on the legacy of free divisors ii: free* divisors and ... · that the simple boundary singularities...

MOSCOW MATHEMATICAL JOURNALVolume 3, Number 2, April–June 2003, Pages 361–395

ON THE LEGACY OF FREE DIVISORS II:FREE* DIVISORS AND COMPLETE INTERSECTIONS

JAMES DAMON

To Vladimir Arnold who has had such an immense influence on mathematics

Abstract. We provide a criterion that for an equivalence group G onholomorphic germs, the discriminant of a G-versal unfolding is a freedivisor. The criterion is in terms of the discriminant being Cohen–Macaulay and generically having Morse-type singularities. When eitherof these conditions fails, we provide a criterion that the discriminanthave a weaker free* divisor structure. For nonlinear sections of a free*divisor V , we obtain a formula for the number of singular vanishingcycles by modifying an earlier formula obtained with David Mond andtaking into account virtual singularities.

2000 Math. Subj. Class. Primary: 14B07, 14M12, 32S30; Secondary: 13C12,

14B10.

Key words and phrases. Discriminants, versal unfoldings, free divisors, free*divisors, liftable vector fields, Morse-type singularities, Cohen–Macaulay con-dition.

Introduction

The notion of free divisors first appeared in investigations of discriminants ofversal unfoldings of isolated hypersurface singularities. They appeared implicitlyin Arnold’s work on wave front evolution via functions on discriminants of stableAk singularities [A], and they were formally defined and investigated by Saito [Sa].Since then, the notion of a free divisor has played a fundamental role for understand-ing the structure of nonisolated singularities such as discriminants, bifurcation sets,hyperplane arrangements, etc. As just one example, we mention that the results ofArnold [A3] and Brieskorn [B] on the factorization of the Poincare polynomial forthe homology of complement of a reflection hyperplane arrangement were shownby Terao [Ter1], [OT] to follow from the freeness of the arrangement. At the sametime, the richness of the class of free divisors has also become increasingly evident,as researchers have established that a number of natural constructions yield freedivisors (e. g. Saito [Sa], Looijenga [L], Terao [Ter1], [Ter2], Bruce [Br], Van Straten[VS], Mond [Mo3] [MVS], Goryunov [Go], Grandjean [Gr] and this author [D6] (andsee [D6] for other references).

Received May 15, 2002.Supported in part by the National Science Foundation grant DMS-0103862.

c©2003 Independent University of Moscow

361

362 J. DAMON

Moreover, (V, 0) being a free divisor allows us to investigate the vanishing topol-ogy (of singular Milnor fibers) of nonisolated singularities arising as nonlinear sec-tions [DM], [D3], [D4]. This includes the higher multiplicities of V itself (whichincludes the topology of the complex link).

In the first part of this paper [D6], we considered how a free divisor (V, 0) ⊂(Cp, 0) passes on its freeness to divisors arising as KV -discriminants DV (F ) forversal unfoldings F of nonlinear sections f0 : (Cn, 0) → (Cp, 0) of V . This generalconstruction encompasses a number of the specific constructions mentioned above.For DV (F ) to be a free divisor, it is sufficient that V generically has “Morse-typesingularities” in dimension n where n < hn(V ), the holonomic codimension of V .This means that there are sections through points of V which exhibit many of thesame properties as classical Morse singularities (appropriately understood). Thereare many basic examples which satisfy this condition. They include bifurcationsets for a certain well-defined class of finitely A-determined complete intersectiongerms, smoothings of certain classes of isolated complete intersection singularitiesand space curve singularities, discriminants for generic hyperplane arrangements,etc.

In this continuation of Part I [D6] we address questions concerning the structureof discriminants and their vanishing topology for many different groups of equiv-alences involving nonisolated singularities. These include: nonisolated completeintersections defined as nonlinear sections of fixed complete intersections, singular-ities of functions, mappings, or divisors on nonisolated complete intersections, orequivalence of functions and mappings preserving a hypersurface V in the source(which can be viewed as boundary singularities for the singular boundary V in thesense of Arnold [A2], Siersma [Si], Lyashko [Ly], Bruce and Giblin [BG1] [BG2],and Tari [Ta], or singularities at infinity with V denoting the divisor at infinity).

We address three questions concerning discriminants for such equivalences.

(1) Is there a general criterion ensuring that for an equivalence group G, thediscriminants for G-versal unfoldings are free divisors?

(2) If this criterion fails, is there a weaker substitute for the notion of freedivisor which still generally applies?

