd−1

r ,Wr, Ed−1r )

is a resolution (F d−1N = ∅), then (6.36) induces a permissible se-

quence of transformations

(FdN ,WN , EdN )→ · · · → (Fdr ,Wr, E

dr )

such that the center Yi of πi+1 satisfies

Yi ⊂ Max ti ⊂ Maxw-ordi ⊂ F difor r ≤ i ≤ N − 1, max tr = · · · = max tN−1, Max ti = F d−1

i fori = r, . . . , N − 1, and one of the following holds:

a. F dN = ∅.b. F dN 6= ∅ and max w-ordN = 0.c. F dN 6= ∅, max w-ordN > 0 and max tN−1 > max tN .


We will now prove Theorem 6.30. Let E0 = D1, . . . , Ds. Let Di be theexceptional divisor of πi, where i = s+ i. Set bλr = bλ(max w-ordr) > 0. Letr0 be the smallest integer such that maxw-ordr0 = maxw-ordr in (6.35).

There exist an open cover W λ0 of W0 and a d-dimensional structure

Bλ0 λ∈Λ of (F0,W0, E0), where Bλ

0 = (W λ0 , (a

λ0 , b

λ), Eλ0 ) for λ ∈ Λ. Thusfor λ ∈ Λ, we have sequences of transformations

(W λr , (a

λr , b

λ), Eλr ) πr→ · · · π1→ (W λ0 , (a

λ0 , b

λ), Eλ0 ),

where the center Y λi of W λ

i+1 → W λi is contained in Max ti. There is an

expression

aλr = Icλ1

Dλ1

· · · Icλr

Dλraλr .

We now define a basic object

(B′r)λ = (W λ

r , ((a′r)λ, (b′)λ), Eλr )

on W λr by

(6.37) (a′r)λ =

aλr if bλr ≥ bλ,(aλr )

bλ−bλr + (Icλ1

Dλ1

· · · Icλr

Dλr)bλr if bλr < bλ,

(b′)λ =bλr if bλr ≥ bλ,bλr (b

λ − bλr ) if bλr < bλ.

Lemma 6.31.

1. Sing((a′r)λ, (b′)λ) = (Maxw-ordr) ∩W λ

r .

2. (W λr , ((a

′r)λ, (b′)λ), Eλr ) is a simple basic object.

3. A transformation

(W λr+1, ((a

′r+1)

λ, (b′)λ), Eλr+1)→ (W λr , ((a

′r)λ, (b′)λ), Eλr )

with center Y λr induces a transformation

(W λr+1, (a

λr+1, b

λ), Eλr+1)→ (W λr , (a

λr , b

λ), Eλr )

such that the center Y λr ⊂ Maxw-ordr ∩W λ

r .

4. Sing((a′r+1)λ, (b′)λ) = ∅ if and only if one of the following holds:

a. maxw-ordr > max w-ordr+1, orb. max w-ordr = maxw-ordr+1 and (Maxw-ordr) ∩W λ

r = ∅.5. If max w-ordr = maxw-ordr+1 then

Sing((a′r+1)λ, (b′)λ) = Maxw-ordr+1 ∩W λ

r+1,

and (a′r+1)λ is defined in terms of aλr+1 by (6.37).


Proof. 1. For q ∈ W λr we have q ∈ Maxw-ordr if and only if νq(aλr ) ≥ bλr

and νq(aλr ) ≥ bλ. These conditions hold if and only if νq((a′r)λ) ≥ (b′)λ.

2 follows since νq(aλr ) ≤ bλr for q ∈ W λr .

3 is immediate from 1.If bλr ≥ bλ, we have

(6.38) aλr+1 = I−bλr

Dλr+1

ar = I−(b′)λ

Dλr+1

(a′r)λ = (a′r+1)

λ.

Suppose that bλr < bλ. Then

(6.39)(aλr+1)

bλ−bλr +(Ic

λ1

Dλ1

· · · Icλr+1

Dλr+1

)bλr

= (I−bλr

Dλr+1

ar)bλ−bλr +

(Ic

λ1

Dλ1

· · · Icλr+1

Dλr+1

)bλr

,

where

cλr+1(q) =r∑j=1

cj(η)νη(IDλj) + bλr − bλ

if η is the generic point of the component of Y λr containing πr+1(q).

If we set K = Icλ1

Dλ1

· · · Icλr+1

Dλr+1

(the ideal sheaf on W λr ), we see that (6.39)

is equal to

(6.40) (ar +K)I(bλ−bλr )bλrDλr+1

= (a′r+1)λ.

Now 4 and 5 follow from 1 and equations (6.38), (6.39) and (6.40).

At q ∈ Max tr ∩ W λr there exist N = η(q) hypersurfaces Dλ

i1, . . . , Dλ

iN∈

(E−r )λ such that

q ∈ Dλi1 ∩ · · · ∩D

λin ∩Maxw-ordr 6= ∅.

Thus there exists an affine neighborhood Uλ,φr of q such that

(6.41) Max tr ∩ Uλ,φr = Dλ,φi1∩ · · · ∩Dλ,φ

in∩Maxw-ordλ,φr

(The superscript λ, φ denotes restriction to Uλ,φr .) Now define a basic object

(B′′r )λ,φ = (Uλ,φr , ((a′′r )

λ,φ, (b′′)λ,φ), (E+r )λ,φ)

by

(6.42)(a′′r )

λ,φ = (a′r)λ,φ + (I

Dλ,φi1)(b

′′)λ,φ + · · ·+ (IDλ,φiN

)(b′′)λ,φ ,

(b′′)λ,φ = (b′)λ.


Lemma 6.32.

1. Sing((a′′r )λ,φ, (b′′)λ,φ) = Max tr ∩ Uλ,φr .

2. (Uλ,φr , ((a′′r )λ,φ, (b′′)λ,φ), (Er+)λ,φ) is a simple basic object.

3. A transformation

(Uλ,φr+1, ((a′′r+1)

λ,φ, (b′′)λ,φ), (E+r+1)

λ,φ)→ (Uλ,φr , ((a′′r )λ,φ, (b′′)λ,φ), (E+

r )λ,φ)

with center Y λ,φr induces a transformation

(Uλ,φr+1, (aλ,φr+1, b

λ), Eλ,φr+1)→ (Uλ,φr , (aλ,φr , bλ), Eλ,φr )

such that Y λ,φr ⊂ Max tr.

4. Sing((a′′r+1)λ,φ, (b′′)λ,φ) = ∅ if and only if one of the following con-

ditions holds:a. max tr > max tr+1, orb. max tr = max tr+1 and Max tr+1 ∩ Uλ,φr+1 = ∅.

5. If max tr = max tr+1 and Max tr+1 ∩W λr+1 6= ∅, then

Sing((a′′r+1)λ,φ, (b′′)λ,φ) = Maxw-ordr+1 ∩Uλ,φr+1

and (a′′r+1)λ,φ is defined in terms of aλr+1 and (a′r+1)

λ by (6.35) and(6.37).

Proof. 1. For q ∈ Uλ,φ1 we have q ∈ Max tr if and only if q ∈ Maxw-ordrand η(q) = N , which hold if and only if νq((a′r)

λ) ≥ (b′)λ (by 1 of Lemma6.31) and νq(Iλ,φDij ) ≥ 1 for 1 ≤ j ≤ N , which holds if and only if q ∈Sing((a′′r )

λ,φ, (b′′)λ,φ).

2 follows since νq((a′′r )λ,φ) ≤ (b′′)λ,φ for q ∈ Uλ,φr .

3 is immediate from 1.4 and 5 follow from 4 and 5 of Lemma 6.31, and the observation that if

max tr+1 = max tr and q ∈ Max tr+1 ∩ Uλ,φr+1, then q must be on the stricttransforms of the hypersurfaces Dλ,φ

i1, . . . , Dλ,φ

iN.

We now observe that the conclusions of Lemmas 6.31 and 6.32, which areformulated for transformations, can be naturally formulated for restrictionsalso. Then it follows that the (B′′

r )λ,φ define a GBO with d-dimensional

structure on (Wr, (Edr )+):

(6.43) (F ′′r ,Wr, (Edr )+),

with associated closed sets F ′′i = Max ti.

If Y is a permissible center for (6.43), then Y makes simple normalcrossings with E+

r . The local description (6.41) and (6.42) then implies that


Y makes simple normal crossings with Er. Thus Y ⊂ Max tr is a permissiblecenter for (Fdr ,Wr, E

dr ). If the induced transformations are

(F ′′r+1,Wr+1, (Edr+1)+)→ (F ′′r ,Wr, (Edr )

+)

and(Fdr+1,Wr+1, E

dr+1)→ (Fdr ,Wr, E

dr ),

then either max tr > max tr+1, in which case F ′′r+1 = ∅, or max tr = max tr+1

and F ′′r+1 = Max tr+1.

We will now verify that the assumptions of Theorem 6.24 hold for thesimple basic object (Uλ,φr , ((a′′r )

λ,φ, (b′′)λ,φ), (E+r )λ,φ).

Define (by (6.37)) (B′r0)

λ in the same way that we defined (B′r)λ. Now

5 of Lemma 6.31 implies (B′r)λ is obtained by a sequence of transformations

of the simple basic object (B′r0)

λ. (Eλr )+ is the exceptional divisor of theproduct of these transformations.

Since (B′r0)

λ is a simple basic object, we can find an open cover V λ,Ψr0

of W λr0 with non-singular hypersurfaces (Vh)

λ,Ψr0 in V λ,Ψ

r0 such that

I(Vh)λ,Ψr0

⊂ 4(b′)λ−1((a′r0)λ,Ψ).

Let (Vh)λ,Ψr be the strict transform of (Vh)

λ,ψr0 in V λ,ψ

r . By Lemma 6.21,(Vh)

λ,ψr is non-singular,

I(Vh)λ,ψr

⊂ 4(b′)λ−1((a′r)λ,ψ),

and we have that (Vh)λ,ψr makes SNCs with (E+

r )λ,ψ. Let Uλ,φ,ψr = V λ,ψr ∩

Uλ,φr . For fixed λ and φ, Uλ,φ,ψr is an open cover of Uλ,φr . For q ∈ Uλ,φ,ψr ,

(I(Vh)λ,ψr

)q ⊂(4(b′)λ−1((a′r)

λ,ψ))q⊂ 4(b′′)λ,φ−1

((a′′r )

λ,φ)q

by (6.42). Thus the assumptions of Theorem 6.24 hold for

(B′′r )λ,φ = (Uλ,φr , ((a′′r )

λ,φ, (b′′)λ,φ), (E+r )λ,ψ),

and Uλ,φ,ψr . Assertion 1 of Theorem 6.30 now follows from Theorem 6.24applied to our GBO of (6.43), with d-dimensional structure (B′′

r )λ,φ.

If R(1)(Max tr) is empty, then by Theorem 6.24, applied to the GBOof (6.43), and its d-dimensional structure, (6.43) has a (d − 1)-dimensionalstructure, and we set

(Fd−1r ,Wr, E

d−1r ) = (F ′′r ,Wr, (Edr )

+).

Given a resolution (6.36), by Lemma 6.32, we have an induced sequence

(FdN ,WN , EdN )→ · · · → (Fdr ,Wr, E

dr )

such that the conclusions of 2 of Theorem 6.30 hold.


Lemma 6.33. Let I1 and I2 be totally ordered sets. Suppose that

(6.44) (WN , EN ) πN→ · · · π1→ (W0, E0)

is a sequence of transformations of pairs together with closed sets Fi ⊂ Wi

such that the center Yi of πi+1 is contained in Fi for all i, πi+1(Fi+1) ⊂ Fifor all i, and FN = ∅. Assume that for 0 ≤ i ≤ N the following conditionshold:

1. There is an upper semi-continuous function gi : Fi → I1.2. There is an upper semi-continuous function g′i : Max gi → I2.3. Yi = Max g′i in (6.44).4. For any q ∈ Fi+1, with 0 ≤ i ≤ N − 2,

gi(πi+1(q)) ≥ gi+1(q) if πi+1(q) ∈ Yi,gi(πi+1(q)) = gi+1(q) if πi+1(q) 6∈ Yi.

This condition implies that

max g0 ≥ max g1 ≥ · · · ≥ max gN−1,

and if max gi = max gi+1, then πi+1(Max gi+1) ⊂ Max gi.5. If max gi = max gi+1 and q ∈ Max gi+1, with 0 ≤ i ≤ N − 2, then

g′i(πi+1(q)) > g′i+1(q) if πi+1(q) ∈ Yi,g′i(πi+1(q)) = g′i+1(q) if πi+1(q) 6∈ Yi.

Then there are upper semi-continuous functions

fi : Fi → I1 × I2defined by

(6.45) fi(q) =

(max gi,max g′i) if q ∈ Yi,fi+1(q) if q 6∈ Yi,

where in the second case we can view q as a point on Wi+1, as Wi+1 → Wi

is an isomorphism in a neighborhood of q. For q ∈ Fi+1,

(6.46)fi(πi+1(q)) > fi(q) if πi+1(q) ∈ Yi,fi(πi+1(q)) = fi+1(q) if πi+1(q) 6∈ Yi,max fi = (max gi,max g′i) and Max fi = Max g′i.

Proof. We first show, by induction on i, that the fi of (6.45) are well definedfunctions. Since FN = ∅, YN−1 = FN−1. By assumption 3,

Max g′N−1 = Max gN−1 = FN−1,

so we can define

fN−1(q) = (gN−1(q), g′N−1(q)) = (max gN−1,max g′N−1)

for q ∈ FN−1.


Suppose that q ∈ Fr. If q ∈ Yr, we define

fr(q) = (gr(q), g′r(q)) = (max gr,max g′r).

Suppose that q 6∈ Yr. As πi+1(Fi+1) ⊂ Fi for all i, we have an isomorphismFr − Yr ∼= Fr+1 − Dr+1, where Dr+1 is the reduced exceptional divisor ofWr+1 →Wr. Since FN = ∅, we can define fr(q) = fr+1(q) for q ∈ Fr − Yr.

The properties (6.46) are immediate from the assumptions and (6.45).It remains to prove that fr is upper semi-continuous. We prove this by

descending induction on r. Fix α = (α1, α2) ∈ I1 × I2. We must prove thatq ∈ Fr | fr(q) ≥ α is closed. If max fr < α, then the set is empty. Ifmax fr ≥ α, then

q ∈ Fr | fr(q) ≥ α = Max g′r ∪ πr+1(q′ ∈ Fr+1 | fr+1(q′) ≥ α),

which is closed by upper semi-continuity of fr+1 and properness of πr+1.

Theorem 6.34. Fix an integer d ≥ 0. There are a totally ordered set Idand functions fdi with the following properties:

1. For each GBO (Fd0 ,W0, Ed0) with a d-dimensional structure and

associated closed subsets F di there is a function

fd0 : F d0 → Id

with the property that Max fd0 is a permissible center for (W0, Ed0).

2. If a sequence of transformations with permissible centers Yi

(6.47) (Fdr ,Wr, Edr )→ · · · → (Fd0 ,W0, E

d0)

and functions fdi : F di → Id, i = 0, . . . , r−1, have been defined withthe property that Yi = Max fdi , then there is a function fdr : F dr → Idsuch that Max fdr is permissible for (Wr, E

dr ).

3. For each GBO (Fd0 ,W0, Ed0) with a d-dimensional structure, there

is an index N so that the sequence of transformations

(6.48) (FdN ,WN , EdN )→ · · · → (Fd0 ,W0, E

d0).

constructed by 1 and 2 is a resolution (FN = ∅).4. The functions fdi in 2 have the following properties:

a. If q ∈ Fi with 0 ≤ i ≤ N − 1, and if q 6∈ Yi, then fdi (q) =fdi+1(q).

b. Yi = Max fdi for 0 ≤ i ≤ N − 1, and

max fd0 > max fd1 > · · · > max fdN−1

c. For 0 ≤ i ≤ N − 1 the closed set Max fdi is smooth, equidi-mensional, and its dimension is determined by the value ofmax fdi .


Proof. We prove Theorem 6.34 by induction on d.First assume that d = 1. Set I1 to be the disjoint union I1 = Q×Zt∞,

where Q× Z is ordered lexicographically, and ∞ is the maximum of I1.Suppose that (F1

0 ,W0, E10) is a GBO with one-dimensional structure, and

with associated closed sets F 1i . Define f1

0 = t10 (t10 is defined by Theorem6.28). The closed set F 1

0 is (locally) a codimension ≥ 1 subset of a one-dimensional subvariety of W0, so dimF0 = 0. Thus R(1)(Max t10) 6= ∅, andwe are in the situation of 1 of Theorem 6.30. If we perform the permissibletransformation with center Max t10,

(F11 ,W1, E

11)→ (F1

0 ,W0, E10),

we have max t10 > max t11. We can thus define a sequence of transformations

(F1r ,Wr, E

1r )→ · · · → (F1

0 ,W0, E10),

where each transformation has center Max t1i , f1i = t1i : F 1

i → I1 and

max t10 > · · · > max t1r .

Since there is a natural number b such that max t1i ∈ 1b!Z×Z for all i, there

is an r such that the above sequence is a resolution.Now assume that d > 1 and that the conclusions of Theorem 6.34 hold

for GBOs of dimension d− 1. Thus there are a totally ordered set Id−1 andfunctions fd−1

i satisfying the conclusions of the theorem.Let I ′d be the disjoint union

I ′d = Q× Z t IM t ∞,

where IM is the ordered set of Definition 6.16. We order I ′d as follows: Q×Zhas the lexicographic order. If α ∈ Q× Z and β ∈ IM , then β < α, and ∞is the maximal element of I ′d. Define Id = I ′d × Id−1 with the lexicographicordering.

Suppose that (Fd0 ,W0, Ed0) is a GBO of dimension d, with associated

closed sets F di . Define g0 : F d0 → I ′d by

g0(q) = td0(q),

and g′0 : Max g0 → Id−1 by

g′0(q) =∞ if q ∈ R(1)(Max td0),fd−10 (q) if q 6∈ R(1)(Max td0),

where fd−10 is the function defined by the GBO of dimension d− 1

(Fd−10 ,W0, E

d−10 )

of 2 of Theorem 6.30.


Assume that we have now inductively defined a sequence of transforma-tions

(6.49) (Fdr ,Wr, Ed1) πr→ · · · π1→ (Fd0 ,W0, E

d0)

and we have defined functions

gi : F di → I ′d and g′i : Max gi → Id−1

satisfying the assumptions 1-5 of Lemma 6.33 (with I1 = I ′d and I2 = Id−1)for i = 0, . . . , r − 1. In particular, the center of πi+1 is Yi = Max g′i.

If F dr 6= ∅, define gr : F dr → I ′d by

gr(q) =

Γ(Br)(q) if w-orddr(q) = 0,tdr(q) if w-orddr(q) > 0,

where Γ(Br) is the function defined in Proposition 6.29 for the GBO Br =(Fdr ,Wr, E

dr ). Define g′r : Max gr → Id−1 by

g′r(q) =

∞ if w-orddr(q) = 0,∞ if q ∈ R(1)(Max tdr) and w-ordr(q) > 0,fd−1r (q) if q 6∈ R(1)(Max tdr) and w-ordr(q) > 0,

where fd−1r is the function defined by the GBO of dimension d− 1

(Fd−1r ,Wr, E

d−1r )

in 2 of Theorem 6.30. Observe that if max w-ordr > 0, then the center Yi ofπi+1 in (6.49) is defined by Yi = Max g′i ⊂ Max tdi for 0 ≤ i ≤ r − 1. Thus(if max w-ordr > 0) we have that (6.49) is a sequence of the form of (6.35)of Theorem 6.30.

Now Theorem 6.30, our induction assumption, and Lemma 6.17 showthat the sequence (6.49) extends uniquely to a resolution

(FN ,WN , EN )→ · · · → (F0,W0, E0)

with functions gi and g′i satisfying assumptions 1-5 of Lemma 6.33. ByLemma 6.33, the functions (gi, g′i) extend to functions

F di → Id = I ′d × Id−1

satisfying the conclusions of Theorem 6.34.Consider the values of the functions fdi which we constructed at a point

q ∈ Fi. fdi (q) has d coordinates, and has one of the following three types:

1. fdi (q) = (td(q), td−1(q), . . . , td−r(q),∞, · · · ,∞).

2. fdi (q) = (td(q), td−1(q), . . . , td−r(q),Γ(q),∞, · · · ,∞).

3. fdi (q) = (td(q), td−1(q), . . . , t1(q)).


Each coordinate is a function defined for a GBO of the correspondingdimension. We have ti(q) = (w-ordi(q), ηi(q)) with w-ordi(q) > 0.

In case 1, td−r is such that q ∈ R(1)(Max td−r); that is, Max td−r hascodimension 1 in a d− r dimensional GBO, and dim(Max fdi ) = d− r − 1.

In case 2, w-ordd−(r+1)(q) = 0 and Γ is the function defined in Proposi-tion 6.29 for a monomial GBO and dim(Max fdi ) = d− r−Γ1− 1, where Γ1

is the first coordinate of max Γ.In case 3, dim(Max fdi ) = 0.Thus Max fdi is non-singular and equidimensional, and 4 c follows.

Theorem 6.35. Suppose that d is a positive integer. Then there are atotally ordered set I ′d and functions pdi with the following properties:

1. For each pair (W0, E0) with d = dimW0 and each ideal sheaf J0 ⊂OW0 there is a function

pd0 : V (J0)→ I ′d

with the property that Max pd0 ⊂ V (J0) is permissible for (W0, E0).

2. If a sequence of transformations of pairs with centers Yi ⊂ V (J i)

(Wr, Er)→ · · · → (W0, E0)

and functions pdi : V (J i) → I ′d, for 0 ≤ i ≤ r − 1, has been definedwith the property that Yi = Max pdi and V (Jk) 6= ∅, then there isa function pdr : V (Jr) → I ′d such that Max pdr is permissible for(Wr, Er).

3. For each pair (W0, E0) and J0 ⊂ OW0 there is an index N suchthat the sequence of transformations

(WN , EN )→ · · · → (W0, E0)

constructed inductively by 1 and 2 is such that V (JN ) = ∅. Thecorresponding sequence will be called a “strong principalization” ofJ0.

4. Property 4 of Theorem 6.34 holds.

Proof. Let J0 = J0, and set b0 = max νJ0. By Theorem 6.34, there exists

a resolution

(WN1 , (JN1 , b0), EN1)→ · · · → (W0, (J0, b0), E0)

of the simple basic object (W0, (J0, b0), E0). Thus we have that JN1 (whichis the weak transform of J0 on OWN1

) satisfies b1 = max νJN1< b0. We now

can construct by Theorem 6.34 a resolution

(WN2 , (JN2 , b1), EN2)→ · · · → (WN1 , (JN1 , b1), EN1)

6.8. Resolution of singularities in characteristic zero 99

of the simple basic object (WN1 , (JN1 , b1), EN1).After constructing a finite number of resolutions of simple basic objects

in this way, we have a sequence of transformations of pairs

(6.50) (WN , EN )→ · · · → (W0, E0)

such that the strict transform JN of J0 on WN is JN = OWN.

We have upper semi-continuous functions

fdi : Max νJi → Id

satisfying the conclusions of Theorem 6.34 on the resolution sequences

(WNi+1 , (JNi+1 , bi), ENi+1)→ · · · → (WNi , (JNi , bi), ENi).

Now apply Lemma 6.33 to the sequence (6.50), with Fi = V (J i), gi =νJi , g

′i = fdi . We conclude that there exist functions pdi : V (J i)→ I ′d = Z×Id

with the desired properties.

6.8. Resolution of singularities in characteristic zero

Recall our conventions on varieties in the notation part of Section 1.1.

Theorem 6.36 (Principalization of Ideals). Suppose that I is an ideal sheafon a non-singular variety W over a field of characteristic zero. Then thereexists a sequence of monodial transforms

π : W1 →W

which is an isomorphism away from the closed locus of points where I is notlocally principal, such that IOW1 is locally principal.

Proof. We can factor I = I1I2, where I1 is an invertible sheaf and V (I2)is the set of points where I is not locally principal. Then we apply Theorem6.35 to J0 = I2.

Theorem 6.37 (Embedded Resolution of Singularities). Suppose that X isan algebraic variety over a field of characteristic zero which is embedded in anon-singular variety W . Then there exists a birational projective morphism

π : W1 →W

such that π is a sequence of monodial transforms, π is an embedded resolu-tion of X, and π is an isomorphism away from the singular locus of W .

Proof. Set X0 = X, W0 = W , J0 = IX0 ⊂ OW0 . Let

(WN , EN )→ (W0, E0)


be the strong principalization of J0 constructed in Theorem 6.35, so thatthe weak transform of J0 on WN is JN = OWN

.With the notation of the proofs of Theorem 6.34 and Theorem 6.35, ifX0

has codimension r in the d-dimensional variety W0, then for q ∈ Reg(X0),

pd0(q) = (1, (1, 0), . . . , (1, 0), 0, . . . , 0) ∈ I ′d,where there are r copies of (1, 0) followed by d − r zeros. The function pd0is thus constant on the non-empty open set Reg(X0) of non-singular pointsof X0. Let this constant be c ∈ I ′d. By property 4 of Theorem 6.34 thereexists a unique index r ≤ N − 1 such that max pdr = c. Since Wr → W0 isan isomorphism over the dense open set Reg(X0), the strict transform Xr ofX0 must be the union of the irreducible components of the closed set Max pdrof Wr. Since Max pdr is a permissible center, Xr is non-singular and makessimple normal crossings with the exceptional divisor Er.

Theorem 6.38 (Resolution of Singularities). Suppose that X is an algebraicvariety over a field of characteristic zero. Then there is a resolution ofsingularities

π : X1 → X

such that π is a projective morphism which is an isomorphism away fromthe singular locus of X.

Proof. This is immediate from Theorem 6.37, after choosing an embeddingof X into a projective space W .

Theorem 6.39 (Resolution of Indeterminacy). Suppose that K is a fieldof characteristic zero and φ : W → V is a rational map of projective K-varieties. Then there exist a projective birational morphism π : W1 → Wsuch that W1 is non-singular and a morphism λ : W1 → V such that λ =φ π. If W is non-singular, then π is a product of monoidal transforms.

Proof. By Theorem 6.38, there exists a resolution of singularities ψ : W1 →W . After replacing W with W1 and φ with ψ φ, we may assume that Wis non-singular. Let Γφ be the graph of φ, with projections π1 : Γφ → Wand π2 : Γφ → V . As π1 is birational and projective, there exists an idealsheaf I ⊂ OW such that Γφ ∼= B(I) is the blow-up of I (Theorem 4.5).By Theorem 6.36, there exists a principalization π : W1 → W of I. Theuniversal property of blowing up (Theorem 4.2) now shows that there is amorphism λ : W1 → V such that λ = φ π.

Theorem 6.40. Suppose that π : Y → X is a birational morphism ofprojective non-singular varieties over a field K of characteristic 0. Then

H i(Y,OY ) ∼= H i(X,OX) ∀ i.


Proof. By Theorem 6.39, there exists a projective morphism f : Z → Ysuch that g = π f is a product of blow-ups of non-singular subvarieties,

g : Z = Zngn→ Zn−1

gn−1→ . . .g2→ Z1

g1→ Z0 = X.

We have ([65] or Lemma 2.1 [31])

Rigj∗OZj =

0, i > 0,OZj−1 , i = 0.

Thus, by the Leray spectral sequence,

Rig∗OZ =

0, if i > 0,OX , if i = 0,

(6.51)

andg∗ : H i(X,OX) ∼= H i(Z,OZ)

for all i. Since g∗ = f∗ π∗, we conclude that π∗ is one-to-one. To show thatπ∗ is an isomorphism we now only need to show that f∗ is also one-to-one.

Theorem 6.40 also gives a projective morphism γ : W → Z such thatβ = f γ is a product of blow-ups of non-singular subvarieties, so we have

β∗ : H i(Y,OY ) ∼= H i(W,OW )

for all i. This implies that f∗ is one-to-one, and the theorem is proved.

For the following theorem, which is stronger than Theorem 6.38, we givea complete proof only the case of a hypersurface. In the embedded resolutionconstructed in Theorem 6.37, which induces the resolution of 6.38, if X isnot a hypersurface, there may be monodial transforms Wi+1 → Wi whosecenters are not contained in the strict transform Xi ofX, so that the inducedmorphism Xi+1 → Xi of strict transforms of X will in general not be amonodial transform.

Theorem 6.41 (A Stronger Theorem of Resolution of Singularities). Sup-pose that X is an algebraic variety over a field of characteristic zero. Thenthere is a resolution of singularities

π : X1 → X

such that π is a sequence of monodial transforms centered in the closed setsof points of maximum multiplicity.

Proof. In the case of a hypersurface X = X0 embedded in a non-singularvariety W = W0, the proof given for Theorem 6.37 actually produces aresolution of the kind asserted by this theorem, since the resolution sequenceis constructed by patching together resolutions of the simple basic objects(Wi, (J i, b), Ei), where J i is the weak transform of the ideal sheaf J0 of X0

in W0. Since J0 is a locally principal ideal, and each transformation in


these sequences is centered at a non-singular subvariety of MaxJ i, the weaktransform J i is actually the strict transform of J0.

For the case of a general variety X, we choose an embedding of X ina projective space W . We then modify the above proof by consideringthe Hilbert-Samuel function of X. It can be shown that there is a simplebasic object (W0, (K0, b), E0) such that Max νK0 is the maximal locus ofthe Hilbert-Samuel function. In the construction of pdi of Theorem 6.35 weuse νK0 in place of νJi . Then the resolution of this basic object induces asequence of monodial transformations of X such that the maximum of theHilbert-Samuel function has dropped.

More details about this process are given, for instance, in [55], [14] and[79].

Exercise 6.42.

1. Follow the algorithm of this chapter to construct an embeddedresolution of singularities of:

a. The singular curve y2 − x3 = 0 in A2.b. The singular surface z2 − x2y3 = 0 in A3.c. The singular curve z = 0, y2 − x3 = 0 in A3.

2. Give examples where the fdi achieve the 3 cases of the proof ofTheorem 6.34.

3. Suppose that K is a field of characteristic zero and X is an integralfinite type K-scheme. Prove that there exists a proper birationalmorphism π : Y → X such that Y is non-singular.

4. Suppose that K is a field of characteristic zero and X is a reducedfinite type K-scheme. Prove that there exists a proper birationalmorphism π : Y → X such that Y is non-singular.

5. Let notation be as in the statement of Theorem 6.35.a. Suppose that q ∈ V (J0) ⊂ W0 is a closed point. Let U =

spec(OW0,q), Di = Di∩U for 1 ≤ i ≤ r and E = D1, . . . , Dr.Consider the function pd0 of Theorem 6.35. Show that pd0(q)depends only on the local basic object (U,E) and the ideal(J0)q ⊂ OW0,q. In particular, pd0(q) is independent of the K-algebra isomorphism of OX,q which takes J0OX,q to J0OX,qand Di to Di for 1 ≤ i ≤ r.

b. Suppose that Θ : W0 → W0 is a K-automorphism such thatΘ∗(J0) = J0. Show that pd0(Θ(q)) = pd0(q) for all q ∈ V (J0).

c. Suppose that Θ : W0 → W0 is a K-automorphism, and Y ⊂W0 is a non-singular subvariety such that Θ(Y ) = Y . Letπ1 : W1 →W be the monoidal transform centered at Y . Showthat Θ extends to a K-automorphism Θ : W1 →W1 such that


Θ(π1(q)) = π1(Θ(q)) for all q ∈W1 and Θ(D) = D if D is theexceptional divisor of π1.

6. Suppose that (W0, E0) is a pair and J0 ⊂ OW0 is an ideal sheaf,with notation as in Theorem 6.35. Consider the sequence in 2 ofTheorem 6.35. Suppose that G is a group of K-automorphisms ofW0 such that Θ(Di) = Di for 1 ≤ i ≤ r and Θ∗(J0) = J0. Showthat G extends to a group of K-automorphisms of each Wi in thesequence 2 of Theorem 6.35 such that Θ(Yi) = Yi for all Θ ∈ G andeach πi is G-equivariant. That is,

πi(Θ(q)) = Θ(πi(q))

for all q ∈ Wi. Furthermore, the functions pdi of Theorem 6.35satisfy pdi (Θ(q)) = pdi (q) for all q ∈Wi.

7. (Equivariant resolution of singularities) Suppose that X is a varietyover a field K of characteristic zero. Suppose that G is a group ofK-automorphisms of X. Show that there exists an equivariantresolution of singularities π : X1 → X of X. That is, G extendsto a group of K-automorphisms of X so that π(Θ(q)) = Θ(π(q))for all q ∈ X1. More generally, deduce equivariant versions of allthe other theorems of Section 6.8, and of parts 3 and 4 of Exercise6.42).

Chapter 7

Resolution of Surfacesin PositiveCharacteristic

7.1. Resolution and some invariants

In this chapter we prove resolution of singularities for surfaces over an al-gebraically closed field of characteristic p ≥ 0. The characteristic 0 proof inSection 5.1 depended on the existence of hypersurfaces of maximal contact,which is false in positive characteristic (see Exercise 7.35 at the end of thissection). The algorithm of this chapter is sketched by Hironaka in his notes[53]. We prove the following theorem:

Theorem 7.1. Suppose that S is a surface over an algebraically closed fieldK of characteristic p ≥ 0. Then there exists a resolution of singularities

Λ : T → S.

Theorem 7.1 in the case when S is a hypersurface follows from inductionon r = maxt | Singt(S) 6= 0 in Theorems 7.6, 7.7 and 7.8. We then obtainTheorem 7.1 in general from Theorem 4.9 and Theorem 4.11.

For the remainder of this chapter we will assume that K is an alge-braically closed field of characteristic p ≥ 0, V is a non-singular 3-dimensionalvariety over K, and S is a surface in V .

It follows from Theorem A.19 that for t ∈ N, the set

Singt(S) = p ∈ S | νp(S) ≥ t

is Zariski closed.

105

106 7. Resolution of Surfaces in Positive Characteristic

Letr = maxt | Singt(S) 6= ∅.

Suppose that p ∈ Singr(S) is a closed point, f = 0 is a local equation ofS at p, and (x, y, z) are regular parameters at p in V (or in OV,p). There isan expansion

f =∑

i+j+k≥raijkx

iyjzk

with aijk ∈ K in OV,p = K[[x, y, z]]. The leading form of f is defined to be

L(x, y, z) =∑

i+j+k=r

aijkxiyjzk.

We define a new invariant, τ(p), to be the dimension of the smallest linearsubspace M of the K-subspace spanned by x, y and z in K[x, y, z] suchthat L ∈ k[M ]. This subspace is uniquely determined. This dimension isin fact independent of the choice of regular parameters (x, y, z) at p (or inOV,p). If x, y, z are regular parameters in OV,p, we will call the subvarietyN = V (M) of spec(OV,p) an “approximate manifold” to S at p. If (x, y, z)are regular parameters in OV,p, we call N = V (M) ⊂ spec(OV,p) a (formal)approximate manifold to S at p. M is dependent on our choice of regularparameters at p. Observe that

1 ≤ τ(q) ≤ 3.

If there is a non-singular curve C ⊂ Singr(S) such that q ∈ C, then τ(q) ≤ 2.

Example 7.2. Let f = x2 + y2 + z2 + x5. If char(K) 6= 2, then τ = 3 andx = y = z = 0 are local equations of the approximate manifold at the origin.

However, if char(K) = 2, then τ = 1 and x+y+z = 0 is a local equationof an approximate manifold at the origin.

Example 7.3. Let f = y2 + 2xy + x2 + z2 + z5, char(K) > 2. Then

L = (y + x)2 + z2,

so that τ = 2 and x + y = z = 0 are local equations of an approximatemanifold at the origin.

Lemma 7.4. Suppose that q ∈ S is a closed point such that νq(S) = r,τ(q) = n (with 1 ≤ n ≤ 3), and L1 = · · · = Ln = 0 are local equations at qof a (possibly formal) approximate manifold N of S at q. Let π : V1 → Vbe the blow-up of q, E = π−1(q) the exceptional divisor of π, S1 the stricttransform of S on V1, and N1 the strict transform of N on V1. Suppose thatq1 ∈ π−1(q). Then νq1(S1) ≤ νq(S), and if νq1(S1) = r, then τ(q) ≤ τ(q1)and q1 ∈ N1 ∩ E.

Moreover, N1 ∩ E = ∅ if τ(q) = 3, N1 ∩ E is a point if τ(q) = 2, andN1 ∩ E is a line if τ(q) = 1.

7.1. Resolution and some invariants 107

Proof. Let f = 0 be a local equation of S at q, and let (x, y, z) be regularparameters at p such that

1. x = y = z = 0 are local equations of N if τ(q) = 3,2. y = z = 0 are local equations of N if τ(q) = 2,3. z = 0 is a local equation of N if τ(q) = 1.

In OV,p = K[[x, y, z]], write

f =∑

aijkxiyjzk = L+G,

where L is the leading form of degree r, and G has order > r. q1 has regularparameters (x1, y1, z1) such that one of the following cases holds:

1. x = x1, y = x1(y1 + α), z = x1(z1 + β) with α, β ∈ K.2. x = x1y1, y = y1, z = y1(z1 + β) with β ∈ K.3. x = x1z1, y = y1z1, z = z1.

Suppose that the first case holds. S1 has a local equation f1 = 0 at q1such that

f1 = L(1, y1 + α, z1 + β) + x1Ω.Thus νq1(f1) ≤ r, and νq1(f1) = r implies

L(1, y1 + α, z1 + β) =∑

i+j+k=r

aijk(y1 + α)j(z1 + β)k =∑j+k=r

bjkyj1zk1

for some bjk ∈ k. Thus, the equality

L(1,y

x,z

x) =

∑j+k=r

(y

x− α)j(

z

x− β)k

implies V (y−αx, z−βx) ⊂ N . We conclude that τ(q) ≤ 2 if νq1(S1) = νq(S).Suppose that τ(q) = 2 and νq1(S1) = νq(S). Then we must have α =

β = 0, so that q1 ∈ N1. We further have

f1 = L(y1, z1) + x1Ω,

so that τ(q1) ≥ τ(q).If τ(q) = 1, then L = czr for some (non-zero) c ∈ K, and we must have

β = 0, so that q1 ∈ N1. We further have

f1 = L(z1) + x1Ω,

so that τ(q1) ≥ τ(q).There is a similar analysis in cases 2 and 3.

Lemma 7.5. Suppose that C ⊂ Singr(S) is a non-singular curve and q ∈ Cis a closed point. Let π : V1 → V be the blow-up of C, E = π−1(C) theexceptional divisor, and S1 the strict transform of S on V1. Suppose that


(x, y, z) are regular parameters in OV,q (or OV,q) such that x = y = 0 arelocal equations of C at q, and x = 0 or x = y = 0 are local equations of a(possible formal) approximate manifold N of S at q. Let N1 be the stricttransform of N on V1. Suppose that q1 ∈ π−1(q). Then νq1(S1) ≤ r, and ifνq1(S1) = r, then τ(q) ≤ τ(q1) and q1 ∈ N1 ∩ E.

Furthermore, N1 ∩ E = ∅ if τ(q) = 2, and N1 ∩ E is a curve whichmaps isomorphically onto spec(OC,q) (or spec(OC,q) in the case where N isformal) if τ(q) = 1.

We omit the proof of Lemma 7.5, as it is similar to Lemma 7.4.

Theorem 7.6. There exists a sequence of blow-ups of points in Singr(Si),

Vn → Vn−1 → · · · → V,

where Si is the strict transform of S on Vi, so that all curves in Singr(Sn)are non-singular.

Proof. This is possible by Corollary 4.4 and since (by Lemma 7.4) theblow-up of a point in Singr(Si) can introduce at most one new curve intoSingr(Si+1), which must be non-singular.

Theorem 7.7. Suppose that all curves in Singr(S) are non-singular, and

· · · → Vn → Vn−1 → · · · → V

is a sequence of blow-ups of non-singular curves in Singr(Si) where Si isthe strict transform of S on Vi. Then Singr(Si) is a union of non-singularcurves and a finite number of points for all i. Further, this sequence is finite.That is, there exists an n such that Singr(Sn) is a finite set.

Proof. If π : V1 → V is the blow-up of the non-singular curve C in Singr(S),and C1 ⊂ Singr(S1) ∩ π−1(C) is a curve, then by Lemma 7.5, C1 is non-singular and maps isomorphically onto C. Furthermore, the strict transformof a non-singular curve must be non-singular, so Singr(Si) must be a unionof non-singular curves and a finite number of points for all i.

Suppose that we can construct an infinite sequence

· · · → Vn → Vn−1 → · · · → V

by blowing up non-singular curves in Singr(Si). Then there exist curvesCi ⊂ Singr(Vi) for all i such that Ci+1 → Ci are dominant morphisms forall i, and

νOVi,Ci (ISi,Ci) = r

for all i. We have an infinite sequence

(7.1) OV,C → OV1,C1 → · · · → OVn,Cn → · · ·

7.2. τ(q) = 2 109

of local rings, where, after possibly reindexing to remove the maps whichare the identity, each homomorphism is a localization of the blow-up of themaximal ideal of a 2-dimensional regular local ring. As explained in thecharacteristic zero proof of resolution of surface singularities (Lemma 5.10),we can view the OSi,Ci as local rings of a closed point of a curve on a non-singular surface defined over a field of transcendence degree 1 over K. ByTheorem 3.15 (or Corollary 4.4),

νOVn,Cn (ISn,Cn) < r

for large n, a contradiction.

We can now state the main resolution theorem.

Theorem 7.8. Suppose that Singr(S) is a finite set. Consider a sequenceof blow-ups

(7.2) · · · → Vn → · · · → V2 → V1 → V,

where Vi+1 → Vi is the blow-up of a curve in Singr(Si) if such a curve exists,and Vi+1 → Vi is the blow-up of a point in Singr(Si) otherwise. Let Si bethe strict transform of S on Vi. Then Singr(Si) is a union of non-singularcurves and a finite number of points for all i. Further, this sequence is finite.That is, there exists Vn such that Singr(Sn) = ∅.

Theorem 7.8 is an immediate consequence of Theorem 7.9 below, andTheorems 7.6 and 7.7.

Theorem 7.9. Let the assumptions be as in the statement of Theorem 7.8,and suppose that q ∈ Singr(S) is a closed point. Then there exists an n suchthat all closed points qn ∈ Sn such that qn maps to q and νqn(Sn) = r satisfyτ(qn) > τ(q).

The proof of this theorem is very simple if τ(q) = 3. By Lemma 7.4the multiplicity must drop at all points above q in the blow-up of q. Wewill prove Theorem 7.9 in the case τ(q) = 2 in Section 7.2 and in the caseτ(q) = 1 in Section 7.3.

7.2. τ(q) = 2

In this section we prove Theorem 7.9 in the case that τ(q) = 2.Suppose that τ(q) = 2, and that there exists a sequence (7.2) of in-

finite length. We can then choose an infinite sequence of closed pointsqi ∈ Singr(Si) such that qi+1 maps to qi for all i. By Lemma 7.4, wehave τ(qi) = 2 for all i, and each qi is isolated in Singr(Si).

Thus each map Vi+1 → Vi is either the blow-up of the unique pointqi ∈ Singr(Si) which maps to q, or it is an isomorphism over q. After


reindexing, we may assume that each map is the blow-up of a point qi suchthat qi+1 maps to qi for all i. Let Ri = OVi,qi . We have a sequence

(7.3) R0 → R1 → · · · → Ri → · · ·

of infinite length, where each Ri+1 is the completion of a local ring of a closedpoint of the blow-up of the maximal ideal of spec(Ri), and νqi(Si) = r andτ(qi) = 2 for all i. We will derive a contradiction. Since τ(q) = 2, thereexist regular parameters (x, y, z) in R0 such that a local equation f = 0 ofS in R0 has the form

f =∑

i+j+k≥raijkx

iyjzk

and the leading form of f is

L(x, y) =∑i+j=r

aij0xiyj .

For an arbitrary series

g =∑

bijkxiyjzk ∈ R0,

define

γxyz(g) = min k

r − (i+ j)| bijk 6= 0 and i+ j < r ∈ 1

r!N ∪ ∞

with the convention that γxyz(g) = ∞ if and only if bijk = 0 wheneveri + j < r. We have γxyz(g) < 1 if and only if νR0(g) < r. Furthermore,γxyz(g) =∞ if and only if g ∈ (x, y)r. Set γ = γxyz(g). Define

(7.4) [g]xyz =∑

(i+j)γ+k=rγ

bijkxiyjzk.

There is an expansion

(7.5) g = [g]xyz +∑

(i+j)γ+k>rγ

bijkxiyjzk.

Returning to our series f , which is a local equation of S, set γ = γxyz(f),and

Tγ = (i, j) | k

r − (i+ j)= γ, i+ j < r, for some k such that aijk 6= 0.

Then

(7.6) [f ]xyz = L+∑

(i,j)∈Tγ

ai,j,γ(r−i−j)xiyjzγ(r−i−j).

Definition 7.10. [f ]xyz is solvable if γ ∈ N and there exists α, β ∈ K suchthat

[f ]xyz = L(x− αzγ , y − βzγ).

7.2. τ(q) = 2 111

Lemma 7.11. There exists a change of variables

x1 = x−n∑i=1

αizi, y1 = y −

n∑i=1

βizi

such that [f ]xyz is not solvable.

Remark 7.12. We can always choose (x, y, z) that are regular parametersin OV0,q0 . Then (x, y, z) are also regular parameters in OV0,q0 . However,since we are only blowing up maximal ideals in (7.2), we may work with anyformal system of regular parameters (x, y, z) in R0 (see Lemma 5.3).

Proof. We prove 7.11. If there doesn’t exist such a change of variables,then there exists an infinite sequence of changes of variables for n ∈ N

xn = x−n∑i=1

αizγi , yn −

n∑i=1

βizγi

such that γn ∈ N for all n, γn+1 = γxnynz(f), and γn+1 > γn for all n. Let

x∞ = x−∞∑i=1

αizγi , y∞ = y −

∞∑i=1

βizγi .

Thusγ∞ = γx∞,y∞,z(f) =∞

and

f ∈ (x−∞∑i=1

αizi, y −

∞∑i−1

βizi)r ⊂ R0.

But by construction of V0, p is isolated in Singr(S). If I ⊂ OV,q is the reducedideal defining Singr(S) at q, then by Remark A.21, I = IR0 is the reducedideal whose support is the locus where f has multiplicity r in spec(R0).Thus (x, y, z) = I ⊂ (x−

∑αiz

i, y −∑βiz

i), which is impossible.

We may thus assume that [f ]xyz is not solvable. Since the leading formof f is L(x, y) and τ(p) = 2, R1 has regular parameters (x1, y1, z1) definedby

(7.7) x = x1z1, y = y1z1, z = z1.

Let f1 = fzr = 0 be a local equation of the strict transform of f in R1.

Lemma 7.13.

1. γx1y1z(f1) = γxyz(f)− 1.

2. [f1]x1y1z = 1zr [f ]xyz, so [f ]xyz not solvable implies [f1]xyz not solv-

able.3. If γx1y1z(f1) > 1, then the leading form of f1 is L(x1, y1) = 1

zrL(x, y).


Proof. The lemma follows from substitution of (7.7) in (7.4), (7.5) and(7.6).

We claim that the case γx1,y1,z(f1) = 1 cannot occur (with our assump-tion that νq1(S1) = r and τ(q1) = 2). Suppose that it does. Then theleading form of f1 is [f1]x1y1z, and there exist a form Ψ of degree r anda, b, c, d, e, f ∈ K such that

[f1]x1y1z = Ψ(ax1 + by1 + cz, dx1 + ey1 + fz).

We have an expression

L(x1, y1) + zΩ = Ψ(ax1 + by1 + cz, dx1 + ey1 + fz)

so thatL(x, y) = Ψ(ax+ by, dx+ ey).

We have ae− bd 6= 0, since τ(q) = 2. By Lemma 7.14 below, [f1]x1y1z is thussolvable, a contradiction to 2 of Lemma 7.13.

Since γx1y1z(f1) < 1 implies νR1(f1) < r, it now follows from Lemma7.13 that after a finite sequence of blow-ups R0 → Ri, where Ri has regularparameters (xi, yi, zi) with

x = xizii , y = yiz

ii , z = zi.

By 3 of Lemma 7.13, we reach a reduction νqi(Si) < r or νqi(Si) = r,τ(qi) = 3, a contradiction to our assumption that (7.3) has infinite length.Thus Theorem 7.9 has been proven when τ(q) = 2.

Lemma 7.14. Let Φ = Φ(x, y), Ψ = Ψ(u, v) be forms of degree r over Kand suppose that

Φ(x, y) = Ψ(ax+ by, dx+ ey)

for some a, b, d, e ∈ K with ae − bd 6= 0. Then for all c, f ∈ K there existα, β ∈ K such that

Φ(x+ αz, y + βz) = Ψ(ax+ by + cz, dx+ ey + fz).

Proof. Indeed,

Φ(x+ αz, y + βz) = Ψ(a(x+ αz) + b(y + βz), d(x+ αz) + e(y + βz))= Ψ(ax+ by + (aα+ bβ)z, dx+ ey + (dα+ eβ)z),

so we need only solve

aα+ bβ = c, dα+ eβ = f

for α, β.

7.3. τ(q) = 1 113

7.3. τ(q) = 1

In this section we prove Theorem 7.9 in the case that τ(q) = 1.We now assume that τ(q) = 1. Suppose that there is a sequence (7.2)

which does not terminate. We can then choose a sequence of points qn ∈ Vn,with q0 = q, such that qn+1 maps to qn for all n and νqn(Sn) = r, τ(qn) = 1for all n. After possibly reindexing, we may assume that for all n, qn is onthe subvariety of Vn which is blown up under Vn+1 → Vn. Let Rn = OVn,qnfor n ≥ 0. We then have an infinite sequence

(7.8) R = R0 → R1 → · · · → Rn → · · · .

Let Ji ⊂ OVi,qi be the reduced ideal defining Singr(Si) at qi. Then byLemmas 7.4 and 7.5, Ji is either the maximal ideal mi of OVi,qi or a regularheight 2 prime ideal pi in OVi,qi . Then the multiplicity r locus of OSi,qi(which is a quotient of Ri by a principal ideal) is defined by the reduced idealJi = JiRi by Remark A.21. We have Ji = mi = miRi or Ji = pi = piRi, aregular prime in Ri. By the construction of the sequence (7.2), we see thatOVi+1,qi+1 is a local ring of the blow-up of mi if Ji = mi, and a local ring ofthe blow-up of pi otherwise. Thus Ri+1 is the completion of a local ring ofthe blow-up of mi or pi.

We are thus free to work with formal parameters and equations (whichdefine the ideal ISi,qi = ISi,qiRi) in the Ri, since the ideals mi and pi aredetermined in Ri by the multiplicy r locus of Singr(OSi,qi).

Suppose that T = K[[x, y, z]] is a power series ring, r ∈ N and

g =∑

bijkxiyjzk ∈ T.

We can construct a polygon in the following way.Define

∆ = ∆(g;x, y, z) = ( i

r − k,

j

r − k) ∈ Q2 | k < r and bijk 6= 0.

Let |∆| be the smallest convex set in R2 such that ∆ ⊂ |∆| and (a, b) ∈ |∆|implies (a + c, b + d) ∈ |∆| for all c, d ≥ 0. For a ∈ R, let S(a) be the linethrough (a, 0) with slope −1, and V (a) the vertical line through (a, 0).

Suppose that |∆| 6= ∅. We define αxyz(g) to be the smallest a appearingin any (a, b) ∈ |∆|, and βxyz(g) to be the smallest b such that

(αxyz(g), b) ∈ |∆|.

Let γxyz(g) be the smallest number γ such that S(γ)∩|∆| 6= ∅ and let δxyz(g)be such that (γxyz(g)−δxyz(g), δxyz(g)) is the lowest point on S(γxyz(g))∩|∆|.Then (αxyz(g), βxyz(g)) and (γxyz(g) − δxyz(g), δxyz(g)) are vertices of |∆|.


Define εxyz(g) to be the absolute value of the largest slope of a line through(αxyz(g), βxyz(g)) such that no points of |∆(g;x, y, z)| lie below it.

Lemma 7.15.

1. The vertices of |∆| are points of ∆, which lie on the lattice 1r!Z×

1r!Z.

2. νR(g) < r holds if and only if |∆| contains a point on S(c) withc < 1, which in turn holds if and only if there is a vertex (a, b) witha+ b < 1.

3. αxyz(g) < 1 if and only if g 6∈ (x, z)r.

4. A vertex of |∆| lies below the line b = 1 if and only if g 6∈ (y, z)r.

Proof. Lemma 7.15 follows directly from the definitions.

Suppose that g ∈ T is such that νT (g) = r. Let

L =∑

i+j+k=r

bijkxiyjzk

be the leading form of g. (x, y, z) will be called good parameters for g ifb00r = 1.

We now suppose that νT (g) = r, and (x, y, z) are good parameters forg. Let ∆ = ∆(g;x, y, z) and suppose that (a, b) is a vertex of |∆|. Let

S(a,b) = k | ( i

r − k,

j

r − k) = (a, b) and bijk 6= 0.

Definegabxyz = zr +

∑k∈S(a,b)

ba(r−k),b(r−k),kxa(r−k)yb(r−k)zk.

We will say that gabxyz is solvable (or that the vertex (a, b) is not preparedon |∆(g;x, y, z)|) if a, b are integers and

gabxyz = (z − ηxayb)r

for some η ∈ K. We will say that the vertex (a, b) is prepared on|∆(g;x, y, z)| if gabx,y,z is not solvable. If gabxyz is solvable, we can makean (a, b) preparation, which is the change of parameters

z1 = z − ηxayb.

If all vertices (a, b) of |∆| are prepared, then we say that (g;x, y, z) is wellprepared.

Lemma 7.16. Suppose that T is a power series ring over K, g ∈ T is suchthat νT (g) = r, τ(g) = 1, (x, y, z) are good parameters of g, and (g;x, y, z)is well prepared. Then z = 0 is an approximate manifold of g = 0.

7.3. τ(q) = 1 115

Proof. νT (g) = r implies all vertices (a, b) of |∆(g;x, y, z)| satisfy a+b ≥ 1.The leading form of g is

L =∑

i+j+k=r

bijkxiyjzk,

where b00r = 1. τ(g) = 1 implies there exists a linear form ax+ by+ cz suchthat

(ax+ by + cz)r = L.

As b00r = 1, we must have c = 1. If a 6= 0, then (1, 0) is a vertex of|∆(g;x, y, z)|, which is not prepared since

g(1,0)xyz = (z + ax)r.

Thus a = 0. If b 6= 0, then (0, 1) is a vertex of |∆(g;x, y, z)|, which is notprepared since

g(0,1)xyz = (z + by)r.Thus L = zr, and z = 0 is an approximate manifold of g = 0.

Lemma 7.17. Consider the terms in the expansion

(7.9) h =k∑

λ=0

ηk−λ(k

λ

)xi+(k−λ)ayj+(k−λ)bzλ1

obtained by substituting z1 = z − ηxayb into the monomial xiyjzk. Define aprojection for (a, b, c) ∈ N3 such that c < r and

π(a, b, c) = (a

r − c,

b

r − c).

Then:

1. Suppose that k < r. Then the exponents of monomials in (7.9) withnon-zero coefficients project into the line segment joining (a, b) to( ir−k ,

jr−k ).

a. If (a, b) = ( ir−k ,

jr−k ), then all these monomials project to

(a, b).b. If (a, b) 6= ( i

r−k ,j

r−k ), then xiyjzk1 is the unique monomial in(7.9) which projects onto ( i

r−k ,j

r−k ). No monomial in (7.9)projects to (a, b).

2. Suppose that r ≤ k, and (i, j, k) 6= (0, 0, r). Then all exponents in(7.9) with non-zero coefficients and z1 exponent less that r projectinto (

(a, b) + Q2≥0

)− (a, b).

3. Suppose that (i, j, k) = (0, 0, r). Then all exponents in (7.9) withnon-zero coefficients and z1 exponent less than r project to (a, b).


Proof. If k 6= r, we havei+(k−λ)ar−λ = a+ (r−k)

(r−λ)

(i

r−k − a),

j+(k−λ)br−λ = b+ (r−k)

(r−λ)

(j

r−k − b).

Suppose that k < r. Since 0 ≤ λ ≤ k, we have 0 < r−kr−λ ≤ 1. Thus(

i+ (k − λ)ar − λ

,j + (k − λ)b

r − λ

)is on the line segment joining (a, b) to ( i

r−k ,j

r−k ), and 1 follows.

If k > r and 0 ≤ λ < r ≤ k, then r−kr−λ < 0, i

r−k − a < 0 and jr−k − b < 0.

Thus 2 follows if r 6= k.Suppose that k = r and 0 ≤ λ < r. Then(

i+ (k − λ)ar − λ

,j + (k − λ)b

r − λ

)=

(i

r − λ+ a,

j

r − λ+ b

),

and the last case of 2 and 3 follow.

We deduce from this lemma that

Lemma 7.18. Suppose that z1 = z − ηxayb is an (a, b) preparation. Then:

1. |∆(g;x, y, z1)| ⊂ |∆(g;x, y, z)| − (a, b).2. If (a′, b′) is another vertex of |∆(g;x, y, z)|, then (a′, b′) is a vertex

of |∆(g;x, y, z1)| and ga′,b′x,y,z1 is obtained from ga

′,b′x,y,z by substi-

tuting z1 for z.

Example 7.19. It is not always possible to well prepare after a finite num-ber of vertex preparations. Consider over a field of characteristic 0 (or p > r)

gm = y(y − x)(y − 2x) · · · (y − rx) + (z − xm + xm+1 − · · · )r

with m ≥ 2. The vertices of |∆(g;x, y, z)| are (0, 1 + 1r ), (1, 1

r ) and (m, 0).We can remove the vertex (m, 0) by the (m, 0) preparation z1 = z − xm.Thus

g = y(y − x)(y − 2x) · · · (y − rx) + (z − x2

1+x)r

= y(y − x)(y − 2x) · · · (y − rx) + (z − x2 + x3 − · · · )r

can only be well prepared by the formal substitution

z1 = z −∞∑i=2

(−1)ixi.

Lemma 7.20. Suppose that g is reduced, νT (g) = r, τ(g) = 1 and (x, y, z)are good parameters for g. Then there is a formal series φ(x, y) ∈ K[[x, y]]such that under the substitution z = z1 + φ(x, y), (x, y, z1) are good param-eters for g and (g;x, y, z1) is well prepared.

7.3. τ(q) = 1 117

Proof. Let vL be the lowest vertex of |∆(g;x, y, z)|. Let h(vL) be the secondcoordinate of vL. Set b = h(vL). If vL is not prepared, make a vL preparationz1 = z − ηxayb (where (a, b) = vL) to remove vL in |∆(g;x, y, z1)|.

Let vL1 be the lowest vertex of |∆(g;x, y, z1)|. If vL1 is not preparedand h(vL1) = h(vL), we can again prepare vL1 by a vL1 preparation z2 =z1 − η1x

a1yb. We can iterate this procedure to either achieve |∆(g;x, y, zn)|such that the lowest vertex vLn is solvable or h(vLn) > h(vL), or we canconstruct an infinite sequence of vLn preparations (with h(vLn) = b for alln)

zn+1 = zn − ηnxanyb,

where (an, b) = vLn , such that the lowest vertex vLn of |∆(g;x, y, zn)| isnot prepared and h(vLn) = h(vL) for all n. Since an+1 > an for all n,we can then make the formal substitution z′ = z −

∑∞i=0 ηix

aiyb, to get|∆(g;x, y, z′)| whose lowest vertex vL′ satisfies h(vL′) > h(vL).

In summary, there exists a series Φ′(x) such that if we set z′ = z −yh(vL)Φ′(x), and vL′ is the lowest vertex of |∆(g;x, y, z′)|, then either vL′ isprepared or h(vL′) > h(vL).

By iterating this procedure, we construct a series Φ(x, y) such that ifz = z−Φ(x, y), then either the lowest vertex vL of |∆(g;x, y, z)| is prepared,or |∆(g;x, y, z)| = ∅. This last case only occurs if g = unit zr, and thuscannot occur since g is assumed to be reduced.

We can thus assume that vL is prepared. We can now apply the sameprocedure to vT , the highest vertex of |∆(g;x, y, z)|, to reduce to the casewhere vT and vL are both prepared. Then after a finite number of prepara-tions we find a change of variables z′ = z − Φ(x, y) such that (g;x, y, z′) iswell prepared.

We will also consider change of variables of the form

y1 = y − ηxn

for η ∈ K, n a positive integer, which we will call translations.

Lemma 7.21. Consider the expansion

(7.10) h =j∑

λ=0

ηj−λ(j

λ

)xi+(j−λ)nyλ1 z

k

obtained by substituting y1 = y − ηxn into the monomial xiyjzk. Considerthe projection for (a, b, c) ∈ N3 with c < r defined by

π(a, b, c) = (a

r − c,

b

r − c).


Suppose that k < r. Set (a, b) = ( ir−k ,

jr−k ). Then xiyj1z

k is the uniquemonomial in (7.10) whose exponents project onto (a, b). All other monomialsin (7.10) with non-zero coefficient project to points below (a, b) on the linethrough (a, b) with slope − 1

n .

Proof. The slope of the line through the points (a, b) and(i+ (j − λ)n

r − k,

λ

r − k

)is

j − λi− (i+ (j − λ)n)

= − 1n.

Since λr−k <

jr−k for λ < j, the lemma follows.

Definition 7.22. Suppose that g ∈ T is reduced, νT (g) = r, τ(g) = 1and (x, y, z) are good parameters for g. Let α = αxyz(g), β = βxyz(g),γ = γxyz(g), δ = δxyz(g), ε = εxyz(g). Then (g;x, y, z) will be said to bevery well prepared if it is well prepared and one of the following conditionsholds.

1. (γ − δ, δ) 6= (α, β), and if we make a translation y1 = y − ηx, withsubsequent well preparation z1 = z − Φ(x, y), then

αxy1z1(g) = α, βxy1z1(g) = β, γxy1z1(g) = γ

andδxy1z1(g) ≤ δ.

2. (γ − δ, δ) = (α, β), and one of the following cases holds:a. ε = 0.b. ε 6= 0 and 1

ε is not an integer.c. ε 6= 0 and n = 1

ε is a (positive) integer and for any η ∈ K, ify1 = y − ηxn is a translation, with subsequent well prepara-tion z1 = z − Φ(x, y), then εxy1z(g) = ε. Further, if (c, d) isthe lowest point on the line through (α, β) with slope −ε in|∆(g;x, y, z1)| and (c1, d1) is the lowest point on this line in|∆(g;x, y1, z1)|, then d1 ≤ d.

Lemma 7.23. Suppose that g ∈ T is reduced, νT (g) = r, τ(g) = 1 and(x, y, z) are good parameters for g. Then there are formal substitutions

z1 = z − φ(x, y), y1 = y − ψ(x),

where φ(x, y), ψ(x) are series such that (g;x, y1, z1) is very well prepared.

Proof. By Lemma 7.20, we may suppose that (g;x, y, z) is well prepared.Let α = αxyz(g), β = βxyz(g), γ = γxyz(g), δ = δxyz(g), ε = εxyz(g). Let(γ − t, t) be the highest vertex on |∆(g;x, y, z)| on the line a + b = γ. Let

7.3. τ(q) = 1 119

η ∈ K and consider the translation y1 = y − ηx. By Lemma 7.21, (γ − t, t)and (α, β) are prepared vertices of |∆(g;x, y1, z)|. Thus if z1 = z − Φ(x, y)is a well preparation of (g;x, y1, z), then (γ − t, t) and (α, β) are vertices of|∆(g;x, y1, z1)|.

We can thus choose η ∈ K so that after the translation y1 = y − ηx,and a subsequent well preparation z1 = z − Φ(x, y), we have αxy1z1 = α,βxy1z1 = β, γxy1z1 = γ and δxy1z1 ≥ δ is a maximum for substitutionsy1 = y − ηx, z1 = z − Φ(x, y) as above. If (γ − δxy1z1 , δxy1z1) 6= (α, β), thenafter replacing (x, y, z) with (x, y1, z1), we have achieved condition 1 of thelemma, so that (g;x, y, z) is very well prepared.

We now assume that (g;x, y, z) is well prepared, and (α, β) = (γ − δ, δ).If ε = 0 or 1

ε is not an integer, then (g;x, y, z) is very well prepared.

Suppose that n = 1ε is an integer. We then choose η ∈ K such that with

the translation y1 = y−ηxn, and subsequent well preparation, we maximized for points (c, d) of the line through (α, β) with slope −ε on the boundaryof |∆(f ;x, y1, z1)|. By Lemma 7.21 α, β and γ are not changed. If we nowhave d 6= β, we are very well prepared.

If this process does not end after a finite number of iterations, then thereexists an infinite sequence of changes of variables (translations followed bywell preparations)

yi = yi+1 + ηi+1xni+1 , zi = zi+1 + φi+1(x, yi+1)

such that ni+1 > ni for all i. Given n ∈ N, there exists σ(n) ∈ N suchthat σ(n) > σ(n − 1) for all n, and i ≥ σ(n) implies all vertices (a, b) of|∆(g;x, yi, zi)| below (α, β) have a ≥ n. This last condition follows since allvertices must lie in the lattice 1

r!Z ×1r!Z, and there are only finitely many

points common to this lattice and the region 0 ≤ a ≤ n, 0 ≤ b ≤ β. Thusxn | φi(x, yi) if i ≥ σ(n). Let

Ψi(x, y) =i∑

j=1

φj(x, y −j∑

k=1

ηkxnk),

so that zi = z − Ψi(x, y) for all i. Ψi(x, y) is a Cauchy sequence inK[[x, y]]. Thus z′ = z −

∑∞i=1 φi(x, yi) is a well defined series in K[[x, y, z]].

Set y′ = y −∑∞

i=1 ηixni . Then |∆(g;x, y′, z′)| has the single vertex (α, β),

and (g;x, y′, z′) is thus very well prepared.

Definition 7.24. Suppose that g ∈ T , g is reduced, νT (g) = r, τ(g) = 1,and (x, y, z) are good parameters for g. We consider 4 types of monodialtransforms T → T1, where T1 is the completion of the local ring of a monoidaltransform of T , and T1 has regular parameters (x1, y1, z1) related to theregular parameters (x, y, z) of T by one of the following rules.


T1. Singr(g) = V (x, y, z),

x = x1, y = x1(y1 + η), z = x1z1,

with η ∈ K. Then g1 = gxr1

is the strict transform of g in T1, andif νT1(g1) = r and τ(g1) = 1, then (x1, y1, z1) are good parametersfor g1.

T2. Singr(g) = V (x, y, z),

x = x1y1, y = y1, z = y1z1.

Then g1 = gyr1

is the strict transform of g in T1, and if νT1(g1) = r

and τ(g1) = 1, then (x1, y1, z1) are good parameters for g1.

T3. Singr(g) = V (x, z),

x = x1, y = y1, z = x1z1.

Then g1 = gxr1



T4. Singr(g) = V (y, z),

x = x1, y = y1, z = y1z1.

Then g1 = gyr1



Lemma 7.25. With the assumptions of Definition 7.24, suppose that g ∈ Tis reduced, νT (g) = r, τ(g) = 1, (x, y, z) are good parameters for g, (x, y, z)and (x1, y1, z1) are related by a transformation of one of the above typesT1-T4, and νT1(g) = r1, τ(g1) = 1. Then there is a 1-1 correspondence

σ : ∆(g, x, y, z)→ ∆(g1, x1, y1, z1)

defined by

1. σ(a, b) = (a+ b− 1, b) if the transformation is a T1 with η = 0,

2. σ(a, b) = (a, a+ b− 1) if the transformation is a T2,

3. σ(a, b) = (a− 1, b) if the transformation is a T3,

4. σ(a, b) = (a, b− 1) if the transformation is a T4.

The proof of Lemma 7.25 is straightforward, and is left to the reader.

Lemma 7.26. In each of the four cases of the preceeding lemma, if σ(a, b) =(a1, b1) is a vertex of |∆(g1;x1, y1, z1)|, then (a, b) is a vertex of

|∆(g;x, y, z)|,

and if (g;x, y, z) is (a, b) prepared then (g1;x1, y1, z1) is (a1, b1) prepared. Inparticular, (g1;x1, y1, z1) is well prepared if (g;x, y, z) is well prepared.

7.3. τ(q) = 1 121

Proof. In all cases, σ (when extended to R2) takes line segments to linesegments and interior points of |∆(g;x, y, z)| to interior points of

σ(|∆(g;x, y, z)|).

Thus the boundary of σ(|∆(g;x, y, z)|) is the union of the image by σ of theline segments on the boundary of σ(|∆(g;x, y, z)|). If (a1, b1) is a vertex ofσ(|∆(g;x, y, z)|), it must then necessarily be σ(a, b) with (a, b) a vertex ofσ(|∆(g;x, y, z)|).

To see that (g1;x1, y1, z1) is (a1, b1) prepared if (g;x, y, z) is (a, b) pre-pared, we observe that

g1a1,b1x1,y1,z1 =

1xr g

a,bx,y,z if we perform a T3 transformation

or a T1 transformation with η = 0,1yr g

a,bx,y,z if we perform a T2 or T4 transformation.

Lemma 7.27. Suppose that assumptions are as in Definition 7.24 and(x, y, z) and (x1, y1, z1) are related by a T3 transformation. Suppose thatνT1(g1) = r and τ(g1) = 1. If (g;x, y, z) is very well prepared, then(g1;x1, y1, z1) is very well prepared. Moreover,

βx1y1z1(g1) = βxyz(g), δx1y1z1(g1) = δxyz(g), εx1y1z1(g1) = εxyz(g)

andαx1y1z1(g1) = αxyz(g)− 1, γx1y1z1(g1) = γxyz(g)− 1.

We further have Singr(g1) ⊂ V (x1, z1).

Proof. Well preparation is preserved by Lemma 7.26. Suppose (g;x, y, z)is very well prepared and

(αxyz, βxyz) 6= (γxyz − δxyz, δxyz).

Let γ = γxyz(g), γ1 = γxyz(g1) = γ − 1. The terms in g which contribute tothe line S(γ) are

(7.11)∑

i+ j + γk = rγk < r

bijkxiyjzk.

They are transformed to

(7.12)∑

(i+ k − r) + j + γ1k = rγ1

k < r

bijkxi+k−r1 yj1z

k1 ,


which are the terms in g1 which contribute to the line S(γ1). We have

δxyz(g) = δx1,y1,z1(g1) = min j

r − k| bijk 6= 0 in (7.11).

If a translation y = y + ηx transforms (7.11) into

(7.13)∑

i+ j + γk = rγk < r

cijkxiyjzk,

then the translation y1 = y1 + ηx transforms (7.12) into

(7.14)∑

(i+ k − r) + j + γ1k = rγ1

k < r

cijkxiyj1z

k1 .

Since (7.13) and (7.14) are the terms which contribute to δxyz(g) andδx1y1z1(g1), we have δxyz(g) = δx1y1z1(g1) and (g;x, y, z) well prepared im-plies (g1;x1, y1, z1) well prepared.

We can make a similiar argument if

(αxyz, βxyz) = (γxyz − δxyz, δxyz).

The statement Singr(g1) ⊂ V (x1, z1) follows since Singr(g) = V (x, y, z),so that Singr(g1) ⊂ V (x1), the exceptional divisor of T3, and Singr(g1) ⊂V (z1) since z = 0 is an approximate manifold of g = 0 (Lemmas 7.16 and7.5).

Lemma 7.28. Suppose that assumptions are as in Definition 7.24, (x, y, z)and (x1, y1, z1) are related by a T1 transformation with η = 0, (g;x, y, z) isvery well prepared and νT1(g1) = r, τ(g1) = 1. Let (g1;x1, y

′, z′) be a verywell preparation of (g1;x1, y1, z1). Let

α = αxyz(g), β = βxyz(g), γ = γxyz(g), δ = δxyz(g), ε = εxyz(g),

α1 = αx1y′z′(g1), β1 = βx1y′z′(g1), γ1 = γx1y′z′(g1),

δ1 = δx1y′z′(g1), ε1 = εx1y′z′(g1).

Then

(β1, δ1,1ε 1, α1) < (β, δ,

1ε, α)

in the lexicographical order, with the convention that if ε = 0, then 1ε = ∞.

If β1 = β, δ1 = δ and ε 6= 0, then 1ε1

= 1ε − 1.

We further have Singr(g1) ⊂ V (x1, z′).

7.3. τ(q) = 1 123

Proof. Let σ : ∆(g;x, y, z)→ ∆(g1;x1, y1, z1) be the 1-1 correspondence ofLemma 7.25 1, which is defined by σ(a, b) = (a+b−1, b). σ transforms linesof slope m 6= −1 to lines of slope m

m+1 , and transforms lines of slope −1 tovertical lines.

First suppose that −ε ≤ −1. Then (α, β) 6= (γ−δ, δ), and we are in case1 of Definition 7.22. Then the line segment through (α, β) and (γ − δ, δ)(which has slope ≤ −1) is transformed to a segment with positive slope, ora vertical line. Thus βx1,y1,z1 = δ < β. Since (g;x1, y1, z1) is well prepared,after very well preparation we have β1 < β.

Suppose that −1 < −ε ≤ −12 . We have (α, β) = (γ − δ, δ). Then the

line segment through (α, β) and (c, d) (with the notation of Definition 7.22)is transformed to the line segment through (α+ β − 1, β) and (c+ d− 1, d),which has slope

−εx1y1z1 =−ε−ε+ 1

< −1.

Thus(αx1y1z1 , βx1y1z1) = (α+ β − 1, β)

and

(γx1y1z1(g1)− δx1y1z1(g1), δx1y1z1(g1)) 6= (αx1y1z1(g1), βx1y1z1(g1)).

Let (γx1y1z1(g1) − t, t) be the highest point on the line S(γx1y1z1(g1)) in|∆(g1;x1, y1, z1)|. By Lemma 7.26, (g1;x1, y1, z1) is well prepared. Thus (byLemma 7.21) very well preparation does not affect the vertices(αx1y1z1(g1), βx1y1z1(g1)) and (γx1y1z1(g1) − t, t) of |∆(g1;x1, y1, z1)| whichare below the horizontal line of points whose second coordinate is d. Thusβ1 = β and δ1 ≤ t ≤ d < δ.

Suppose that −12 < −ε < 0. In this case we have (α, β) = (γ−δ, δ). Then

the line segment through (α, β) and (c, d) (with the notation of Definition7.22) is transformed to the line segment through (α+β−1, β) and (c+d−1, d)which has slope

−εx1y1z1 =−ε−ε+ 1

,

so that −1 < −εx1y1z1 < 0. Thus

(αx1y1z1 , βx1y1z1) = (γx1y1z1 − δx1y1z1 , δx1y1z1) = (α+ β − 1, β),

and (c1, d) = (c + d − 1, d) is the lowest point on the line through thepoint (αx1y1z1 , βx1y1z1) with slope −εx1y1z1 which is on |∆(g1;x1, y1, z1)|.By Lemma 7.26, (g1;x1, y1, z1) is well prepared. We will now show that(g1;x1, y1, z1) is very well prepared. If 1

εx1,y1,z16∈ N, then this is immediate,

so suppose that

n =1

εx1y1z1

∈ N.


We necessarily have n > 1, and 1ε = n + 1. If we can remove the ver-

tex (c1, d) from |∆(g1;x1, y1, z1)| by a translation y1 = y1 − ηxn1 , then thetranslation y = y − ηxn+1 will remove the vertex (c, d) from |∆(g;x, y, z)|,a contradiction to the assumption that (g;x, y, z) is very well prepared.

Thus (g1;x1, y1, z1) is very well prepared, with β1 = β, δ1 = δ and1ε1

= 1ε − 1.

The final case is when ε = 0. Then β = δ, (g1;x1, y1, z1) is very wellprepared,

(α1, β1) = (γ1 − δ1, δ1) = (α+ β − 1, β)

and ε1 = 0. Thus β1 = β, δ1 = δ and 1ε1

= 1ε = ∞. Also, α1 < α, since

Singr(g) = V (x, y, z) (by assumption), so ε = 0 implies β < 1.By Lemmas 7.4 and 7.16, Singr(g1) ⊂ V (x1, z1). If Singr(g1) = V (x1, z1),

then all vertices of |∆(g1;x1, y1, z1)| have the first cordinate ≥ 1, and thusSingr(g1) = V (x1, z

′).

Lemma 7.29. Suppose that assumptions are as in Definition 7.24, (x, y, z)and (x1, y1, z1) are related by a T2 or a T4 transformation, and (g;x, y, z)is well prepared. Suppose that νT1(g1) = r and τ(g1) = 1. We have that(g1;x1, y1, z1) is well prepared and Singr(g1) ⊂ V (y1, z1). Let (g1;x1, y

′, z′)be a very well preparation. Then

βx1,y1,z1 = βx1,y′,z′(g1) < βxyz(g).

Proof. First suppose that T → T1 is a T2 transformation. The correspon-dence σ : ∆(g;x, y, z) → ∆(g1;x1, y1, z1) of Lemma 7.25 2. is defined byσ(a, b) = (a, a+ b−1). σ takes lines of slope m to lines of slope m+1. Thus

(αx1y1z1(g1), βx1y1z1(g1)) = (αxyz(g), αxyz(g) + βxyz(g)− 1).

Since Singr(g) = V (x, y, z) by assumption, we have αxyz(g) < 1, and thus

βx1y1z1(g1) < βxyz(g).

(g1;x1, y1, z1) is well prepared by Lemma 7.26. Therefore, the vertex

(αx1y1z1(g1), βx1y1z1(g1))

is not affected by very well preparation. We thus have βx1y′z′(g1) < βxyz(g).Now suppose that T → T1 is a T4 transformation. The 1-1 correspon-

dence σ : ∆(g;x, y, z) → ∆(g1;x1, y1, z1) of Lemma 7.25 4 is defined byσ(a, b) = (a, b − 1). Thus βx1y1z1(g1) < βxyz(g). Also, (g1;x1, y1, z1) is wellprepared by Lemma 7.26, and the vertex (αx1y1z1(g1), βx1y1z1(g1)) is notaffected by very well preparation. Thus βx1y′z′(g1) = βx1y1z1 < βxyz(g).

Lemma 7.30. Suppose that p is a prime, s is a non–negative integer andr0 is a positive integer such that p - r0. Let r = r0p

s. Then:

7.3. τ(q) = 1 125

1.(rλ

)≡ 0 mod p, if ps - λ, 0 ≤ λ ≤ r is an integer.

2.(rλps

)≡

(r0λ

)mod p, if 0 ≤ λ ≤ r0 is an integer.

Proof. Compare the expansions over Zp of (x+ y)r = (xps+ yp

s)r0 .

Theorem 7.31. Suppose that assumptions are as in Definition 7.24, (x, y, z)and (x1, y1, z1) are related by a T1 transformation with η 6= 0, and (g;x, y, z)is very well prepared. Suppose that νT1(g1) = r, τ(g1) = 1 and βxyz(g) > 0.Then there exist good parameters (x, y′1, z

′1) in T1 such that (g1;x1, y

′1, z

′1) is

very well prepared, with βx1,y′1,z′1(g1) < βx,y,z(g) and Singr(g1) ⊂ V (x1, z

′1).

Proof. Let α = αxyz(g), β = βxyz(g), γ = γxyz(g), δ = δxyz(g). Lemma7.16 implies z = 0 is an approximate manifold of g = 0. Thus

(7.15) α+ β > 1.

Apply the translation y′ = y − ηx and well prepare by some substitutionz′ = z − Φ(x, y′). This does not change (α, β) or γ. Set δ′ = δx,y′,z′(g).

First assume that δ′ < β. We have regular parameters (x1, y1, z1) in T1

such that x = x1, y′ = x1y1, z

′ = x1z1 by Lemma 7.16, so we have a 1-1correspondence (by Lemma 7.25 1)

σ : ∆(g;x, y′, z′)→ ∆(g1;x1, y1, z1)

defined by σ(a, b) = (a+b−1, b). (g1;x1, y1, z1) is well prepared (by Lemma7.26). We are essentially in case 1 (−ε ≤ −1) of the proof of Lemma 7.28(this case only requires well preparation), so that (α1, β1) = σ(γ − δ′, δ′)and β1 = δ′ < β. After very well preparation, we thus have regular pa-rameters x1, y

′1, z

′1 in T1 such that (g1;x1, y

′1, z

′1) is very well prepared and

βx1,y′1,z′1(g1) < βxyz(g).

We have thus reduced the proof to showing that δ′ < β. If (α, β) 6=(γ − δ, δ), we have δ′ ≤ δ < β since (g;x, y, z) is very well prepared.

Let g =∑aijkx

iyjzk. Suppose that (α, β) = (γ − δ, δ). Set

W = (i, j, k) ∈ N3 | k < r and (i

r − k,

j

r − k) = (α, β),

F =∑

(i,j,z)∈W

aijkxiyjzk.

By assumption, zr + F is not solvable. Moreover,

(7.16) F (x, y, z) = F (x, y′ + ηx, z) =∑

(i,j,k)∈W

j∑λ=0

aijkηλ

(j

λ

)xi+λ(y′)j−λzk.


By Lemma 7.21, the terms in the expansion of g(x, y′ + ηx, z) contributingto (γ, 0) in |∆(g;x, y′, z)|, where γ = α+ β, are

F(γ,0) =∑

(i,j,k)∈W

aijkηjxi+jzk 6= 0.

If (g;x, y′, z) is (γ, 0) prepared, then δ′ = 0 < β. Suppose that

gγ0x,y′,z = zr + F(γ,0)

is solvable. Then γ ∈ N, and there exists ψ ∈ K such that

(7.17) (z − ψxγ)r = zr + F(γ,0),

so that, with ω = −ψηβ∈ K, for 0 ≤ k < r, we have:

1. If(r

r−k)6= 0 (in K), then i = α(r − k), j = β(r − k) ∈ N and

(7.18) aijk =(

r

r − k

)ωr−k.

2. If i = α(r − k), j = β(r − k) ∈ N and(r

r−k)

= 0 (in K), thenaijk = 0.

Thus (by Lemma 7.30) aijk = 0 if ps - k.Suppose that K has characteristic 0. By Remark 7.32, we obtain a

contradiction to our assumption that g(γ,0)xy′z is solvable. Thus 0 = δ′ < β

if K has characteristic 0.Now we consider the case where K has characteristic p > 0, and r = psr0

with p - r0, r0 ≥ 1. By (7.18) and Lemma 7.30, we have i = αps, j = βps ∈ Nand ai,j,(r0−1)ps 6= 0.

We have an expression βps = ept, where p - e. Suppose that t ≥ s. Thenβ ∈ N, which implies α = γ − β ∈ N, so that

(z + ωxαyβ)r = zr + F,

a contradiction, since zr + F is by assumption not solvable. Thus t < s.Suppose that e = 1. Then β = pt−s < 1 and α < 1 (since we must haveSingr(g) = V (x, y, z)) imply γ = α+ β = 1, a contradiction to (7.15). Thuse > 1. Also,

zr + F = (zps+ ωp

sxαp

syβp

s)r0

=(zp

s+ ωp

sxαp

s(y′ + ηx)βp

s)r0=

(zp

s+ ωp

sxαp

s[(y′)p

t+ ηp

txp

t]e)r0

.

7.3. τ(q) = 1 127

Now make the (γ, 0) preparation z = z′ − ηβωxγ (from (7.17)) so that(g;x, y′, z′) is (γ, 0) prepared. Let G = zr + F . Then

G =((z′)p

s+ eωp

sηp

t(e−1)(y′)ptxαp

s+pt(e−1) + (y′)2ptΩ)

)r0= (z′)p

sr0 + r0

[eωp

sηp

t(e−1)(y′)ptxαp

s+pt(e−1) + (y′)2ptΩ

](z′)p

s(r0−1)

+Λ2(x, y′)(z′)ps(r0−2) + · · ·+ Λr0(x, y

′)

for some polynomials Ω(x, y′),Λ2, . . . ,Λr0 , where (y′)ipt | Λi for all i. Since

all contributions to S(γ) ∩ |∆(g;x, y′, z′)| must come from this polynomial(recall that we are assuming (α, β) = (γ − δ, δ)), the term of lowest secondcoordinate on S(γ) ∩ |∆(g;x, y′, z′)| is

(a, b) =(αps + pt(e− 1)

ps,pt

ps

),

which is not in N2 since t < s, and is not (α, β) since e > 1. Thus (g;x, y′, z′)is not (a, b) solvable, and

δ′ =pt

ps<ept

ps= β.

Remark 7.32. In Theorem 7.31 suppose that K has characteristic 0. From(7.18) we see that if gγ0xy′z′ is solvable, then for 0 ≤ k ≤ r − 1 there existi, j ∈ N with i = α(r− k), j = β(r− k) so that α, β ∈ N. Comparing (7.18)with the binomial expansion

(z + ωxαyβ)r =r∑i=0

(r

r − i

)(ωxαyβ)r−izi = zr + F,

we see that zr+F is solvable, a contradiction. Thus (γ, 0) is not solvable on|∆(g;x, y′, z)| if K has characteristic 0, and after well preparation, (γ, 0) isa vertex of |(g;x, y′, z′)|.

Example 7.33. With the notation of Theorem 7.31, it is possible to have(γ, 0) solvable in |∆(g;x, y′, z)| if K has positive characteristic, as is shownby the following simple example. Suppose that the characteristic p of K isgreater than 5.

Let g = zp + xyp2−1 + xp

2−1y2.The origin is isolated in Singp(g = 0), and (g;x, y, z) is very well pre-

pared. The vertices of |∆(g;x, y, z)| are (α, β) = (γ − δ, δ) = (1p , p−

1p) and


(p− 1

p ,2p

). Set y = y′ + ηx with 0 6= η ∈ K. Then

g = zp + x(y′ + ηx)p2−1 + xp

2−1(y′ + ηx)2

= zp +(∑p2−1

i=0

(p2−1i

)ηp

2−1−ixp2−i(y′)i

)+η2xp

2+1 + 2ηy′xp2+ xp

2−1(y′)2

The vertices of |∆(g;x, y′, z)| are (α, β) =(

1p , p−

1p

)and (γ, 0) = (p, 0).

However, (γ, 0) is solvable. Set z′ = z + ηp− 1

pxp. Then

g = (z′)p+

p2−1∑i=1

(p2 − 1i

)ηp

2−1−ixp2−i(y′)i

+η2xp2+1+2ηy′xp

2+xp

2−1(y′)2.

Since (p2 − 1

1

)≡ −1 mod p,

it follows that δ1 = δxy′z′(g) = 1p , and the vertices of |∆(g;x, y′, z′)| are

(α, β) = (1p , p−

1p), (γ1 − δ1, δ1) = (p− 1

p ,1p) and (p+ 1

p , 0).

Theorem 7.34. For all n ∈ N, there are fn ∈ Rn such that fn is a generatorof ISn,qnRn, and good parameters (xn, yn, zn) for fn in Rn such that if

βn = βxnynzn(fn), εn = εxnynzn(fn),

αn = αxnynzn(fn), δn = δxnynzn(fn),

σ(n) = (βn, δn,1εn, αn),

then σ(n + 1) < σ(n) for all n in the lexicographical order in 1r!N ×

1r!N ×

(Q+ ∪ ∞)× 1r!N. If βn+1 = βn, δn+1 = δn and 1

εn6=∞, then

1εn+1

=1εn− 1.

As an immediate corollary, we find a contradiction to the assumptionthat (7.8) has infinite length. We have now verified Theorem 7.9 for thefinal case τ(q) = 1.

Proof. We inductively construct regular parameters (xn, yn, zn) in Rn, agenerator fn of ISn,qnRn such that (xn, yn, zn) are good parameters for fn,Singr(fn) ⊂ V (xn, zn) or Singr(fn) ⊂ V (yn, zn) and (fn;xn, yn, zn) is wellprepared. If Singr(fn) ⊂ V (xn, zn), we will further have that (fn;xn, yn, zn)is very well prepared.

Let f be a generator of IS,qR0. We first choose (possibly formal) regularparameters (x, y, z) in R which are good for f = 0, such that |∆(f ;x, y, z)|

7.3. τ(q) = 1 129

is very well prepared. By Lemma 7.16, z = 0 is an approximate manifoldfor f = 0. Let α = αx,y,z(f), β = βx,y,z(f).

By assumption, q is isolated in Singr(f).Suppose that fi and regular parameters (xi, yi, zi) in Ri have been de-

fined for i ≤ n as specified above.If Singr(fn) = V (yn, zn), then Rn → Rn+1 must be a T4 transformation,

xn = x′n+1, yn = y′n+1, zn = y′n+1z′n+1.

Thus (fn+1;x′n+1, y′n+1, z

′n+1) is prepared by Lemma 7.29. Further,

Singr(fn+1) ⊂ V (y′n+1, z′n+1).

If Singr(fn+1) = V (xn+1, y′n+1, z

′n+1), make a change of variables to very

well prepare. Otherwise, set yn+1 = y′n+1, zn+1 = z′n+1. We have βn+1 < βnby Lemma 7.29.

If Singr(fn) = V (xn, zn), then Rn → Rn+1 must be a T3 transformation

xn = xn+1, yn = yn+1, zn = xn+1zn+1,

and (fn;xn, yn, zn) is very well prepared. Thus (fn+1;xn+1, yn+1, zn+1) isalso very well prepared. Moreover,

(βn+1, δn+1,1

εn+1, αn+1) < (βn, δn,

1εn, αn)

by Lemma 7.27. Further, Singr(fn+1) ⊂ V (xn+1, zn+1).Now suppose that Singr(fn) = V (xn, yn, zn). Then (f ;xn, yn, zn) is very

well prepared, and Rn+1 must be a T1 or T2 transformation of Rn.If Rn+1 is a T2 transformation of Rn,

xn = xn+1y′n+1, yn = y′n+1, zn = yn+1z

′n+1,

then (fn+1;xn+1, y′n+1, z

′n+1) is well prepared with βn+1 < βn by Lemma

7.29. Further, Singr(fn+1) ⊂ V (y′n+1, z′n+1). If

Singr(fn+1) = V (xn+1, y′n+1, z

′n+1),

then make a further change of variables to very well prepare. Otherwise, setyn+1 = y′n+1, zn+1 = z′n+1.

If Rn+1 is a T1 transformation of Rn,

xn = xn+1, yn = xn+1(y′n+1 + η), zn = xn+1z′n+1,

then Singr(fn+1) ⊂ V (xn+1, z′n+1) by Lemma 7.5.

If βn 6= 0 and η 6= 0 in the T1 transformation relating Rn and Rn+1,we can change variables to yn+1, zn+1 to very well prepare, with βn+1 < βnand Singr(fn+1) ⊂ V (xn+1, zn+1) by Theorem 7.31.


If η = 0 in the T1 transformation of Rn, and (xn+1, yn+1, zn+1) areregular parameters of Rn+1 obtained from (xn+1, y

′n+1, z

′n+1) by very well

preparation, then (fn+1;xn+1, yn+1, zn+1) is very well prepared,

(βn+1, δn+1,1

εn+1, αn+1) < (βn, δn,

1εn, αn),

and Singr(fn+1) ⊂ V (xn+1, zn+1) by Lemma 7.28.If βn = 0, we can make a translation y′n = yn + ηxn, and have that

(g′nxn, y′n, zn) is very well prepared, and βn, γn, δn, εn, αn are unchanged.

Thus we are in the case of η = 0.

7.4. Remarks and further discussion

The first proof of resolution of singularities of surfaces in positive character-istic was given by Abhyankar ([1], [3]). The proof used valuation-theoreticmethods to prove local uniformization (defined in Section 2.5).

The most general proof of resolution of surfaces, which includes excellentsurfaces, was given by Lipman [63]. This lucid article gives a self-containedgeometric proof.

This proof of resolution of surfaces which are hypersurfaces over an al-gebraically closed field (Theorem 7.1) extends to give a proof of resolutionof excellent surfaces. Some general remarks about the proof are given in thefinal 3 pages of Hironaka’s notes [53]. The first part of our proof of Theorem7.1 generalizes to the case of an excellent surface S, and we reduce to provinga generalization of Theorem 7.9, when V = spec(R), where R is a regularlocal ring with maximal ideal m and S = spec(R/I) is a two-dimensionalequidimensional local ring with Singr(R/I) = V (m).

There are two main new points which must be considered. The firstpoint is to find a generalization of the polygon |∆(f ;x, y, z)| to the casewhere OS,q ∼= R/I is as above. This is accomplished by the general theoryof characteristic polyhedra [54]. The second point is to extend the theoryto the case of a non-closed residue field. This is accomplished in [23].

We will now restrict to the case when our surface S is a hypersurface overan arbitrary field K. The proof of Theorem 7.9 in the case when τ(q) = 3is the same, and if τ(q) = 2, the proof is really the same, since we mustthen have equality of residue fields K(qi) = K(qi+1) for all i in the sequence(7.3), under the assumption that τ(qi) = 2, νqi(Si) = r for all i. This isthe analogue of our proof of resolution of curves embedded in a non-singularsurface, over an arbitrary field.

In the case when τ(q) = 1, some extra care is required. It is possible tohave non-trivial extensions of residue fields k(qi)→ k(qi+1) in the sequence

7.4. Remarks and further discussion 131

(7.8). However, if k(qi+1) 6= k(qi), we must have βi+1 < βi (This is provenin the theorem on page 26 of [23]).

Thus (βi, δi, 1εi, αi) drops after every transformation in (7.8), and we see

that we must eventually have ν drop or τ increase.The essential difficulty in resolution in positive characteristic, as opposed

to characteristic zero, is the lack of existence of a hypersurface of maximalcontact (see Exercise 7.35 at the end of this section).

Abhyankar is able to finesse this difficulty to prove resolution for 3-dimensional varieties X, over an algebraically closed field K of characteristicp > 5 (13.1, [4]). We reduce to proving local uniformization by a patchingargument (9.1.7, [4]). The starting point of the proof of local uniformizationis to use a projection argument (12.4.3, 12.4.4 [4]) to reduce the problem tothe case of a point on a normal variety of multiplicity ≤ 3! = 6. A simpleexposition of this projection method is given on page 200 of [62]. We willsketch the proof of resolution, in the case of a hypersurface singularity ofmultiplicity ≤ 6. We then have that a local equation of this singularity is

(7.19) wr + ar−1(x, y, z)wr−1 + · · ·+ a0(x, y, z) = 0.

Since r ≤ 6 < char(K), it follows that 1r ∈ K, and we can perform a

Tschirnhausen transformation to reduce to the case

wr + ar−2(x, y, z)wr−2 + · · ·+ a0(x, y, z) = 0.

Now w = 0 is a hypersurface of maximal contact, and we have a good the-orem for embedded resolution of 2-dimensional hypersurfaces, so we reduceto the case where each ai is a monomial in x, y, z times a unit. Now we havereduced to a combinatorial problem, which can be solved in a characteristicfree way.

A resolution in the case where (7.19) has degree p = charK is found byCossart [24]. The proof is extraordinarily difficult.

Suppose that X is a variety over a field K. In [36], de Jong has shownthat there is a dominant proper morphism of K-varieties X ′ → X such thatdimX ′ = dimX and X ′ is non-singular. If K is perfect, then the finiteextension of function fields K(X) → K(X ′) is separable. This is weakerthan a resolution of singularities, since the map is in general not birational(the extension of function fields K(X ′)→ K(X) is finite). This proof relieson sophisticated methods in the theory of moduli spaces of curves.

Exercise 7.35.

1. (Narasimhan [69]) Let K be an algebraically closed field of char-acterisitc 2, and let X be the 3-fold in A4

K with equation

f = w2 + xy3 + yz3 + zx7 = 0.


a. Show that the maximal multiplicity of a singular point on Xis 2, and the locus of singular points on X is the monomialcurve C with local equations

y3 + zx6 = 0, xy2 + z3 = 0,

yz2 + x7 = 0, w2 + zx7 = 0,which has the parameterization t → (t7, t19, t15, t32). Thus Chas embedding dimension 4 at the origin, so there cannot exista hypersurface (or formal hypersurface) of maximal contact forX at the origin.

b. Resolve the singularities of X.2. (Hauser [50]) Let K be an algebraically closed field of characteristic

2, and let S be the surface in A2K with equation f = x2 + y4z +

y2z4 + z7 = 0.a. Show that the maximal multiplicity of points on S is 2, and the

singular locus of S is defined by the singular curve y2 +z3 = 0,x+ yz2 = 0.

b. Suppose that X is a hypersurface on a non-singular variety W ,and p ∈ X is a point in the locus of maximal multiplicity rof X. A hypersurface H through p is said to have permanentcontact with X if under any sequence of blow-ups π : W1 →W of W , with non-singular centers, contained in the locusof points of multiplicity r on the strict transform of X, thestrict transform of H contains all points of the intersection ofthe strict transform of X with multiplicity r which are in thefiber π−1(p). Show that there does not exist a non-singularhypersurface of permanent contact at the origin for the abovesurface S. Conclude that a hypersurface of maximal contactdoes not exist for S.

c. Resolve the singularities of S.

Chapter 8

Local Uniformizationand Resolution ofSurfaces

In this chapter we present a proof of resolution of surface singularitiesthrough local uniformization of valuations. Our presentation is a modern-ization of Zariski’s original proof in [86] and [88]. An introduction to thisapproach was given earlier in Section 2.5

8.1. Classification of valuations in function fields ofdimension 2

Let L be an algebraic function field of dimension two over an algebraicallyclosed field K of characteristic zero. That is, L is a finite extension of arational function field in two variables over K. A valuation of the functionfield L is a valuation ν of L which is trivial on K. ν is a homomorphismν : L∗ → Γ from the multiplicative group of L onto an ordered abelian groupΓ such that

1. ν(ab) = ν(a) + ν(b) for a, b ∈ L∗,2. ν(a+ b) ≥ minν(a), ν(b) for a, b ∈ L∗,3. ν(c) = 0 for 0 6= c ∈ K.

We formally extend ν to L by setting ν(0) =∞. We will assume throughoutthis section that ν is non-trivial, that is, Γ 6= 0.

Some basic references to the valuation theory of algebraic function fieldsare [92] and [3].

133

134 8. Local Uniformization and Resolution of Surfaces

A basic invariant of a valuation is its rank. A subgroup Γ′ of Γ is said tobe isolated if it has the property that if 0 ≤ α ∈ Γ′ and β ∈ Γ is such that0 ≤ β < α, then β ∈ Γ′. The isolated subroups of Γ form a single chain ofsubgroups. The length of this chain is the rank of the valuation. Since L is atwo-dimensional algebraic function field, we have that rank(ν) ≤ 2 (cf. thecorollary and the note to the lemma of Section 10, Chapter VI of [92]). If Γis of rank 1 then Γ is embeddable as a subgroup of R (cf. page 45 of Section10, Chapter VI [92]). If Γ is of rank 2, then there is a non-trivial isolatedsubgroup Γ1 ⊂ Γ such that Γ1 and Γ/Γ1 are embeddable as subgroups of R.If these subgroups of R are discrete, then ν is called a discrete valuation. Inparticular, a discrete valuation (in an algebraic function field of dimensiontwo) can be of rank 1 or of rank 2. The ordered chain of prime ideals p ofV corresponds to the isolated subgroups Γ′ of Γ by

P = f ∈ V | ν(f) ∈ Γ− Γ′ ∪ 0

(cf. Theorem 15, Section 10, Chapter VI [92]).In our analysis, we will find a rational function field of two variables

L′ over which L is finite, and consider the restricted valuation ν ′ = ν | L′.We will consider L′ = K(x, y), where x, y ∈ L are algebraically independentelements such that ν(x) and ν(y) are non-negative. Let [L′ : L] = r. Supposethat w ∈ L, and suppose that

(8.1) F (x, y, w) = A0(x, y)wr + · · ·+Ar(x, y) = 0

is the irreducible relation satisfied by w, with Ai ∈ K[x, y] for all i. Sinceν(F ) = ∞, there must be two distinct terms Aiwr−i, Ajwr−j of the samevalue, where we can assume that i < j. Thus (j − i)ν(w) = ν(AjAi ), and wehave that r!Γ′ ⊂ Γ. Thus Γ and Γ′ have the same rank, and Γ is discrete ifand only if Γ′ is discrete.

Associated to the valuation ν is the valuation ring

V = f ∈ L | ν(f) ≥ 0.

V has a unique maximal ideal

mV = f ∈ V | ν(f) > 0.

The residue field K(V ) = V/mV of V contains K and has transcendencedegree r∗ ≤ r over K. ν is called an r∗-dimensional valuation. In our case(ν non-trivial) we have r∗ = 0 or 1.

If R is local ring contained in L, we will say that ν dominates R if R ⊂ Vand mV ∩ R is the maximal ideal of R. If S is a subring of V , we will saythat the center of ν on S is the prime ideal mV ∩ S. We will say that R isan algebraic local ring of L if R is essentially of finite type over K (R is alocalization of a finite type K-algebra) and R has quotient field L.

8.1. Classification of valuations in function fields of dimension 2 135

We again consider our algebraic extension L′ = K(x, y) → L. Thevaluation ring of ν ′ is V ′ = V ∩ L, with maximal ideal mV ′ = mV ∩ L. Theresidue field K(V ′) is thus a subfield of K(V ). Suppose that w∗ ∈ K(V ),and let w be a lift of w∗ to V . We have an irreducible equation of algebraicdependence (8.1) of w over K[x, y]. Let Ah be such that

ν(Ah) = minν(A1), . . . , ν(Ar).

We may assume that h 6= r, because if ν(Ar) < ν(Ai) for i < r, then wewould have ν(Ar) < ν(Aiwr−i) for 1 ≤ i ≤ r − 1, since ν(w) ≥ 0, so thatν(F ) = ν(Ar) < ∞, which is impossible. Thus ν( AiAh ) ≥ 0 for 0 ≤ i ≤ r, sothat Ai

Ah∈ V ′ for all i. Let a∗i be the corresponding residues in K(V ′). We

have a relation

a∗0(w∗)r + · · ·+ a∗h−1(w

∗)r−h+1 + (w∗)r−h + a∗h+1(w∗)r−h−1 + · · ·+ a∗r = 0.

Since r − h > 0, w∗ is algebraic over K(V ′). Thus ν and ν ′ = ν | L′ havethe same dimension.

We will now give a classification of the various types of valuations whichoccur on L.

8.1.1. One-dimensional valuations. Let x ∈ L be the lift of an elementof K(V ) which is transcendental over K. Suppose that every y ∈ L whichis transcendental over K(x) satisfies ν(y) = 0. Then ν must be trivial whenrestricted to K(x, y) for any y which is transcendental over K(x). Since Lis finite over K(x, y), we would have that ν has rank 0 on L, so that ν istrivial on L, a contradiction. We may thus choose y ∈ L such that x, y arealgebraically independent over K, ν(x) = 0, ν(y) > 0 and the residue of xin K(V ) is transcendental over K. Let L′ = K(x, y), and let ν ′ = ν | L′with valuation ring V ′ = V ∩ L′. By construction, if mV ′ is the maximalideal of V ′, there exists c ∈ K such that mV ′ ∩K[x, y] ⊂ (x− c, y). Suppose(if possible) that p = mV ′ ∩K[x, y] = (x − c, y). Then the residue of x inK(V ) is c, a contradiction. Thus p is a height one prime in K[x, y] andp = (y). It follows that the valuation ring V ′ is Rp (cf. Theorem 9, Section5, Chapter VI [92]) and the value group of ν ′ is Z. The valuation ring V isa localization of the integral closure of the local Dedekind domain V ′ in L.Thus V is itself a local Dedekind domain which is a localization of a ring offinite type over K. In particular, V is itself a regular algebraic local ring ofL.

8.1.2. Zero-dimensional valuations of rank 2. Let Γ1 be the non-trivial isolated subgroup of Γ, with corresponding prime ideal p ⊂ V . Thecomposition of ν with the order-preserving homomorphism Γ → Γ/Γ1 de-fines a valuation ν1 of L of rank 1. Let V1 be the valuation ring of ν1. V1

consists of the elements of K whose value by ν is in Γ1 or is positive. Hence


V1 = Vp. ν induces a valuation ν2 of K(Vp) with value group Γ1, whereν2(f) = ν(f∗) if f∗ ∈ Vp is a lift of f to Vp. Since K(Vp) has a non-trivialvaluation, which vanishes on K, K(Vp) cannot be an algebraic extension ofK. Thus K(Vp) has positive transcendence degree over K, so it must havetranscendence degree 1. Let x∗ ∈ K(Vp) be such that x∗ is transcendentalover K. Let x be a lift of x∗ to Vp. Then K(x) ⊂ Vp, so that ν1 is trivial onK(x), and ν1 is a valuation of the one-dimensional algebraic function fieldL over K(x).

Let V2 ⊂ K(VP ) be the valuation ring of ν2, and let π : Vp → K(Vp)be the residue map. Then V = π−1(V2). ν1 is a 1-dimensional valuation ofL, so Vp is an algebraic local ring which is a local Dedekind domain. Theresidue field K(Vp) is an algebraic function field of dimension 1 over K, soV2 is also a local Dedekind domain. Thus Γ is a discrete group. Let w be aregular parameter in Vp.

Let w be a regular parameter in Vp. Given any non-zero element f ∈ L,we have a unique expression f = wmg, where m = ν1(f) ∈ Z, and g is aunit in Vp. Let g∗ be the residue of g in K(Vp). Let n = ν2(g∗) ∈ Z. Then,up to an order-preserving isomorphism of Γ, ν(f) = (m,n) ∈ Γ = Z2, whereZ2 has the lexicographic order.

8.1.3. Discrete zero-dimensional valuations of rank 1. Let x ∈ L besuch that ν(x) = 1. If y ∈ L and ν(y) = n1, then there exists a uniquec1 ∈ K such that n2 = ν(y − c1xn1) > ν(y). If we iterate this construc-tion infinitely many times, we have a uniquely determined Laurent seriesexpansion

c1xn1 + c2x

n2 + . . .

with ν(y − c1xn1 − · · · − cmxnm) = nm+1 > nm for all m. This constructiondefines an embedding of K-algebras L ⊂ K〈〈x〉〉, where K〈〈x〉〉 is the field ofLaurent series in x. The order valuation on K〈〈x〉〉 restricts to the valuationν on L.

8.1.4. Non-discrete zero-dimensional valuations of rank 1. We sub-divide this case by the rational rank of ν. The rational rank is the dimensionof Γ⊗Q as a rational vector space. Since ν has rank 1, we may assume thatΓ ⊂ R. If 1 6∈ Γ, we can choose 0 < λ ∈ Γ and define an order-preservinggroup isomorphism φ : Γ→ 1

λΓ by φ(α) = αλ . Then ν = φ ν is a valuation

of L with the same valuation ring

f ∈ L | ν(f) > 0 = f ∈ L | ν(f) > 0 = V

as ν. We may thus normalize ν (by replacing ν with the equivalent valuationν) so that Γ is an ordered subgroup of R and 1 ∈ Γ.

8.2. Local uniformization of algebraic function fields of surfaces 137

Suppose that ν has rational rank larger that 1. Choose x, y ∈ L suchthat ν(x) = 1 and ν(y) = τ is a positive irrational number. Suppose thatF (x, y) =

∑aijx

iyj is a polynomial in x and y with coefficients in K. Thenν(aijxiyj) = i + jτ for any term with aij 6= 0, so all the monomials in theexpansion of F have distinct values. It follows that ν(F ) = mini + τj |aij 6= 0. We conclude that x and y are algebraically independent, since arelation F (x, y) = 0 has infinite value. Furthermore, if L1 = K(x, y) andν1 = ν | L1, we have that the value group Γ1 of ν1 is Z + Zτ . Since Γ/Γ1 isa finite group, we have that Γ is a group of the same type. In particular, Γhas rational rank 2.

Now suppose that ν has rational rank 1. We can normalize Γ so thatit is an ordered subgroup of Q, whose denominators are not bounded, as Γis not discrete. In Example 3, Section 15, Chapter VI [92], examples aregiven of two-dimensional algebraic function fields with value group equal toany given subgroup of the rationals. There is a nice interpretation of thiskind of valuation in terms of certain “formal” Puiseux series (cf. MacLaneand Shilling [64]) where the denominators of the rational exponents in thePuiseux series expansion are not bounded.

Exercise 8.1. Let K be a field, R = K[x, y] a polynomial ring and L thequotient field of R.

1. Show that the rule ν(x) = 1, ν(y) = π defines a K-valuation on Lwhich satisfies

ν(f) = mini+ jπ | aij 6= 0

if f =∑aijx

iyj ∈ K[x, y].

2. Compute the rank, rational rank and value group Γ of ν.

3. Let V be the valuation ring of ν. Suppose that τ ∈ Γ. Show that

Iτ = f ∈ V | ν(f) ≥ τ

is an ideal of V .

4. Show that V is not a Noetherian ring.

8.2. Local uniformization of algebraic function fields ofsurfaces

In this section, we prove the following local uniformization theorem.

Theorem 8.2. Suppose that K is an algebraically closed field of character-istic zero, L is a two-dimensional algebraic function field over K and ν is avaluation of L. Then there exists a regular local ring A which is essentiallyof finite type over K with quotient field L such that ν dominates A.


Our proof is by a case by case analysis of the different types of valuationsof L which were classified in the previous section. If ν is 1-dimensional, wehave that R = V satisfies the conclusions of the theorem. We must provethe theorem in the case when ν has dimension zero, and one of the followingthree cases hold: ν has rational rank 1, or ν has rank 1 and rational rank 1,or ν has rank 2. The case when ν has rank 1 includes the cases when ν isdiscrete and ν is not discrete. We will follow the notation of the preceedingsection.

Suppose that K[x1, . . . , xn] is a polynomial ring, and m = (x1, . . . , xn).For f ∈ K[x1, . . . , xn], ord(f) will denote the order of f in K[x1, . . . , xn]m(cf. Section A.4). In particular, ord(f) = r if and only if f ∈ mr andf 6∈ mr+1.

8.2.1. Valuations of rational rank 1 and dimension 0.

Lemma 8.3. Suppose that S = K[x1, . . . , xn] is a polynomial ring, ν is arank 1 valuation with valuation ring V and maximal ideal mV which containsS such that K(V ) = K and mV ∩S = (x1, . . . , xn). Suppose that there existfi ∈ S for i ∈ N such that

ν(f1) < ν(f2) < · · · < ν(fn) < · · ·

is a strictly increasing sequence. Then limi→∞ ν(fi) =∞.

Proof. Suppose that ρ > 0 is a positive real number. We will show thatthere are only finitely many real numbers less than ρ which are the valuesof elements of S. Let τ = minν(f) | f ∈ (x1, . . . , xn). τ is well definedsince S is Noetherian. In fact, if there were not a minimum, we would havean infinite descending sequence of positive real numbers

τ1 > τ2 > · · · τi > · · ·

and elements gi ∈ (x1, . . . , xn) such that ν(gi) = τi. Let

Iτi = f ∈ S | ν(f) ≥ τi.

The Iτi are distinct ideals of S which form a strictly ascending chain

Iτ1 ⊂ Iτ2 ⊂ · · · ⊂ Iτi ⊂ · · · ,

which is impossible.Choose a positive integer r such that r > ρ

τ . If f ∈ S is such thatν(f) < ρ, then write f = g + h with h ∈ (x1, . . . , xn)r and deg(g) < r.ν(h) > ρ implies ν(g) = ν(f). Thus there exist a finite number of elementsg1, . . . , gm ∈ S (the monomials of degree < r) such that if 0 < λ < ρ is thevalue of an element of S, then there exist c1, . . . , cm ∈ K such that

(8.2) ν(c1g1 + · · ·+ cmgm) = λ.


We will replace the gi with appropriate linear combinations of the gi sothat

ν(g1) < ν(g2) < · · · < ν(gm).

Then λ = ν(gk) in (8.2), where ck is the first non-zero coefficient.By induction, it suffices to make a linear change in the gi so that

(8.3) ν(g1) < ν(gi) if i > 1.

After reindexing the gi, we can suppose that there exists an integer l > 1such that

ν(g1) = ν(g2) = · · · = ν(gl)

andν(gi) > ν(g1) if l < i.

The equality ν( gig1 ) = 0 for 2 ≤ i ≤ l implies gig1∈ V and there exist

ci ∈ K(V ) = K such that gig1

has residue ci in K(V ). Then gig1− ci ∈ mV

implies ν( gig1 − ci) > 0, so that ν(gi − cig1) > ν(g1) for 2 ≤ i ≤ l. Afterreplacing gi with gi − cig1 for 2 ≤ i ≤ l, we have that (8.3) is satisfied.

Theorem 8.4. Suppose that S = K[x, y] is a polynomial ring over an alge-braically closed field K of characteristic zero, ν is a rational rank 1 valuationof K(x, y) with valuation ring V and maximal ideal mV , K(V ) = V/mV =K such that S ⊂ V and the center of V on S is the maximal ideal (x, y).Further suppose that f ∈ K[x, y] is given. Then there exists a birationalextension K[x, y]→ K[x′, y′] (K[x, y] and K[x′, y′] have a common quotientfield) such that K[x′, y′] ⊂ V , the center of ν on K[x′, y′] is (x′, y′), andf = (x′)lδ, where l is a non-negative integer and δ ∈ K[x′, y′] is not in(x′, y′).

Proof. Set r = ord f(0, y). We have 0 ≤ r ≤ ∞. If r = 0 we are done, sosuppose that r > 0. We have an expansion

(8.4) f =d∑i=1

xαiγi(x)yβi +∑i>d

xαiγi(x)yβi ,

where the first sum is over terms with minimal value ρ = ν(xαiγi(x)yβi)for 1 ≤ i ≤ d, γi(x) ∈ K[x] are polynomials with non-zero constant term,β1 < · · · < βd, βd+1 < βd+2 < · · · and ν(xαiγi(x)yβi) > ρ if i > d.

Set ν(x)ν(y) = a

b with a, b ∈ N, (a, b) = 1. There exist non-negative integers

a′, b′ such that ab′ − ba′ = 1. Then ν(ya

xb) = 0 implies there exists 0 6= c ∈

K(V ) = K such that ν(ya

xb− c) > 0. Set

x = xa1(y1 + c)a′, y = xb1(y1 + c)b

′.


We have thatx1 = xb

′y−a

′, y1 + c = x−bya,

so that ν(y1) > 0 and

ν(x1) =ν(y)b

[ab′ − a′b] =ν(y)b

> 0.

Set α = α1a + β1b. We have that αia + βib = α for 1 ≤ i ≤ d. Thus thereexist ei ∈ Z such that(

a ba′ b′

) (αi − α1

βi − β1

)=

(0ei

)for 1 ≤ i ≤ d. By Cramer’s rule, we have

βi − β1 = Det(a 0a′ ei

)= aei.

Thus ei = βi−β1

a .We have a factorization f = xα1 f1(x1, y1), where

f1(x1, y1) = (y1 + c)a′α1+b′β1(

d∑i=1

γi(y1 + c)βi−β1a ) + x1Ω

is the strict transform of f in the birational extension K[x1, y1] of K[x, y].We have r1 <∞. ord f(0, y) = r implies there exists an i such that αi = 0,βi = r. Thus ν(yr) ≥ ρ, and βi ≤ r for 1 ≤ i ≤ d. We have

r1 = ord f1(0, y1) ≤βd − β1

a≤ r.

If r1 < r, then we have a reduction, and the theorem follows from inductionon r, since the conclusions of the theorem hold in K[x, y] if r = 0. We thusassume that r1 = r. Then βd = r, β1 = 0, a = 1 and αd = 0. Further, thereis an expression

d∑i=1

γi(0)(y1 + c)βi = γd(0)yr1.

Let u be an indeterminate, and set

g(u) =d∑i=1

γi(0)uβi = γd(0)(u− c)r.

The binomial theorem implies that βd−1 = r − 1, and thus

ν(y) = ν(xαd−1).

Let 0 6= λ ∈ k(V ) = k be the residue of yxαd−1 . Then ν(y − λxαd−1) >

ν(y). Set y′ = y − λxαd−1 and f ′(x, y′) = f(x, y). We have ord f ′(0, y′) =ord f(0, y) = r.


yr is a minimal value term of f(x, y) in the expansion (8.4), so ν(y) ≤ν(f).

Replacing y with y′ and f with f ′ and iterating the above procedureafter (8.4), either we achieve a reduction ord f1(0, y1) < r, or we find thatthere exists a′(x) ∈ K[x] such that ν(y′ − a′(x)) > ν(y′). We further havethat ν(y′) ≤ ν(f). Set y′′ = y′ − a′(x) and repeat the algorithm following(8.4) until we either reach a reduction ord f1(0, y1) < r or show that thereexist polynomials a(i)(x) ∈ K[x] for i ∈ N such that we have an infiniteincreasing bounded sequence

ν(a(x)) < ν(a′(x)) < · · · .

Since each ν(a(i)(x)) is an integral multiple of ν(x), this is impossible. Thusthere is a polynomial q(x) ∈ K[x] such that after replacing y with y −q(x), and following the algorithm after (8.4), we achieve a reduction r1 <r in K[x1, y1]. By induction on r, we can achieve the conclusions of thetheorem.

We now give the proof of Theorem 8.2 in the case of this subsection,that is, ν has dimension 0 and rational rank 1.

Let x, y ∈ L be algebraically independent over K and of positive value.Let w be a primitive element of L over K(x, y). We can assume that w haspositive value, for if ν(w) = 0, there exists c ∈ K such that the residue ofw − c is non-zero in K(V ) = K, and we can replace w with w − c. LetR = K[x, y, w]. Then R ⊂ V , the center of ν on R is the maximal ideal(x, y, w), and R has quotient field L. Let z be algebraically independent overK(x, y), and let f(x, y, z) in the polynomial ring K[x, y, z] be the irreduciblepolynomial such that R ∼= K[x, y, z]/(f) (with w mapping to z+(f)). Afterpossibly making a change of variables, replacing x with x + βz and y withy + γz for suitable β, γ ∈ K, we may assume that

r = ord f(0, 0, z) <∞.

If r = 1, then RmV ∩R is a regular local ring which is dominated by ν(by Exercise 3.4), so the conclusions of the theorem are satisfied. We thussuppose that r > 1. Let π : K[x, y, z]→ K[x, y, w] be the natural surjection.For g ∈ K[x, y, z], we will denote ν(π(g)) by ν(g). Write

(8.5) f(x, y, z) =d∑i=1

ai(x, y)zσi +∑i>d

ai(x, y)zσi ,

where ai(x, y) ∈ K[x, y] for all i, the ai(x, y)zσi are the terms of minimalvalue ρ = ν(ai(x, y)zσi) for 1 ≤ i ≤ d, ν(ai(x, y)zσi) > ρ for i > d andσ1 < σ2 < · · · < σd.


By Theorem 8.4, there exists a birational extension K[x, y]→ K[x1, y1]with K[x1, y1] ⊂ V and mV ∩K[x1, y1] = (x1, y1) such that

f =d∑i=1

xλi1 ai(x1, y1)zσi +∑i>d

xλi1 ai(x1, y1)zσi ,

where ai(0, 0) 6= 0 for all i. R1 = K[x1, y1, z1]/(f) is a domain with quotientfield L, R → R1 is birational, R1 ⊂ V and mV ∩ R1 = (x1, y1, z). Writeν(z)ν(x1) = s

t with s, t relatively prime positive integers. Let s′, t′ be positiveintegers such that s′t− st′ = 1. Set

x1 = xt2(z2 + c2)t′, y1 = y2, z = xs2(z2 + c2)s

′,

where 0 6= c2 ∈ K is determined by the condition ν(z2) > 0. Let α =λ1t + σ1s. We have α = λit + σis for 1 ≤ i ≤ d. We have (as in the proofof Theorem 8.4) that σi−σ1

t is a non-negative integer for 1 ≤ i ≤ d andf = xα2 f2(x2, y2, z2), where

f2(x2, y2, z2) = (x2 + c2)t′λ1+s′σ1(

d∑i=1

ai(z2 + c2)σi−σ1t ) + x2Ω

is the strict transform of f in k[x2, y2, z2]. Set R2 = K[x2, y2, z2]/(f2). ThenR1 → R2 is birational, R2 ⊂ V and mV ∩R2 = (x2, y2, z2).

Setr1 = ord f2(0, 0, z2) ≤

σd − σ1

t≤ r.

If r1 < r, we have a reduction, so we may assume that r1 = r. Then wehave σd = r, σ1 = 0, t = 1 and ad(0, 0) 6= 0. We further have a relation

d∑i=1

ai(0, 0)(z2 + c2)σi = ad(0, 0)zr2.

Set

g(u) =d∑i=1

ai(0, 0)uσi = ad(0, 0)(u− c2)r.

The binomial theorem shows that σd−1 = r − 1 and ν(z) = ν(ad−1(x, y)).There exists c ∈ k such that ν(z − cad−1(x, y)) > ν(z). Then radz

r−1 is aminimal value term of ∂f

∂z , so that ν(∂f∂z ) ≥ ν(z). Set z′ = z − cad−1(x, y),f ′(x, y, z′) = f(x, y, z). We have ord(f ′(0, 0, z′)) = r and ∂f ′

∂z′ = ∂f∂z .

We repeat the analysis following (8.5), with f replaced by f ′ and zreplaced with z′. Either we get a reduction r1 = ord(f2(0, 0, z2)) < r, orthere exists a′(x, y) ∈ K[x, y] such that ν(z′ − a′(x, y)) > ν(z′) and

ν(z′) ≤ ν(∂f′

∂z′) = ν(

∂f

∂z).


Set z′′ = z′−a′(x, y) and repeat the algorithm following (8.5). If we do reacha reduction r1 < r after a finite number of iterations, there exist polynomialsa(i)(x, y) ∈ K[x, y] for i ∈ N such that

ν(a(x, y)) < ν(a′(x, y)) < · · ·

is an increasing bounded sequence of real numbers. By Lemma 8.3 this isimpossible.

Thus there exists a polynomial a(x, y) ∈ K[x, y] such that after replacingz with z−a(x, y), the algorithm following (8.5) results in a reduction r1 < r.We then repeat the algorithm, so that by induction on r, we construct abirational extension of local rings R→ R1 so that R1 is a regular local ringwhich is dominated by V .

8.2.2. Valuations of rank 1, rational rank 2 and dimension 0. Afternormalizing ν, we have that its value group is Z + Zτ , where τ is a positiveirrational number. Choose x, y ∈ L such that ν(x) = 1 and ν(y) = τ .By the analysis of Subsection 8.1.4, x and y are algebraically independentover K. Let w be a primitive element of L over K(x, y) which has positivevalue. There exists an irreducible polynomial f(x, y, z) in the polynomialring K[x, y, z] such that T = K[x, y, w] ∼= K[x, y, z]/(f) (where w maps toz+(f)) and mV ∩T = (x, y, w). After possibly replacing x with x+βzm andy with y+γzm, where m is sufficiently large that mν(w) > maxν(x), ν(y),and β, γ ∈ K are suitably chosen, we may assume that

r = ord(f(0, 0, z)) <∞.

If r = 1, TmV ∩T is a regular local ring (by Exercise 3.4), so we assumethat r > 1. Let π : K[x, y, z] → K[x, y, w] be the natural surjection. Forg ∈ K[x, y, z], we will denote ν(π(g)) by ν(g). Write

f =d∑i=1

aixαiyβizγi +

∑i>d

aixαiyβizγi ,

where all ai ∈ K are non-zero, aixαiyβizγi for 1 ≤ i ≤ d are the minimalvalue terms, and γ1 < γ2 < · · · < γd.

There exist integers s, t such that ν(z) = s+ τt, and there exist integersM,N such that

ν(xαiyβizγi) = M +Nτ

for 1 ≤ i ≤ d. We thus have equalities

(8.6)M = α1 + sγ1 = · · · = αd + sγd,N = β1 + tγ1 = · · · = βd + tγd.

We further have

(αi + sγi) + (βi + tγi)τ > M +Nτ


if i > d. Since ord(f(0, 0, z)) = r and adxαdyβdzγd is a minimal value term,we have γd ≤ r. We expand τ into a continued fraction:

τ = h1 +1

h2 + 1h3+···

.

Let fjgj

be the convergent fractions of τ (cf. Section 10.2 [46]). Since lim fjgj

=τ , we have

(αi + sγi) + (βi + tγi)fjgj> M +N

fjgj

for j = p − 1 and j = p, whenever i > d and p is sufficiently large. Wefurther have that sgp+ tfp > 0 and sgp−1 + tfp−1 > 0 for p sufficiently large.Since ν(xsyt) = ν(z), there exists c ∈ K = K(V ) such that ν( z

xsyt − c) > 0.Consider the extension

(8.7) K[x, y, z]→ K[x1, y1, z1],

where

(8.8)x = x

gp1 y

gp−1

1 ,

y = xfp1 y

fp−1

1 ,

z = xsyt(z1 + c) = xsgp+tfp1 y

sgp−1+tfp−1

1 (z1 + c).

We have fp−1gp− fpgp−1 = ε = ±1 (cf. Theorem 150 [46]), and ε,−τ + fp−1

gp−1

and τ − fpgp

have the same signs (cf. Theorem 154 [46]). From (8.8), we seethat

xε1 =xfp−1

ygp−1, yε1 =

ygp

xfp,

from which we deduce that (8.7) is birational, and

εν(x1) = gp−1(fp−1

gp−1− τ), εν(y1) = gp(τ −

fpgp

).

Hence ν(x1), ν(y1), ν(z1) are all > 0. We have

f(x, y, z) = xMgp+Nfp1 y

Mgp−1+Nfp−1

1 f1(x1, y1, z1),

where

f1 = (z1 + c)γ1(d∑i=1

ai(z1 + c)γi−γ1) + x1y1H

is the strict transform of f in K[x1, y1, z1]. We thus have constructed abirational extension T → T1 = K[x1, y1, z1]/(f1) such that T1 ⊂ V andmV ∩ T1 = (x1, y1, z1).

We have f1(0, 0, 0) = 0, so u = c is a root of

φ(u) =d∑i=1

aiuγi−γ1 = 0.


Let r1 = ord f1(0, 0, z1). Then r1 ≤ γd − γ1 ≤ r. If r1 < r, we haveobtained a reduction, so we may assume that r1 = r. Thus γ1 = 0, γd = r,αd = βd = 0 (since γi ≤ r for 1 ≤ i ≤ d) and

φ(u) = ad(u− c)r.

Hence γd−1 = r − 1 and ad−1 6= 0, by the binomial theorem. We haveν(z) = αd−1+βd−1τ . Our conclusion is that there exist non-negative integersm and n such that ν(z) = m+ nτ . There exists c ∈ K such that

ν(z − cxmyn) > ν(z).

We make a change of variables in K[x, y, z], replacing z with z′ = z−cxmyn,and let f ′(x, y, z′) = f(x, y, z). We have ν(z′) > ν(z) and ord f ′(0, 0, z′) = r.Since radzr−1 is a minimal value term of ∂f

∂z , we have ν(z) ≤ ν(∂f∂z ). Wefurther have ∂f

∂z = ∂f ′

∂z′ . We now apply a transformation of the kind (8.8) withrespect to the new variables x, y, z′. If we do not achieve a reduction r1 < r,we have a relation ν(z′) = m1 + n1τ with m1, n1 non-negative integers, andwe have

ν(z) < ν(z′) ≤ ν(∂f∂z

).

We have that there exists c2 ∈ K such that

ν(z′ − c2xm1yn1) > ν(z′).

Set z′′ = z′ − c2xm1yn1 . We now iterate the procedure starting with thetransformation (8.8). If we do not achieve a reduction r1 < r after a finitenumber of iterations, we construct an infinite bounded sequence (since ∂f

∂z 6=0 in T )

m+ nτ < m1 + n1τ < · · · < mi + niτ < · · · ,where mi, ni are non-negative integers, which is impossible. We thus mustachieve a reduction r1 < r after a finite number of interations. By inductionon r, we can construct a birational extension T → T1 such that T1 ⊂ V and(T1)mV ∩T1 is a regular local ring.

8.2.3. Valuations of rank 2 and dimension 0. Suppose that ν has rank2 and dimension 0. Let 0 ⊂ p ⊂ mV be the distinct prime ideals of V . Letx, y ∈ L be such that x, y are a transcendence basis of L over K, ν(x) andν(y) are positive and the residue of x in K(Vp) is a transcendence basis ofK(Vp) over K. Let w be a primitive element of L over K(x, y) such thatν(w) > 0. Let T = K[x, y, w] ⊂ V . Let f(x, y, z) be an irreducible elementin the polynomial ring K[x, y, z] such that

(8.9) K[x, y, z]/(f) ∼= K[x, y, w],

where z + (f) is mapped to w.


By our construction, the quotient field of T is L and trdegK(T/p ∩ T ) =1. Thus p ∩ T is a prime ideal of a curve on the surface spec(T ).

Set R0 = RmV ∩T . Let C0 be the curve in spec(T ) with ideal p ∩ R0

in R0. Corollary 4.4 implies there exists a proper birational morphism π :X → spec(T ) which is a sequence of blow-ups of points such that the stricttransform C0 of C0 on X is non-singular. Let X1 → X be the blow-up ofC0. Since X1 → spec(T ) is proper, there exists a unique point a1 ∈ X1 suchthat V dominates R1 = OX1,a1 . R1 is a quotient of a 3-dimensional regularlocal ring (as explained in Lemma 5.4). Let C1 be the curve in X1 withideal p ∩ R1 in R1. (R1)p∩R1 dominates Rp∩R and (R1)p∩R1 is a local ringof a point on the blow-up of the maximal ideal of Rp∩R. We can iterate thisprocedure to construct a sequence of local rings (with quotient field L)

R0 → R1 → · · · → Ri → · · ·

such that V dominates Ri for all i, Ri is a quotient of a regular local ring ofdimension 3 and (Ri+1)p∩Ri+1 is a local ring of the blow-up of the maximalideal of (Ri)p∩Ri for all i. Furthermore, each Ri is a localization of a quotientof a polynomial ring in 3 variables.

Since (R0)p∩R0 can be considered as a local ring of a point on a planecurve over the field K(x) (as is explained in the proof of Lemma 5.10), byTheorem 3.15 (or Corollary 4.4) there exists an i such that (Ri)p∩Ri is aregular local ring.

After replacing the T in (8.9) with a suitable affine ring, we can nowassume that Tp∩T is a regular local ring. If TmV ∩T is a regular local ring weare done, so suppose TmV ∩T is not a regular local ring. Let p = π−1(p∩ T ),where π : K[x, y, z] → T is the natural surjection. If g ∈ K[x, y, z] is apolynomial, we will denote ν(π(g)) by ν(g). Let I ⊂ K[x, y, z] be the idealI = (f, ∂f∂x ,

∂f∂y ,

∂f∂z ) defining the singular locus of T . Since I 6⊂ p (as Tp∩T is

a regular local ring), one of π(∂f∂x ), π(∂f∂y ), π(∂f∂z ) 6∈ p ∩ T . Thus

minν(∂f∂x

), ν(∂f

∂y), ν(

∂f

∂z) = (0, n)

for some n ∈ N. Write

f = Fr + Fr+1 + · · ·+ Fm,

where Fi is a form of degree i in x, y, z and r = ord(f). After possiblyreindexing the x, y, z, we may assume that 0 < ν(x) ≤ ν(y) ≤ ν(z).

Since K(V ) = K, there exist c, d ∈ k (which could be 0) such that

ν(y − cx) > ν(x) and ν(z − dx) > ν(x).


Thus we have a birational transformation K[x, y, z] → K[x1, y1, z1] (a qua-dratic transformation), where

x1 = x, y1 =y − cxx

, z1 =z − dxx

,

with inverse

x = x1, y = x1(y1 + c), z = x1(z1 + d).

In K[x1, y1, z1] we have f(x, y, z) = xr1f1(x1, y1, z1), where

f1(x1, y1, z1) = Fr(1, y1 + c, z1 + d) + x1Fr+1(1, y1 + c, z1 + d)+ · · ·+ xm−r1 Fm(1, y1 + c, z1 + d)

is the strict transform of f . Let T1 = K[x1, y1, z1]/(f1). Then T → T1 isbirational, T1 ⊂ V and m ∩ T1 = (x1, y1, z1). We calculate

∂f∂x = xr−1

1 (x1∂f1∂x1− (y1 + c)∂f1∂y1

− (z1 + d) ∂f∂z1 + rf1),∂f∂y = xr−1

1∂f1∂y1

,∂f∂z = xr−1

1∂f1∂z1

.

Hence

minν(∂f∂x

), ν(∂f

∂y), ν(

∂f

∂z) ≥ (r − 1)ν(x1) + minν(∂f1

∂x1), ν(

∂f1

∂y1), ν(

∂f1

∂z1).

Since we have assumed that TmV ∩T is not regular, we have r > 1. Wefurther have that ν(x1) > 0, so we conclude that

minν(∂f1

∂x1), ν(

∂f1

∂y1), ν(

∂f1

∂z1) = (0, n1) < (0, n).

Let r1 = ord(f1) ≤ r. (T1)mV ∩T1 is a regular local ring if and only if r1 = 1.Thus we may assume that r1 > 1.

We now apply to T1 a new quadratic transform, to get

minν(∂f2

∂x2), ν(

∂f2

∂y2), ν(

∂f2

∂z2) = (0, n2) < (0, n1).

Thus after a finite number of quadratic transforms T → Tl, we construct aTl such that (Tl)mV ∩Tl is a regular local ring which is dominated by V .

Remark 8.5. Zariski proves local uniformization for arbitrary characteristiczero algebraic function fields in [87].


8.3. Resolving systems and the Zariski-Riemann manifold

Let L/K be an algebraic function field. A projective model of L is a pro-jective K-variety whose function field is L.

Let Σ(L) be the set of valuation rings of L. If X is a projective model ofL, then there is a natural morphism πX : Σ(L)→ X defined by πX(V ) = pif p is the (unique) point of X whose local ring is dominated by V . We willsay that p is the center of V on X. There is a topology on Σ(L), wherea basis for the topology are the open sets π−1

X (U) for an open set U of aprojective model X of L. Zariski shows that Σ(L) is quasi-compact (everyopen cover has a finite subcover) in [89] and Section 17, Chapter VI [92].Σ(L) is called the Zariski-Riemann manifold of L.

Let N be a set of zero-dimensional valution rings of L. A resolvingsystem of N is a finite collection S1, . . . , Sr of projective models of Lsuch that for each V ∈ N , the center of V on at least one of the Si is anon-singular point.

Theorem 8.6. Suppose that L is a two-dimensional algebraic function fieldover an algebraically closed field K of characteristic zero. Then there existsa resolving system for the set of all zero-dimensional valuation rings of L.

Proof. By local uniformization (Theorem 8.2), for each valuation W of Lthere exists a projective model XW of L such that the center of W on XW

is non-singular. Let UW be an open neighborhood of the center of W inXW which is non-singular, and let AW = π−1

W (UW ), an open neighborhoodof W in Σ(L). The set AW | W ∈ Σ(L) is an open cover of Σ(L). SinceΣ(L) is quasi-compact, there is a finite subcover AW1 , . . . , AWr of Σ(L).Then XW1 , . . . , XWr is a resolving system for the set of all zero-dimensionalvaluation rings of L.

Lemma 8.7. Suppose that L is a two-dimensional algebraic function fieldover an algebraically closed field K of characteristic zero, and

R0 ⊂ R1 ⊂ · · · ⊂ Ri ⊂ · · ·

is a sequence of distinct normal two-dimensional algebraic local rings ofL. Let Ω =

⋃Ri. If every one-dimensional valuation ring V of L which

contains Ω intersects Ri in a height one prime for some i, then Ω is thevaluation ring of a zero-dimensional valuation of L.

Proof. Ω is by construction integrally closed in L. Thus Ω is the intersectionof the valuation rings of L containing Ω (cf. the corollary to Theorem 8,Section 5, Chapter VI [92]). Observe that Ω 6= L. We see this as follows.If mi is the maximal ideal of Ri, then m =

⋃mi is a non-zero ideal of Ω.

Suppose that 1 ∈ m. Then 1 ∈ mi for some i, which is impossible.

8.3. Resolving systems and the Zariski-Riemann manifold 149

We will show that Ω is an intersection of zero-dimensional valuationrings. To prove this, it suffices to show that if V is a one-dimensionalvaluation ring containing Ω, then there exists a zero-dimensional valuationring W such that V is a localization of W and Ω ⊂ W . Since V is one-dimensional, V is a local Dedekind domain and is an algebraic local ringof L. Let K(V ) be the residue field of V . Let mV be the maximal idealof V , and let pi = Ri ∩mV . By assumption, there exists an index j suchthat i ≥ j implies that pi is a height one prime in Ri. (Ri)pi is a normallocal ring which is dominated by V . Since both (Ri)pi and V are localDedekind domains, and they have the common quotient field L, we havethat (Ri)pi = V for i ≥ j (cf. Theorem 9, Section 5, Chapter VI [92]).Thus the residue field Ki of (Ri)pi is K(V ) for i ≥ j. The one-dimensionalalgebraic local rings Si = Ri/pi for i ≥ j have quotient field K(V ), and Si+1

dominates Si.⋃Si is not K(V ) by the same argument we used to show that

Ω is not L. Thus there exists a valuation ring V ′ of K(V ) which dominates⋃Si. Let π : V → K(V ) be the residue map. Then W = π−1(V ′) is a

zero-dimensional valuation ring of V and W contains Ω.Since we have established that Ω is the intersection of zero-dimensional

valuation rings, we need only show that Ω is not contained in two distinctzero-dimensional valuation rings. Suppose that Ω is contained in two distinctzero-dimensional valuations, V1 and V2.

We will first construct a projective surface Y with function field L suchthat V1 and V2 dominate (unique) distinct points q1 and q2 of Y . Let x, ybe a transcendence basis of L over K. Let ν1, ν2 be valuations of L whoserespective valuation rings are V1 and V2. If ν1(x) and ν2(x) are both ≤ 0,replace x with 1

x . Suppose that ν1(x) ≥ 0 and ν2(x) < 0. Let c ∈ K be suchthat the residue of x in the residue field K(V1) ∼= K of V1 is not c. Afterreplacing x with x− c, we have that ν1(x) = 0 and ν2(x) < 0. We can nowreplace x with 1

x to get that ν1(x) ≥ 0 and ν2(x) ≥ 0. In this way, afterpossibly changing x and y, we can assume that x and y are contained inthe valuation rings V1 and V2. Since V1 and V2 are distinct zero-dimensionalvaluation rings, there exist f ∈ V1 such that f 6∈ V2, and g ∈ V2 such thatg 6∈ V1 (cf. Theorem 3, Section 3, Chapter VI [92]). As above, we canassume that ν1(f) = 0, ν2(f) < 0, ν1(g) < 0 and ν2(g) = 0. Let a = 1

f ,b = 1

g , and let A be the integral closure of K[x, y, a, b] in L. By construction,A ⊂ V1 ∩ V2. Let m1 = A ∩ mV1 , m2 = A ∩ mV2 . We have that a ∈ m2

but a 6∈ m1, and b ∈ m1 but b 6∈ m2. Thus m1 6= m2. We now let Ybe the normalization of a projective closure of spec(A). Y has the desiredproperties.

Since Ri is essentially of finite type over K, it is a localization of afinitely generated K-algebra Bi which has quotient field L. Let Xi be the


normalization of a projective closure of spec(Bi). Since Ri is integrallyclosed, the projective surface Xi has a point ai with associated local ringOXi,ai = Ri.

There is a natural birational map Ti : Xi → Y . We will now establishthat Ti cannot be a morphism at ai. If Ti were a morphism at ai, therewould be a unique point bi on Y such that Ri = OXi,ai dominates OYi,bi .Since V1 and V2 both dominate OXi,ai , V1 and V2 both dominate OYi,bi , acontradiction.

By Zariski’s main theorem (cf. Theorem V.5.2 [47]) there exists a one-dimensional valuation ring W of L such that W dominates OXi,ai and Wdominates the local ring of the generic point of a curve on Y . Let Γi bethe set of one-dimensional valuation rings W of K such that W dominatesOXi,ai and W dominates the local ring of the generic point of a curve on Y .Γi is a finite set, since the valuation rings W of Γi are in 1-1 correspondencewith the curves contained in π−1

1 (ai), where π1 : ΓTi → Xi is the projectionof the graph of Ti onto Xi. Since Ri+1 = OXi+1,ai+1 dominates Ri = OXi,ai ,we have Γi+1 ⊂ Γi for all i. Since each Γi is a non-empty finite set, thereexists a one-dimensional valuation ring W of L such that W dominates Rifor all i, a contradiction.

Theorem 8.8. Suppose that L is a two-dimensional algebraic function fieldover an algebraically closed field K of characteristic zero, R is a two-dimen-sional normal algebraic local ring with quotient field L, and

R→ R1 → R2 → · · · → Ri → · · ·

is a sequence of normal local rings such that Ri+1 is obtained from Ri byblowing up the maximal ideal, normalizing, and localizing at a maximal idealso that Ri+1 dominates Ri. Then Ω =

⋃Ri is a zero-dimensional valuation

ring of L.

Proof. By Lemma 8.7, we are reduced to showing that there does not exista one-dimensional valuation ring V of L such that Ω is contained in V andmV ∩Ri is the maximal ideal for all i.

Suppose that V is a one-dimensional valuation ring of L which containsΩ. Let ν be a valuation of L whose associated valuation ring is V . Since Vis one-dimensional, there exists α ∈ V such that the residue of α in K(V )is transcendental over K.

Let mi be the maximal ideal of Ri. Assume that mV ∩R is the maximalideal m0 of R. Write α = ξ

η with ξ, η ∈ R. Then α 6∈ R, since R/m0∼= K.

Hence η ∈ m0, and also ξ ∈ m0 since ν(α) ≥ 0. Since R1 is a local ring ofthe normalization of the blow-up of m0, m0R1 is a principal ideal. In fact if


ζ ∈ m0 is such that ν(ζ) = minν(f) | f ∈ m0, we have that ζR1 = m0R1.Thus ξ1 = ξ

ζ ∈ R1 and η1 = ηζ ∈ R1.

If we assume that mV ∩R1 = m1, we have that ξ1, η1 ∈ m1, α = ξ2η2

withξ2, η2 ∈ R2 and

ν(ξ) > ν(ξ1) > 0, ν(ξ1) > ν(ξ2),

ν(η) > ν(η1) > 0, ν(η1) > ν(η2).

More generally, if we assume that mi = mV ∩V for i = 1, . . . , n, we can findelements ξi, ηi ∈ Ri for 1 ≤ i ≤ n such that α = ξi

ηiand

ν(ξ) > ν(ξ1) > · · · > ν(ξn) > 0,

ν(η) > ν(η1) > · · · > ν(ηn) > 0.

Since the value group of V is Z, it follows that n ≤ minν(ξ), ν(η). Hence,for all i sufficiently large, mV ∩Ri is not the maximal ideal. mV ∩Ri mustthen be a height one prime in Ri, for otherwise we would have mV ∩Ri = (0),which would imply that V = L, against our assumption.

Remark 8.9. The conclusions of this theorem are false in dimension 3(Shannon [76]).

Theorem 8.10. Suppose that S, S′ are normal projective surfaces over analgebraically closed field K of characteristic zero and T is a birational mapfrom S to S′. Let S1 → S be the projective morphism defined by taking thenormalization of the blow-up of the finitely many points of T , where T isnot a morphism. If the induced birational map T1 from S1 to S′ is not amorphism, let S2 → S1 be the normalization of the blow-up of the finitelymany fundamental points of T1. We then iterate to produce a sequence ofbirational morphisms of surfaces

S ← S1 ← S2 ← · · · ← Si ← · · · .

This sequence must be of finite length, so that the induced rational mapSi → S′ is a morphism for all i sufficiently large.

Proof. Since Si is a normal surface, the set of points where Ti is not amorphism is finite (cf. Lemma 5.1 [47]). Suppose that no Ti : Si → S′ is amorphism. Then there exists a sequence of points pi ∈ Si for i ∈ N such thatpi+1 maps to pi and Ti is not a morphism at pi for all i. Let Ri = OSi,pi .The birational maps Ti give an identification of the function fields of theSi and S′ with a common field L. By Theorem 8.8, Ω =

⋃Ri is a zero-

dimensional valuation ring. Hence there exists a point q ∈ S′ such that Ωdominates OS′,q. OS′,q is a localization of a K-algebra B = K[f1, . . . , fr] forsome f1, . . . , fr ∈ L. Since B ⊂ Ω, there exists some i such that B ⊂ Ri.Since Ω dominates Ri, we necessarily have that Ri dominates OS′,q. Thus Ti


is a morphism at pi, a contradiction to our assumption. Thus our sequencemust be finite.

Theorem 8.11. Suppose that S and S′ are projective surfaces over an alge-braically closed field K of characteristic zero which form a resolving systemfor a set of zero-dimensional valuations N . Then there exists a projectivesurface S∗ which is a resolving system for N .

Proof. By Theorem 8.10, there exists a birational morphism S → S ob-tained by a sequence of normalizations of the blow-ups of all points wherethe birational map to S′ is not defined. Now we construct a birational mor-phism S

′ → S′, applying the algorithm of Theorem 8.10 by only blowing upthe points where the birational map to S is not defined and which are non-singular points. The algorithm produces a birational map S

′ → S which isa morphism at all non-singular points of S′.

Let S∗ be the graph of the birational map from S to S′, with naturalprojections π1 : S∗ → S and π2 : S∗ → S

′. We will show that S∗ is aresolving system for N . Suppose that V ∈ N . Let p, p′, p′ and p∗ be thecenters of V on the respective projective surfaces S, S′, S′ and S∗.

First suppose that p′ is a singular point. Then p must be non-singular,as S, S′ is a resolving system for N . The birational map from S to S′ isa morphism at p, and π1 is an isomorphism at p. Thus the center of V onS∗ is a non-singular point.

Now suppose that p′ is a non-singular point. Then p′ is a non-singularpoint of S′ and the birational map from S

′ to S is a morphism at p′. Thusπ2 is an isomorphism at p′, and the center of V on S∗ is a non-singularpoint.

Theorem 8.12. Suppose that L is a two-dimensional algebraic function fieldover an algebraically closed field of characteristic zero. Then there exists anon-singular projective surface S with function field L.

Proof. The theorem follows from Theorem 8.6 and Theorem 8.10, by in-duction on r applied to a resolving system S1, . . . , Sr for the set of zero-dimensional valuation rings of L.

Remark 8.13. Zariski proves the generalization of Theorem 8.11 to di-mension 3 in [90], and deduces resolution of singularities for characteristiczero 3-folds from his proof of local uniformization for characteristic zeroalgebraic function fields [87]. Abhyankar proves local uniformization in di-mension three and characteristic p > 5, from which he deduces resolution ofsingularities for 3-folds of characteristic p > 5 [4].


Exercise 8.14.

1. Prove that all the birational extensions constructed in Section 8.2are products of blow-ups of points and non-singular curves.

2. Identify where characteristic zero is used in the proofs of this chap-ter. All but one of the cases of Section 8.2 extend without greatdifficulty to characteristic p > 0. Some care is required to ensurethat K(x, y)→ L is separable. In [1], Abhyankar accomplishes thisand gives a very different proof in the remaining case to prove localuniformization in characteristic p > 0 for algebraic surfaces.

Chapter 9

Ramification ofValuations andSimultaneousResolution

Suppose that L is an algebraic function field over a field K, and trdegK L <∞ is arbitrary. We will use the notation for valuation rings of Section 8.1.

Suppose that R is a local ring contained in L. We will say that R isalgebraic (or an algebraic local ring of L) if L is the quotient field of R andR is essentially of finite type over K. Let K(S) denote the residue field of aring S containing K with a unique maximal ideal. We will also denote themaximal ideal of a ring S containing a unique maximal ideal by mS .

Suppose that L→ L∗ is a finite separable extension of algebraic functionfields over K, and V ∗ is a valuation ring of L∗/K associated to a valuationν∗ with value group Γ∗. Then the restriction ν = ν∗ | L of ν∗ to L is avaluation of L/K with valuation ring V = L∩V ∗. Let Γ be the value groupof ν.

There is a commutative diagram

L → L∗

↑ ↑V = L ∩ V ∗ → V ∗.

155

156 9. Ramification of Valuations and Simultaneous Resolution

There are associated invariants (cf. page 53, Section 11, Chapter VI [92])namely, the reduced ramification index of ν∗ relative to ν,

e = [Γ∗ : Γ] <∞,and the relative degree of ν∗ with respect to ν,

f = [K(V ∗) : K(V )].

The classical case is when V ∗ is an algebraic local Dedekind domain. Inthis case V is also an algebraic local Dedekind domain. If mV ∗ = (y) andmV = (x), then there is a unit γ ∈ V ∗ such that

x = γye.

Since we thus haveΓ∗/Γ ∼= Z/eZ,

we see that e is the reduced ramification index.Still assuming that V ∗ is an algebraic local Dedekind domain, if we fur-

ther assume that K = K(V ∗) is algebraically closed, and (e, char(K)) = 1,then we have that the induced homomorphism on completions with respectto the respective maximal ideals

V → V ∗

is given by the natural inclusion of K algebras

K[[x]]→ K[[y]],

where y = e√γy and x = ye. We thus have a natural action of Γ∗/Γ on V ∗

(a generator multiplies y by a primitive e-th root of unity), and the ring ofinvariants by this action is

V ∼= (V ∗)Γ∗/Γ.

In this chapter we find analogs of these results for general valuationrings. The basic approach is to find algebraic local rings R of L and S of L∗

such that V ∗ dominates S (S is contained in V ∗ and mV ∗ ∩S = mS) and Vdominates R, and such that the ramification theory of V → V ∗ is capturedin

(9.1) R→ S,

and we have a theory for R→ S comparable to that of the above analysis ofV → V ∗ in the special case when V ∗ is an algebraic local Dedekind domain.

This should be compared with Zariski’s Local Uniformization Theorem[87] (also Section 2.5 and Chapter 8 of this book). It is proven in [87]that if char(K) = 0 and V is a valuation ring of an algebraic function fieldL/K, then V dominates an algebraic regular local ring of L. The case whentrdegK L = 2 is proven in Section 8.2.

9. Ramification of Valuations and Simultaneous Resolution 157

We first consider the following diagram:

(9.2)L → L∗

↑S

where L∗/L is a finite separable extension of algebraic function fields overL, and S is a normal algebraic local ring of L∗. We say that S lies above analgebraic local ring R of L if S is a localization at a maximal ideal of theintegral closure of R in L∗.

Abhyankar and Heinzer have given examples of diagrams (9.2) wherethere does not exist a local ring R of L such that S lies above R (cf. [35]).To construct an extension of the kind (9.1), we must at least be able to findan S dominated by V ∗ such that S lies over an algebraic local ring R of L.To do this, we consider the concept of a monoidal transform.

Suppose that V is a valuation ring of the algebraic function field L/K,and suppose that R is an algebraic local ring of L such that V dominates R.Suppose that p ⊂ R is a regular prime; that is, R/p is a regular local ring.If f ∈ p is an element of minimal value, then R[ pf ] is contained in V , andif Q = mV ∩ R[ pf ], then R1 = R[ pf ]Q is an algebraic local ring of L whichis dominated by V . We say that R→ R1 is a monoidal transform along V .R1 is the local ring of the blow-up of spec(R) at the non-singular subschemeV (p). If R is a regular local ring, then R1 is a regular local ring. In thiscase there exists a regular system of parameters (x1, . . . , xn) in R such thatif ht(p) = r, then R1 = R[x2

x1, · · · , xrx1

]Q.The main tool we use for our analysis is the Local Monomialization

Theorem 2.18.With the notation of (9.1) and (9.2), consider a diagram

(9.3)

L → L∗

↑ ↑V = V ∗ ∩ L → V ∗

↑S∗

where S∗ is an algebraic local ring of L∗.The natural question to consider is local simultaneous resolution; that

is, does there exist a diagram (constructed from (9.3))

V → V ∗

↑ ↑R → S

↑S∗


such that S∗ → S is a sequence of monoidal transforms, S is a regular localring, and there exists a regular local ring R such that S lies above R?

The answer to this question is no, even when trdegK(L∗) = 2, as wasshown by Abhyankar in [2]. However, it follows from a refinement in Theo-rem 4.8 [35] (strong monomialization) of Theorem 2.18 that if K has char-acteristic zero, trdegK(L∗) is arbitrary and V ∗ has rational rank 1, thenlocal simultaneous resolution is true, since in this case (2.8) becomes

x1 = δ1ya111 , x2 = y2, · · · , xn = yn,

and S lies over R0.The natural next question to consider is weak simultaneous local reso-

lution; that is, does there exist a diagram (constructed from (9.3))

V → V ∗

↑ ↑R → S

↑S∗

such that S∗ → S is a sequence of monoidal transforms, S is a regular localring, and there exists a normal local ring R such that S lies above R?

This was conjectured by Abhyankar on page 144 of [8] (and is implicit in[1]). If trdegK(L∗) = 2, the answer to this question is yes, as was shown byAbhyankar in [1], [3]. If trdegK(L∗) is arbitrary, and K has characteristiczero, then the answer to this question is yes, as we prove in [29] and [35].This is a simple corollary of local monomialization. In fact, weak simulta-neous local resolution follows from the following theorem, which will be ofuse in our analysis of ramification.

Theorem 9.1 (Theorem 4.2 [35]). Let K be a field of characteristic zero,L an algebraic function field over K, L∗ a finite algebraic extension of L,and ν∗ a valuation of L∗/K, with valuation ring V ∗. Suppose that S∗ is analgebraic local ring of L∗ which is dominated by ν∗, and R∗ is an algebraiclocal ring of L which is dominated by S∗. Then there exists a commutativediagram

(9.4)R0 → R → S ⊂ V ∗

↑ ↑R∗ → S∗

where S∗ → S and R∗ → R0 are sequences of monoidal transforms along ν∗

such that R0 → S have regular parameters of the form of the conclusions ofTheorem 2.18, R is a normal algebraic local ring of L with toric singularities


which is the localization of the blow-up of an ideal in R0, and the regularlocal ring S is the localization at a maximal ideal of the integral closure ofR in L∗.

Proof. By resolution of singularities [52] (cf. Theorems 2.6 and 2.9 in [26]),we first reduce to the case where R∗ and S∗ are regular, and then construct,by the Local Monomialization Theorem 2.18, a sequence of monoidal trans-forms along ν∗

(9.5)R0 → S ⊂ V ∗

↑ ↑R∗ → S∗

so that R0 is a regular local ring with regular parameters (x1, . . . , xn), S isa regular local ring with regular parameters (y1, . . . , yn), and there are unitsδ1, . . . , δn in S, and a matrix of natural numbers A = (aij) with non-zerodeterminant d such that

xi = δiyai11 · · · y

ainn

for 1 ≤ i ≤ n. After possibly reindexing the yi, we may assume that d > 0.Let (bij) be the adjoint matrix of A. Set

fi =n∏j=1

xbijj = (

n∏j=1

δbijj )ydi

for 1 ≤ i ≤ n. Let R be the integral closure of R0[f1, . . . , fn] in L, localizedat the center of ν∗. Since

√mRS = mS , Zariski’s Main Theorem (10.9 in

[4]) shows that R is a normal algebraic local ring of L such that S lies aboveR.

We thus have a sequence of the form (9.4).

In the theory of resolution of singularities a modified version of the weaksimultaneous resolution conjecture is important. This is called global weaksimultaneous resolution.

Suppose that f : X → Y is a proper, generically finite morphism ofk-varieties. Does there exist a commutative diagram

(9.6)X1

f1→ Y1

↓ ↓X

f→ Y

such that f1 is finite, X1 and Y1 are complete k-varieties with X1 non-singular, Y1 normal, and the vertical arrows are birational?


This question has been posed by Abhyankar (with the further conditionsthat Y1 → Y is a sequence of blow-ups of non-singular subvarieties and Y ,X are projective) explicitly on page 144 of [8] and implicitly in the paper[2].

We answer this question in the negative, in an example constructedIn Theorem 3.1 of [30]. The example is of a generically finite morphismX → Y of non-singular, projective surfaces, defined over an algebraicallyclosed field K of characteristic not equal to 2. This example also givesa counterexample to the principle that a geometric statement that is truelocally along all valuations should be true globally.

We now state our main theorem on ramification of arbitrary valuations.

Theorem 9.2 (Theorem 6.1 [35]). Suppose that char(K) = 0 and we aregiven a diagram of the form of (9.3)

L → L∗

↑ ↑V = V ∗ ∩ L → V ∗

↑S∗.

Let K be an algebraic closure of K(V ∗). Then there exists a regular algebraiclocal ring R1 of L such that if R0 is a regular algebraic local ring of L whichcontains R1 such that

R0 → R → S ⊂ V ∗

↑S∗

is a diagram satisfying the conclusions of Theorem 9.1, then:

1. There is an isomorphism Zn/AtZn ∼= Γ∗/Γ given by

(b1, . . . , bn) 7→ b1ν∗(y1) + · · ·+ bnν

∗(yn).

2. Γ∗/Γ acts on S⊗K(S)K, and the invariant ring is

R⊗K(R)K ∼= (S⊗K(S)K)Γ∗/Γ.

3. The reduced ramification index is

e = |Γ∗/Γ| = |Det(A)|.

4. The relative degree is

f = [K(V ∗) : K(V )] = [K(S) : K(R)].

It is shown in [35] and [32] that we can realize the valuation rings Vand V ∗ as directed unions of local rings of the form of R and S in the above


theorem, and the union of the valuation rings is a Galois extension withGalois group Γ∗/Γ

We now give an overview of the proof of Theorem 9.2, referring to [35]for details. ν∗ is our valuation of L∗ with valuation ring V ∗, and ν = ν∗ | Lis the restiction of ν∗ to K. In the statement of the theorem Γ∗ is thevalue group of ν∗ and Γ is the value group of ν. By assumption, we haveregular parameters (x1, . . . , xn) in R0 and (y1, . . . , yn) in S which satisfy theequations

xi = δi

n∏j=1

yaijj for 1 ≤ i ≤ n

of (2.8). We have the relations

ν(xi) =n∑j=1

aijν∗(yj) ∈ Γ

for 1 ≤ i ≤ n. Thus there is a group homomorphism

Zn/AtZn → Γ∗/Γ

defined by 1. To show that this homomorphism is an isomorphism we needan extension of local monomialization (proven in Theorem 4.9 [35]). 3 isimmediate from 1

R0 → S is monomial, and R0 → R is a weighted blow-up. Lemma 4.4 of[35] shows that if M∗ is the quotient field of S⊗K(S)K and M is the quotientfield of R⊗K(R)K, then M∗/M is Galois with Galois group Gal(M∗/M) ∼=Zn/AZn ∼= Zn/AtZn. Furthermore, (S⊗K(S)K)Zn/AZn ∼= R⊗K(R)K.

We see that R ⊗K(R) K is a quotient singularity, by a group whoseinvariant factors are determined by Γ∗/Γ. This observation leads to thecounterexample to global weak simultaneous resolution stated earlier in thissection. This example is written up in detail in [30]. We will give a verybrief outline of the construction. With the notation of (9.6), if a valuation νof K(Y ) has at least two distinct extensions ν∗1 and ν∗2 to K(X) which havenon-isomorphic quotients Γ∗1/Γ and Γ∗2/Γ, then this presents an obstructionto the existence of a map X1 → Y1 (with the notation of (9.6)) which isfinite, and X1 is non-singular. If there were such a map, we would havepoints p1 and p2 on X1, whose local rings R1 and R2 are regular and aredominated by ν∗1 and ν∗2 respectively, and a point p on Y1 whose local ringR is dominated by p. If we choose our valuation ν carefully, we can ensurethat R is a quotient of R1 by Γ∗1/Γ and R is also a quotient of R2 by Γ∗2/Γ.We can in fact choose the extended valutions so that we can conclude thatnot only are these quotient groups not isomorphic, but the invariant ringsare not isomorphic, which is a contradiction.


An interesting question is whether the conclusions of local monomializa-tion (Theorem 2.18) hold if the ground field K has positive characteristic.This is certainly true if trdegK(L∗) = 1.

Now suppose that trdegK(L∗) = 2 and K is algebraically closed of pos-itive characteristic. Strong monomialization (a refinement in Theorem 4.8[35] of Theorem 2.18) is true if the value group Γ∗ of V ∗ is a finitely gener-ated group (Theorem 7.3 [35]) or if V ∗/V is defectless (Theorem 7.35 [35]).The defect is an invariant which is trivial in characteristic zero, but is apower of p in an extension of characteristic p function fields.

Theorem 7.38 [35] gives an example showing that strong monomializa-tion fails if trdegK(L∗) = 2 and char(K) > 0. Of course the value group isnot finitely generated, and V ∗/V has a defect. This example does satisfythe less restrictive conclusions of local monomialization (Theorem 2.18).

It follows from strong monomialization in characteristic zero (a refine-ment of Theorem 2.18) that local simultaneous resolution is true (for arbi-trary trdegK(L∗)) if K has characteristic zero and rational rank ν∗ = 1. Weconsider this condition when trdegK(L∗) = 2 and K is algebraically closed ofpositive characteristic. In Theorem 7.33 [35] it is shown that in many caseslocal simultaneous resolution does hold if trdegK(L∗) = 2, char(K) > 0 andthe value group is not finitely generated. For instance, local simultaneousresolution holds if Γ is not p-divisible. The example of Theorem 7.38 [35]discussed above does in fact satisfy local simultaneous resolution, althoughit does not satisfy strong monomialization.

Appendix. Smoothnessand Non-singularity II

In this appendix, we prove the theorems on the singular locus stated inSection 2.2, and prove theorems on upper semi-continuity of order neededin our proofs of resolution. The proofs in Section A.1 are based on Zariski’soriginal proofs in [85].

A.1. Proofs of the basic theorems

Suppose that P ∈ AnK = spec(K[x1, . . . , xn]) has height n − r. Let mP ⊂

OAn,P be the ideal of P . The differential

d : OAn → Ω1An = OAndx1 ⊕ · · · ⊕ OAndxn

is defined by

d(u) =∂u

∂x1dx1 + · · ·+ ∂u

∂xndxn.

d induces a linear transformation of K(P )-vector spaces

dP : mP /(mP )2 → Ω1An,P ⊗K(P ).

Define D(P ) to be the image of dP . Since OAn,P is a regular local ring,dimK(P )mP /m

2P = n−r. Thus dimK(P )D(P ) ≤ n−r, and dimK(P )D(P ) =

n− r if and only ifdP : mP /(mP )2 → D(P )

is a K(P )-linear isomorphism.

Theorem A.1. Suppose that P ∈ AnK = spec(K[x1, . . . , xn]) is a closed

point. Then dimK(P )D(P ) = n if and only if K(P ) is separable over K.

163

164 Appendix. Smoothness and Non-singularity II

Proof. Suppose that P ∈ spec(K[x1, . . . , xn]) is a closed point. Let mP ⊂OAn,P be the ideal of P . Let αi be the image of xi in K(P ) for 1 ≤ i ≤ n.Let f1(z1) ∈ K[z1] be the minimal polynomial of α1 over K. We can theninductively define fi(z1, . . . , zi) in the polynomial ring K[z1, . . . , zi] so thatfi(α1, . . . , αi−1, zi) is the minimal polynomial of αi over K(α1, . . . , αi−1) for2 ≤ i ≤ n. Let I be the ideal

(f1(x1), f2(x1, x2), . . . , fn(x1, . . . , xn)) ⊂ OAn,P .

By construction I ⊂ mP and K[x1, . . . , xn]/I ∼= K(P ), so we have I =mP . Thus f1(x1), f2(x1, x2), . . . , fn(x1, . . . , xn) is a regular system of pa-rameters in OAnK ,P , and the images of f1(x1), f2(x1, x2), . . . , fn(x1, . . . , xn)in mP /m

2P form a K(P )-basis. Thus dimK(P )D(P ) = n if and only if

dP (f1), . . . , dP (fn) are linearly independent over K(P ). This holds if andonly if J(f ;x) has rank n at P , which in turn holds if and only if the deter-minant

|J(f ;x)| =n∏i=1

∂fi∂xi

is not zero in K(P ). This condition holds if and only if K(α1, . . . , αi) isseparable over K(α1, . . . , αi−1) for 1 ≤ i ≤ n, which is true if and only ifK(P ) is separable over K.

Corollary A.2. Suppose that P ∈ AnK = spec(K[x1, . . . , xn]) is a closed

point such that K(P ) is separable over K. Let mP ⊂ OAn,P be the ideal ofP , and suppose that u1, . . . , un ∈ mP . Then u1, . . . , un is a regular systemof parameters in OAnK ,P if and only if J(u;x) has rank n at P .

Proof. This follows since dP : mP /m2P → D(P ) is an isomorphism if and

only if K(P ) is separable over K.

Lemma A.3. Suppose that A is a regular local ring of dimension b with max-imal ideal m, and p ⊂ A is an equidimensional ideal such that dim(A/p) = a.Then

dimA/m(p+m2/m2) ≤ b− a,and dimA/m(p+m2/m2) = b− a if and only if A/p is a regular local ring.

Proof. Let A′ = A/p, m′ = mA′. k = A/m ∼= A′/m′. There is a shortexact sequence of k-vector spaces

(A.1) 0→ p/p∩m2 ∼= p+m2/m2 → m/m2 → m′/(m′)2 = m/(p+m2)→ 0.

The lemma now follows, since

dimk m′/(m′)2 ≥ a

and A′ is regular if and only if dimk m′/(m′)2 = a (Corollary 11.5 [13]).

A.1. Proofs of the basic theorems 165

We now give two related results, Lemma A.4 and Corollary A.5.

Lemma A.4. Suppose that A is a regular local ring of dimension n withmaximal ideal m, p ⊂ A is a prime ideal such that A/p is regular, anddim(A/p) = n−r. Then there exist regular parameters u1, . . . , un in A suchthat p = (u1, . . . , ur) and ur+1, . . . , un map to regular parameters in A/p.

Proof. Consider the exact sequence (A.1). Since A and A′ are regular,there exist regular parameters u1, . . . , un in A such that ur+1, . . . , un mapto regular parameters in A′, and u1, . . . , ur are in p. (u1, . . . , ur) is thus aprime ideal of height r contained in p. Thus p = (u1, . . . , ur).

Corollary A.5. Suppose that Y is a subvariety of dimension t of a varietyX of dimension n. Suppose that q ∈ Y is a non-singular closed point ofboth Y and X. Then there exist regular parameters u1, . . . , un in OX,q suchthat IY,q = (u1, . . . , un−t), and un−t+1, . . . , un map to regular parameters inOY,q. If vn−t+1, . . . , vn are regular parameters in OY,q, there exist regularparameters u1, . . . , un as above such that ui maps to vi for n− t+1 ≤ i ≤ n.Lemma A.6. Let I = (f1, . . . , fm) ⊂ K[x1, . . . , xn] be an ideal. Supposethat P ∈ V (I) and J(f ;x) has rank n at P . Then K(P ) is separable alge-braic over K, and n of the polynomials f1, . . . , fm form a system of regularparameters in OAnK ,P .

Proof. Let m be the ideal of P in K[x1, . . . , xn]. After possibly reindexingthe fi, we may assume that |J(f1, . . . , fn;x)| 6∈ m. By the Nullstellensatz,m is the intersection of all maximal ideals n of K[x1, . . . , xn] containing m.Thus there exists a closed point Q ∈ An

K with maximal ideal n containingm such that |J(f1, . . . , fn;x)| 6∈ n, so that J(f1, . . . , fn;x) has rank n at n.We thus have dimK(Q)D(Q) = n, so that K(Q) is separable over K and(f1, . . . , fn) is a regular system of parameters in OAn,Q by Theorem A.1 andCorollary A.2. Since I ⊂ m ⊂ n, we have P = Q.

Lemma A.7. Suppose that P ∈ AnK = spec(K[x1, . . . , xn]) with ideal

m ⊂ K[x1, . . . , xn]

is such that m has height n − r and t1, . . . , tn−r ∈ mm ⊂ OAn,P . Letαi be the image of xi in K(P ) for 1 ≤ i ≤ r. Then the determinant|J(t1, . . . , tn−r;xr+1, . . . , xn)| is non-zero at P if and only if α1, . . . , αr isa separating transcendence basis of K(P ) over K and t1, . . . , tn−r is a reg-ular system of parameters in OAn,P .

Proof. Suppose that |J(t1, . . . , tn−r;xr+1, . . . , xn)| is non-zero at P . Thereexist s0, s1, . . . , sn−r ∈ K[x1, . . . , xn] such that si ∈ K[x1, . . . , xn] for all i,s0 6∈ m and si

s0= ti for 1 ≤ i ≤ n− r. Then

|J(s1, . . . , sn−r;xr+1, . . . , xn)|


is non-zero at P . Let K∗ = K(α1, . . . , αr), let n be the kernel of

K∗[xr+1, . . . , xn]→ K(P ),

and let s∗i for 1 ≤ i ≤ n− r be the image of si under

K[x1, . . . , xn]→ K∗[xr+1, . . . , xn].

n is the ideal of a point Q ∈ An−rK∗ . We have K(P ) = K∗(Q), and at Q,

|J(s∗1, . . . , s∗n−r;xr+1, . . . , xn)| 6= 0.

By Lemma A.6, K∗(Q) is separable algebraic over K∗ and (s∗1, . . . , s∗n−r)

is a regular system of parameters in OAn−rK∗ ,Q

= K∗[xr+1, . . . , xn]n. SincetrdegK K(P ) = r, α1, . . . , αr is a separating transcendence basis of K(P )over K. Thus K∗ ⊂ K[x1, . . . , xn]m and

K[x1, . . . , xn]m = K∗[xr+1, . . . , xn]n,

so that t1, . . . , tn−r is a regular system of parameters in

OAnK ,P = K[x1, . . . , xn]m.

Now suppose that α1, . . . , αr is a separating transcendence basis of K(P )over K and t1, . . . , tn−r is a regular system of parameters in OAnK ,P =K[x1, . . . , xn]m. Let K∗ = K(α1, . . . , αr), and let n be the kernel of

K∗[xr+1, . . . , xn]→ K(P ).

n is the ideal of a point Q ∈ An−rK∗ . Then K∗(Q) = K(P ) and K∗(Q) is

separable over K∗.We have a natural inclusion K∗ ⊂ K[x1, . . . , xn]m. Thus

K[x1, . . . , xn]m = K∗[xr+1, . . . , xn]n,

and t1, . . . , tn−r is a regular system of parameters in K∗[xr+1, . . . , xn]n. ByCorollary A.2, |J(t1, . . . , tn−r;xr+1, . . . , xn)| 6= 0 at Q, and thus it is also6= 0 at P .

Theorem A.8. Suppose that P ∈ AnK = spec(K[x1, . . . , xn]) and t1, . . . , tn−r

are regular parameters in OAn,P . Then rank(J(t;x)) = n − r at P if andonly if K(P ) is separably generated over K.

Proof. Let αi, 1 ≤ i ≤ n, be the images of xi in K(P ), and let m be theideal of P in K[x1, . . . , xn].

Suppose that rank(J(t;x)) = n− r at P . After possibly reindexing thexi and tj , we have |J(t1, . . . , tn−r;xr+1, . . . , xn)| 6= 0 at P . Now K(P ) isseparably generated over K by Lemma A.7.

Suppose that K(P ) is separably generated over K. Let ζ1, . . . , ζr bea separating transcendence basis of K(P ) over K. There exists Ψj(x) ∈K[x1, . . . , xn] for 0 ≤ j ≤ r with Ψ0(x) 6∈ m such that ζj = Ψj(α)

Ψ0(α) . Let

A.1. Proofs of the basic theorems 167

n be the kernel of the K-algebra homomorphism K[x1, . . . , xn+r] → K(P )defined by xi 7→ αi for 1 ≤ i ≤ n, xi 7→ ζi−n if n < i. n is the ideal ofa point Q ∈ An+r

K . Let Φ0(x),Φ1(x), . . . ,Φn−r(x) ∈ K[x1, . . . , xn] be suchthat Φ0 6∈ m and

ti =Φi(x)Φ0(x)

for 1 ≤ i ≤ n− r.

Let Φn−r+j(x) = xn+jΨ0(x)−Ψj(x) for 1 ≤ j ≤ r. Let I ⊂ K[x1, . . . , xn+r]be the ideal generated by Φ1, . . . ,Φn; thus I ⊂ n. We will show thatΦ1, . . . ,Φn are in fact a regular system of parameters in

OAn+rK ,Q = K[x1, . . . , xn+r]n.

It suffices to show that if F (x) ∈ n, then there exists

A(x) ∈ K[x1, . . . , xn+r]− n

such that AF ∈ I. Given such an F , we have an expansion

Ψ0(x)sF (x) =r∑j=1

Bj(x)Φn−r+j(x) +G(x1, x2, . . . , xn),

where Bj(x) ∈ K[x1, . . . , xn+r] and s is a non-negative integer. Now F ∈ nimplies G ∈ m. Thus there exists an expansion

G =n−r∑i=1

fiti

with fi ∈ K[x1, . . . , xn]m, and there exist h ∈ K[x1, . . . , xn] − m andCi ∈ K[x1, . . . , xn] such that

hG =r∑i=1

CiΦi(x)

and hΨ0(x)sF (x) ∈ I.Since Φ1, . . . ,Φn is a regular system of parameters in K[x1, . . . , xn+r]n

and ζ1, . . . , ζr is a separating transcendence basis of K(Q) over K, it followsthat

|J(Φ1, . . . ,Φn;x1, . . . , xn)|is non-zero at Q by Lemma A.7. Thus J(Φ1, . . . ,Φn−r;x1, . . . , xn) is ofrank n− r at P , and by Lemma A.7, α1, . . . , αn must contain a separatingtranscendence basis of K(P ) over K.

Proof that smoothness at a point is well defined

Suppose that X is a variety of dimension s over a field K and P ∈ X. InDefinition 2.6 we gave a definition for smoothness of X over K at P which


depended on choices of an affine neighborhood U = spec(R) of P and apresentation R ∼= K[x1, . . . , xn]/I with I = (f1, . . . , fm).

We now show that this definition is well defined, so that it is independentof all choices U , f and x.

Consider the sheaf of differentials Ω1X/K . We have, by the “second fun-

damental exact sequence” (Theorem 25.2 [66]), a presentation

(A.2) (K[x1, . . . , xn])m J(f ;x)→ Ω1

K[x1,...,xn]/K ⊗R→ Ω1R/K → 0.

Tensoring (A.2) with K(P ), we see that Ω1R/K ⊗K(P ) has rank s if and

only if J(f ;x) has rank n− s at P .

Proof of Theorem 2.8

Let s = dimX. There exists an affine neighborhood U = spec(R)of P in X such that R = K[x1, . . . , xn]/I, I = (f1, . . . , fm). Let mP ∈spec(K[x1, . . . , xn]) be the ideal of P in An

K . Let

t = dim(K[x1, . . . , xn]/mP ).

Suppose that K(P ) is separably generated over K and P is a non-singular point of X. By Theorem A.8, dP : mP /m

2P → D(P ) is an isomor-

phism. By Lemma A.3, dimK(P ) dP (I) = n−s (take A = OAn,P , p = ImP inthe statement of the lemma, so that b = n− t, a = s− t), which is equivalentto the condition that J(f ;x) has rank n − s at P . Thus X is smooth overK at P .

Suppose that X is smooth at P , so that J(f ;x) has rank n − s at P .Then dimK(P ) dP (I) = n− s, so that dimK(P )(I +m2

P )/m2P ≥ n− s. Thus

P is a non-singular point of X by Lemma A.3.

Remark A.9. With the notation of Theorem 2.8 and it’s proof, for anypoint P ∈ V (I), we have

dimK(P )(I +m2P )/m2

P ≤ n− s

by Lemma A.3, so that J(f ;x) has rank ≤ n− s at P .

Proof of Theorem 2.7

It suffices to prove that the locus of smooth points of X lying on anopen affine subset U = spec(R) of X of the form of Definition 2.6 is open.Let A = In−s(J(f ;x)). Then P ∈ U − V (A) if and only if J(f ;x) has rank

A.2. Non-singularity and uniformizing parameters 169

greater than or equal to n−s at P , which in turn holds if and only if J(f ;x)has rank n− s at P by Remark A.9.

Proof of Theorem 2.10 when K is perfect

Suppose thatK is perfect. The openness of the set of non-singular pointsof X follows from Theorem 2.7 and Corollary 2.9.

Let η ∈ X be the generic point of an irreducible component of X. ThenOX,η is a field which is a regular local ring. Thus η is a non-singular point.We conclude that the non-singular points of X are dense in X.

A.2. Non-singularity and uniformizing parameters

Theorem A.10. Suppose that X is a variety of dimension r over a fieldK, and P ∈ X is a closed point such that X is non-singular at P and K(P )is separable over K.

1. Suppose that y1, . . . , yr are regular parameters in OX,P . Then

Ω1X/K,P = dy1OX,P ⊕ · · · ⊕ dyrOX,P .

2. Let m be the ideal of P in OX,P . Suppose that f1, . . . , fr ∈ m ⊂OX,P and df1, . . . , dfr generate Ω1

X/K,P as an OX,P module. Thenf1, . . . , fr are regular parameters in OX,P .

Proof. We can restrict to an affine neighborhood of X, and assume that

X = spec(K[x1, . . . , xn]/I).

The conclusions of part 1 of the theorem follow from Theorem A.8 whenX = An

K .In the general case of 1, by Corollary A.5 there exist regular param-

eters (y1, . . . , yn) in A = K[x1, . . . , xn]m (where m is the ideal of P inspec(K[x1, . . . , xn]) such that IA = (yr+1, . . . , yn) and (y1, . . . , yr) map tothe regular parameters (y1, . . . , yr) in B = A/IA. By the second funda-mental exact sequence (Theorem 25.2 [66]) we have an exact sequence ofB-modules

IA/I2A→ Ω1A/K ⊗A B → Ω1

B/K → 0.

This sequence is thus a split short exact sequence of free B-modules, withΩ1B/K

∼= dy1B⊕· · ·⊕dyrB, since IA/I2A ∼= yr+1B⊕· · ·⊕ynB and Ω1A/K⊗A

B ∼= dy1B ⊕ · · · ⊕ dynB.Suppose that the assumptions of 2 hold. Let R = OX,P . Let y1, . . . , yr

be regular parameters in R. Then dy1, . . . , dyr is a basis of the R-module


Ω1X/K,P by part 1 of this theorem. There are aij ∈ R such that

fi =r∑j=1

aijyj , 1 ≤ j ≤ r.

We have expressions

dfi =r∑j=1

aijdyj + (r∑j=1

daijyj).

Let wi =∑r

j=1 aijdyj for 1 ≤ i ≤ r. By Nakayama’s Lemma we have thatthe wi generate Ω1

X/K,P as an R-module. Thus |(aij)| 6∈ m and (aij) isinvertible over R, so that f1, . . . , fr generate m.

Lemma A.11. Suppose that X is a variety of dimension r over a perfectfield K. Suppose that P ∈ X is a closed point such that X is non-singularat P , and y1, . . . , yr are regular parameters in OX,P . Then there exists anaffine neighborhood U = spec(R) of P in X such that y1, . . . , yr ∈ R, thenatural inclusion S = K[y1, . . . , yr] → R induces a morphism π : U → Ar

K

such that

(A.3) Ω1R/K = dy1R⊕ · · · ⊕ dyrR,

DerK(R,R) = HomR(Ω1R/K , R) =

∂

∂y1R⊕ · · · ⊕ ∂

∂yrR

(with ∂∂yi

(dyj) = δij), and the following condition holds. Suppose that a ∈ Uis a closed point, b = π(a), and (x1, . . . , xr) are regular parameters in OArK ,b.Then (x1, . . . , xr) are regular parameters in OX,a.

y1, . . . , yr are called uniformizing parameters on U .

Proof. By Theorem A.10, there exists an affine neighborhood U = spec(R)of P in X such that y1, . . . , yr ∈ R and dy1, . . . , dyr is a free basis of Ω1

R/K .Consider the natural inclusion S = K[y1, . . . , yr] → R, with induced mor-phism π : spec(R) → spec(S). Suppose that a ∈ U is a closed point withmaximal ideal m in R, and b = π(a), with maximal ideal n in S. Supposethat (x1, . . . , xr) are regular parameters in B = Sn. Let A = Rm. By con-struction, Ω1

S/K is generated by dy1, . . . , dyr as an S-module. Thus Ω1A/K is

generated by Ω1B/K as an A-module. By 1 of Theorem A.10, dx1, . . . , dxr

generate Ω1B/K as a B-module, and hence they generate Ω1

A/K as an A-module. By 2 of Theorem A.10, x1, . . . , xr is a regular system of parametersin A.

A.3. Higher derivations 171

A.3. Higher derivations

Suppose that K is a field and A is a K-algebra. A higher derivation of Aover K of length m is a sequence

D = (D0, D1, . . . , Dm)

if m <∞, and a sequence

D = (D0, D1, . . .)

if m =∞, of K-linear maps Di : A→ A such that D0 = id and

(A.4) Di(xy) =i∑

j=1

Dj(x)Dj−i(y)

for 1 ≤ i ≤ m and x, y ∈ A.Let t be an indeterminate. D = (D0, D1, . . .) is a higher derivation of

A over K (of length ∞) precisely when the map Et : A→ A[[t]] defined byEt(f) =

∑∞i=0Di(f)ti is a K-algebra homomorphism with D0(f) = f , and

D = (D0, D1, . . . , Dm) is a higher derivation of A over K of length m pre-cisely when the map Et : A→ A[t]/(tm+1) defined by Et(f) =

∑mi=0Di(f)ti

is a K-algebra homomorphism with D0(f) = f .

Example A.12. Suppose that A = K[y1, . . . , yr] is a polynomial ring overa field K. For 1 ≤ i ≤ r, let εj be the vector in Nr with a 1 in the j-th coefficient and 0 everywhere else. For f =

∑ai1,...,iry

i11 · · · yirr ∈ A and

m ∈ N, define

D∗mεj (f) =

∑ai1,...,ir

(ijm

)yi11 · · · y

ij−mj · · · yirr .

Then D∗j = (D∗

0, D∗εj , D

∗2εj, . . .) is a higher derivation of A over K of length

∞ for 1 ≤ j ≤ r. Observe that the D∗j are iterative; that is,

Diεj Dkεj =(i+ k

i

)D(i+k)εj

for all i, k. Define

D∗i1,...,ir = D∗

irεr D∗ir−1εr−1

· · · D∗i1ε1

for (i1, . . . , ir) ∈ Nr. The D∗i1,...,ir

commute with each other.

The D∗i1,...,ir

are called Hasse-Schmidt derivations [56]. If K has charac-teristic zero, then

D∗i1,...,ir =

1i1!i2! · · · ir!

∂i1+···+ir

∂yi11 · · · ∂yirr

.


Lemma A.13. Suppose that X is a variety of dimension r over a perfectfield K. Suppose that P ∈ X is a closed point such that X is non-singularat P , y1, . . . , yr are regular parameters in OX,P , and U = spec(R) is anaffine neighborhood of P in X such that the conclusions of Lemma A.11hold. Then there are differential operators Di1,...,ir on R, which are uniquelydetermined on R by the differential operators D∗

i1,...,iron S of Example A.12,

such that if f ∈ OX,P and

f =∑

ai1,...,iryi11 · · · y

irr

in OX,P ∼= K(P )[[y1, . . . , yr]], where we identify K(P ) with the coefficientfield of OX,P containing K, then ai1,...,ir = Di1,...,ir(f)(P ) is the residue ofDi1,...,ir(f) in K(P ) for all indices i1, . . . , ir.

Proof. If K has characteristic zero, we can take

(A.5) Di1,...,ir =1

i1! · · · ir!∂i1+···+ir

∂yi11 · · · ∂yirr

.

Suppose that K has characteristic p ≥ 0. Consider the inclusion ofK-algebras

S = K[y1, . . . , yr]→ R

of Lemma A.11, with induced morphism

π : U = spec(R)→ V = spec(S).

Let D∗j for 1 ≤ j ≤ r be the higher order derivation of S over K defined in

Example A.12. We have Ω1R/S = 0 by 1 of Theorem A.10, (A.3) and the “first

fundamental exact sequence” (Theorem 25.1 [66]). Thus π is non-ramifiedby Corollary IV.17.4.2 [45]. By Theorem IV.18.10.1 [45] and PropositionIV.6.15.6 [45], π is etale.

We will now prove that D∗j extends uniquely to a higher derivation Dj of

R over K for 1 ≤ j ≤ r. It suffices to prove that (D∗0, D

∗εj , . . . , D

∗mεj ) extends

to a higher derivation of R over K for all m. The extension of D∗0 = id

to D0 = id is immediate. Suppose that an extension (D0, D1, . . . , Dm) of(D∗

0, D∗εj , . . . , D

∗mεj ) has been constructed. Then we have a commutative

diagram of K-algebra homomorphisms

Rg→ R[t]/(tm+1)

↑ ↑S

f→ R[t]/(tm+2)

where

g(a) =m∑i=0

Diεj (a)tj ,

A.3. Higher derivations 173

f(a) =m+1∑i=0

D∗iεj (a)t

j .

Since (tm+1)R[t]/(tm+2) is an ideal of square zero, and S → R is formallyetale (Definitions IV.17.3.1 and 17.1.1 [45]), there exists a unique K-algebrahomomorphism h making a commutative diagram

Rg→ R[t]/(tm+1)

↑h ↑

Sf→ R[t]/(tm+2)

Thus there is a unique extension of (D∗0, D

∗εj , . . . , D

∗(m+1)εj

) to a higherderivation of R.

Set Di1,...,ir = Dirεr · · · Di1ε1 .Let A = OX,P . Since K(P ) is separable over K, and by Exercise 3.9 of

Section 3.2, we can identify the residue field K(P ) of A with the coefficientfield of A which contains K. Thus we have natural inclusions

K[y1, . . . , yr] ⊂ A ⊂ A ∼= K(P )[[y1, . . . , yr]].

We will now show that Dεj extends uniquely to a higher derivation of Aover K for 1 ≤ j ≤ r. Let m be the maximal ideal of A. Let t be anindeterminate. Then A[[t]] is a complete local ring with maximal ideal

n = mA[[t]] + tA[[t]].

Define Et : A → A[[t]] by Et(f) =∑∞

n=0Dnεj (f)tn. Et is a K-algebrahomomorphism, since Dεj is a higher derivation. Et is continuous in them-adic topology, as Et(ma) ⊂ na for all a. Thus Et extends uniquely toa K-algebra homomorphsim A → A[[t]]. Thus Dεj extends uniquely to ahigher derivation of A over K.

We can thus extend Di1,...,ir uniquely to A. Since K(P ) is algebraic overK, and Di1,...,ir extends D∗

i1,...,ir, it follows that Di1,...,ir acts on A as stated

in the theorem.

Remark A.14. Suppose that the assumptions of Lemma A.13 hold, andthat K is an algebraically closed field. Then the following properties hold.

1. Suppose that Q ∈ U is a closed point and a is the correspondingmaximal ideal in R. Then a = (y1, . . . , yr), where yi = yi−αi withαi = yi(Q) ∈ K for 1 ≤ i ≤ r, and K[[y1, . . . , yr]] = Ra = OU,Q.

2. The differential operators Di1,...,in on R are such that if f ∈ R, anda is a maximal ideal in R, then, with the notation of 1,

f =∑

ai1,...,iryi11 · · · y

irr


in K[[y1, . . . , yr]] = Ra, where ai1,...,ir = Di1,...,ir(f)(Q) is theresidue of Di1,...,ir(f) in K(Q) for all indices i1, . . . , ir.

Proof. The ideal of π(Q) in S is (y1−α1, . . . , yr−αr) with αi = yi(Q) ∈ K.Let yi = yi − αi for 1 ≤ i ≤ r. We have a = (y1, . . . , yr) by Theorem A.102, and 1 follows. The derivations ∂

∂y1, . . . , ∂

∂yrare exactly the derivations

∂∂y1

, . . . , ∂∂yr

on S of Example A.12, so they extend to the differential op-erators D∗

i1,...,iron S of Example A.12. Now 2 follows from Lemma A.13

applied to Q and y1, . . . , yr.

Lemma A.15. Let notation be as in Lemmas A.11 and A.13. In particular,we have a fixed closed point P ∈ U = spec(R) and differential operatorsDi1,...,ir on R. Let I ⊂ R be an ideal. Define an ideal Jm,U ⊂ R by

Jm,U = Di1,...,ir(f) | f ∈ I and i1 + · · ·+ ir < m.

Suppose that Q ∈ U is a closed point, with corresponding ideal mQ in R,and suppose that (z1, . . . , zr) is a regular system of parameters in mQ.

Let Di1,...,ir be the differential operators constructed on OW,Q by the con-struction of Lemmas A.11 and A.13. Then

(Jm,U )mQ = Di1,...,ir(f) | f ∈ I and i1 + · · ·+ ir < m.

Proof. When K has characteristic zero, this follows directly from differen-tiation and (A.5).

Assume that K has positive characteristic. By Lemma A.13 and Theo-rem IV.16.11.2 [45], U is differentiably smooth over spec(K) and Di1,...,ir |i1, . . . , ir < m is an R-basis of the differential operators of order less thanm on R. Since dz1, . . . , dzr is a basis of Ω1

RmQ/K(by Lemma A.11),

Di1,...,ir | i1, . . . , ir < m is an RmQ-basis of the differential operators oforder less than m on RmQ by Theorem IV.16.11.2 [45]. The lemma fol-lows.

A.4. Upper semi-continuity of νq(I)

Lemma A.16. Suppose that W is a non-singular variety over a perfectfield K and K ′ is an algebraic field extension of K. Let W ′ = W ×K K ′,with projection π1 : W ′ → W . Then W ′ is a non-singular variety overK ′. Suppose that q ∈ W is a closed point. Then there exists a closed pointq′ ∈W ′ such that π1(q′) = q. Furthermore,

1. Suppose that (f1, . . . , fr) is a regular system of parameters in OW,q.Then (f1, . . . , fr) is a regular system of parameters in OW ′,q′.

A.4. Upper semi-continuity of νq(I) 175

2. If I ⊂ OW,q is an ideal, then

(IOW ′,q′) ∩ OW,q = I.

Proof. The fact that W ′ is non-singular follows from Theorem 2.8 and thedefinition of smoothness. Let q′ ∈W ′ be a closed point such that π1(q′) = q.

We first prove 1. Suppose that U = spec(R) is an affine neighborhoodof q in W . There is a presentation R = K[x1, . . . , xn]/I. We further havethat π−1

1 (U) = spec(R′), with R′ = K ′[x1, . . . , xn]/I(K ′[x1, . . . , xn]). Let mbe the ideal of q in K[x1, . . . , xn], and n the ideal of q′ in K ′[x1, . . . , xn].By the exact sequence of Lemma A.3, there exists a regular system ofparameters f1, . . . , fn in K[x1, . . . , xn]m, such that Im = (f r+1, . . . , fn)mand f i maps to fi for 1 ≤ i ≤ r. By Corollary A.2, the determinant|J(f1, . . . , fn;x1, . . . , xn)| is non-zero at q, so it is non-zero at q′. By Corol-lary A.2, f1, . . . , fn is a regular system of parameters in K ′[x1, . . . , xn]n.Thus f1, . . . , fr is a regular system of parameters in R′

n = OW ′,q′ .We will now prove 2. Since n maps to a maximal ideal of OW,q ⊗K K ′,

OW ′,q′ = (OW,q ⊗K K ′)n

( n denotes completion with respect to nOW,q ⊗K K ′). Thus OW ′,q is faith-fully flat over OW,q, and (IOW ′,q′)∩OW,q = I (by Theorem 7.5 (ii) [66]).

Suppose that (R,m) is a regular local ring containing a field K of char-acteristic zero. Let K ′ be the residue field of R. Suppose that J ⊂ R is anideal. We define the order of J in R to be

νR(J) = maxb | J ⊂ mb.

If J is a locally principal ideal, then the order of J is its multiplicity. How-ever, order and multiplicity are different in general.

Definition A.17. Suppose that q is a point on a variety W and J ⊂ OWis an ideal sheaf. We denote

νq(J) = νOW,q(JOW,q).

If X ⊂W is a subvariety, we denote

νq(X) = νq(IX).

Remark A.18. Let notation be as in Lemma A.16 (except we allow K tobe an arbitrary, not necessarily perfect field). Observe that if q ∈ W andq′ ∈W ′ are such that π(q′) = q and J ⊂ OW is an ideal sheaf, then

νOW,q(Jq) = νOW,q(JqOW,q) = νOW ′,q′ (JqOW ′,q′) = νOW ′,q′(JqOW ′,q′).


Suppose that X is a noetherian topological space and I is a totallyordered set. f : X → I is said to be an upper semi-continuous function iffor any α ∈ I the set

q ∈ X | f(q) ≥ αis a closed subset of X.

Suppose that X is a subvariety of a non-singular variety W . Define

νX : W → N

by νX(q) = νq(X) for q ∈W . The order νq(X) is defined in Definition A.17.

Theorem A.19. Suppose that K is a perfect field, W is a non-singularvariety over K and I is an ideal sheaf on W . Then

νq(I) : W → N

is an upper semi-continuous function.

Proof. Suppose that m ∈ N. Suppose that U = spec(R) is an open affinesubset of W such that the conclusions of Lemma A.11 hold on W . SetI = Γ(U, I).

With the notation of Lemmas A.11 and A.13, define an ideal Jm,U ⊂ Rby

Jm,U = Di1,...,ir(g) | g ∈ I and i1 + · · · ir < m.By Lemma A.15, this definition is independent of the choice of y1, . . . , yr sat-isfying the conclusions of Lemma A.11, so our definition of Jm,U determinesa sheaf of ideals Jm on W .

We will show that

V (Jm) = η ∈W | νη(I) ≥ m,from which upper semi-continuity of νq(I) follows.

Suppose that η ∈ W is a point. Let Y be the closure of η in W , andsuppose that P ∈ Y is a closed point such that Y is non-singular at P .

By Corollary A.5, Lemma A.11 and Lemma A.15, there exists an affineneighborhood U = spec(R) of P in W such that there are y1, . . . , yr ∈ Rwith the properties that y1 = . . . = yt = 0 (with t ≤ r) are local equationsof Y in U and y1, . . . , yr satisfy the conclusions of Lemma A.11.

For g ∈ IP , consider the expansion

g =∑

ai1,...,iryi11 · · · y

irr

in OW,P ∼= K(P )[[y1, . . . , yr]], where we have identified K(P ) with the coef-ficient field of OW,P containing K and (with the notation of Lemma A.13)

ai1,...,ir = Di1,...,ir(g)(P ).

A.4. Upper semi-continuity of νq(I) 177

Then η ∈ V (Jm) is equivalent to Di1,...,ir(g) ∈ (y1, . . . , yt) wheneverg ∈ IP and i1 + · · ·+ ir < m, which is equivalent to

ai1,...,ir = Di1,...,ir(g)(P ) = 0

if g ∈ IP and i1 + · · ·+ it < m, which in turn holds if and only if

g ∈ ImY,P ∩ OW,P = ImY,Pfor all g ∈ IP , where the last equality is by Lemma A.16. Localizing OW,Pat η, we see that νη(I) ≥ m if and only if η ∈ V (Jm).

Definition A.20. Suppose that W is a non-singular variety over a perfectfield K and X ⊂W is a subvariety.

For b ∈ N, define

Singb(X) = q ∈W | νq(W ) ≥ b.

Singb(X) is a closed subset of W by Theorem A.19. Let

r = maxνX(q) | q ∈W.

Suppose that q ∈ Singr(X). A subvariety H of an affine neighborhood U ofq in W is called a hypersurface of maximal contact for X at q if

1. Singr(X) ∩ U ⊂ H, and2. if

Wnπn→ · · · →W1

π1→W

is a sequence of monoidal transforms such that for all i, πi is cen-tered at a non-singular subvariety Yi ⊂ Singr(Xi), where Xi is thestrict transform of X on Wi, then the strict transform Hn of H onUn = Wn ×W U is such that Singr(Xn) ∩ Un ⊂ Hn.

With the above assumptions, a non-singular codimension one subvarietyH of U = spec(OW,q) is called a formal hypersurface of maximal contact forX at q if 1 and 2 above hold for U = spec(OW,q).

Remark A.21. Suppose that W is a non-singular variety over a perfectfield K and X ⊂W is a subvariety, q ∈W . Let T = OW,q, and define

Singb(OX,q) = P ∈ spec(T ) | ν(IX,qTP ) ≥ b.

Then Singb(OX,q) is a closed set. If J ⊂ OW is a reduced ideal sheafwhose support is Singb(X), then JqT is a reduced ideal whose support isSingb(OX,q).

Proof. Let U = spec(R) be an affine neighborhood of q in W satisfying theconclusions of Lemma A.13 (with p = q), and let notation be as in LemmaA.13 (with p = q). With the further notation of Theorem A.19 (and I = IX)we see that Γ(U, J) =

√Jb,U . JqT is a reduced ideal by Theorem 3.6.


Suppose that P ∈ Singb(OX,q). Then νTP (g) ≥ b for all g ∈ IX,q.Thus g ∈ P (b) ⊂ T , and there exists h ∈ T − P such that gh ∈ P b. Nowinduction in the formula (A.4) shows that Di1,...,ir(gh) ∈ P b−i1,...,ir andDi1,...,ir(g) ∈ P (b−i1−···−ir) ⊂ P if i1 + · · ·+ ir < b. Thus JqT ⊂ P .

Now suppose that P ∈ spec(T ) and JqT ⊂ P . Let S = OW,q. IfJq = P1∩· · ·∩Pm is a primary decomposition, then all primes Pi are minimalsince Jq is reduced. By Theorem 3.6 there are primes Pij in T and a functionσ(i) such that PiT = Pi1 ∩ · · · ∩ Piσ(i) is a minimal primary decomposition.Thus JqTPij = PiTPij = PijTPij for all i, j. We have

⋂ij Pij ⊂ P , so Pij ⊂ P

for some i, j. Moreover, IX,q ⊂ P bi SPi implies IX,q ⊂ P bijTPij , which in turnimplies IX,q ⊂ P bTPij . Thus νTP (IX,qTP ) ≥ b and P ∈ Singb(OX,q).

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Index

Abhyankar, 12, 28, 42, 45, 130, 131, 152,153, 157, 158, 160

Abramovich, 9

Albanese, 55

approximate manifold, 106

Atiyah, viii

Bierstone, 61

Bravo, 61

Cano, 13

Chevalley, 24

Cohen, 22

completion of the square, 24

Cossart, 131

curve, 2

de Jong, 61, 131

derivations

Hasse-Schmidt, 171

higher, 171

Eisenbud, viii

Encinas, 61

exceptional divisor, 20, 39, 40

exceptional locus, 39, 40

Giraud, 70

Harris, viii

Hartshorne, viii, 2

Hauser, 13, 61, 132

Heinzer, 13, 157

Hensel’s Lemma, 23

Hironaka, 2, 15, 61, 62, 130

hypersurface, 2

intersection

reduced set-theoretic, 62

scheme-theoretic, 2, 62

isolated subgroup, 134

Jung, 55

Karu, 9

Kawasaki, 11

Kuhlmann, 13

Levi, 55

Lipman, 130

local uniformization, 12, 130, 138, 147, 156

Macaulayfication, 11

Macdonald, viii

MacLane, 137

Matsuki, 9

maximal contact

formal curve, 28

formal hypersurface, 177

hypersurface, 56, 67, 70, 131, 177

Milman, 61

Moh, 61

monoidal transform, 16, 39, 40

monomialization, 13, 157

morphism

birational, 9

monomial, 14

proper, 9

multiplicity, 3, 19, 25, 175

Nagata, 11

Narasimhan, 131

Newton, 1, 3, 6, 12

185

186 Index

Newton Polygon, 34

normalization, 3, 10, 12, 26

ord, 22, 138

order, 3, 19, 25, 138

νJ (q), 63

νR, 175

νX , 63, 176

νq(J), 175

νq(X), 175

upper semi-continuous, 176

Piltant, 13

principalization of ideals, 40, 43, 99

Puiseux, 6

Puiseux series, 6, 11, 137

resolution of differential forms, 13

resolution of indeterminacy, 41, 100

resolution of singularities, 9, 40, 41, 61,100–102

curve, 10, 12, 26, 27, 30, 38, 39, 42

embedded, 41, 56, 99

equivariant, 103

excellent surface, 130

positive characteristic, 10, 12, 13, 30, 42,105, 152

surface, 12, 45, 105, 152

three-fold, 12, 152

resolution of vector fields, 13

ring

affine, 2

regular, 7

Rotthaus, 9, 13

scheme

Cohen-Macaulay, 11

excellent, 9

non-singular, 7

quasi-excellent, 9

smooth, 7

Schilling, 137

Seidenberg, 13

Serre, 10

Shannon, 151

singularity

Sing(J, b), 66

Singb(X), 45, 105, 177

Singb(OX,q), 177

SNCs, 29, 40, 49

Spivakovsky, 13

surface, 2

tangent space, 8

Teissier, 13

three-fold, 2

transformstrict, 20, 25, 38, 65total, 21, 25, 37weak, 65

Tschirnhausen transformation, 24, 28, 51,56, 61, 67, 70

uniformizing parameters, 170upper semi-continuous, 176

valuation, 12, 133dimension, 134discrete, 134rank, 134rational rank, 136ring, 134

variety, 2integral, 2quasi-complete, 15subvariety, 2

Villamayor, 61

Walker, 55Weierstrass preparation theorem, 24Wiegand, 13Wlodarczyk, 9, 61

Zariski, 1, 12, 55, 133, 147, 148, 152, 156

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