(3) With this weaker notion, is it still possible to deduce results about thevanishing topology of nonlinear sections of such divisors?

Our goal in this paper is to give answers to these questions. First we give acriterion for freeness of discriminants which is described by the motto

Cohen–Macaulay of codim 1 + Genericity of Morse Type Singularities =⇒Freeness of Discriminants.

We consider the class of geometrically defined subgroups G of A or K so thebasic theorems of singularity theory are valid. We introduce the notion of G beingCohen–Macaulay. The geometric notion of “genericity of Morse-type singularities”implies a condition of “genericity of G-liftable vector fields”. Then, the above mottois given form by Theorem 1: G being Cohen–Macaulay with genericity of G-liftablevector fields implies the freeness of discriminants for G-versal unfoldings.

LEGACY OF FREE DIVISORS II 363

For the second question, we consider what remains true when either of theseconditions fail. For example, for a free divisor (V, 0), provided n < hn(V ) thegroup KV is Cohen–Macaulay; but there are many examples where genericity ofMorse-type singularities fails (see [D6]). However, in a very well-defined sense, thediscriminants (and bifurcation sets) for these cases “just fail” to be free divisors.We are led to introduce the weaker notion of a “free* divisor” (pronounced “freestar divisor”). A “free* divisor” structure is defined by a free submodule of vectorfields which are tangent to the discriminant, but define the discriminant with anonreduced structure. In this sense, the “*” denotes that the divisor is “free” withan asterisque indicating the lack of reduced structure.

Such a structure is highly nonunique (there is even a trivial free* divisor struc-ture, which carries no new useful information). Hence, the importance of a particu-lar free* divisor structure depends upon: first, its being defined in terms of naturalproperties of the divisor, and second the extent to which it can still be used toobtain useful topological and geometric information about the divisor.

In Theorem 2, we deduce that if G is only Cohen–Macaulay, then discriminantsfor G-versal unfoldings are still free* divisors for the module of G-liftable vectorfields. We deduce Theorem 3 that for a free divisor V , even in the absence ofgenericity of Morse-type singularities, provided n < hn(V ) the KV -discriminantsof versal unfoldings are always free* divisors for the module of KV -liftable vectorfields. We conclude (Corollaries 2.17–2.19) that for finitely A determined germsin the nice dimensions, for complete intersection germs on ICIS, and for “almostfree” nonlinear arrangements, the bifurcation sets are always free* divisors for themodule of liftable vector fields.

These theorems are applied in Part III of this paper to boundary singularitiesfor singular boundary V . A standard group G = A, R+, K which also preservesV in the source is denoted V G. The equivalence group VK is Cohen–Macaulay.Surprisingly whether VK generically has Morse type singularities depends uponwhether V does for KV -equivalence. Hence, Arnold [A2] and Lyashko [Ly] showthat the simple boundary singularities of functions have versal unfoldings whosediscriminants are free divisors. Our results show this fails in general; and when itholds it does so for functions and complete intersection mappings. This leads to thefreeness of the discriminants for VA stable or VR+-versal germs and VA-bifurcationsets.

The second way the motto may fail is if the group G is not Cohen–Macaulay. Itmay still be possible that G-discriminants may still behave as if G were. We see thisby introducing the notion of a Cohen–Macaulay reduction G∗ of G (or more brieflya CM-reduction). We replace the group G by a subgroup G∗ with better algebraicproperties, but with the same discriminant for versal unfoldings. We prove as thesecond parts of Theorems 1 and 2 that if G has such a CM-reduction G∗, then theG-discriminants for versal unfoldings are free* divisors but for the module of G∗-liftable vector fields. Moreover, if in addition “G generically has G∗-liftable vectorfields”, then the G-discriminant is a free divisor.

A basic situation where CM-reduction arises involves nonlinear sections of non-isolated complete intersections. We introduce the notion of “free complete inter-section” which for nonisolated complete intersections is the analogue of the notion

364 J. DAMON

of free divisor (Section 5). For example, products of free divisors are free completeintersections. Nonlinear sections of a free complete intersection give the noniso-lated analogue of ICIS. Moreover, complete intersections may possess Morse-typesingularities with the analogous properties as for divisors. However, unexpectedly,even if a free complete intersection V generically has Morse-type singularities, theKV -discriminants for versal unfoldings of sections of V need not be free becausegenerally KV is not Cohen–Macaulay. We give an explicit example in Section 3where this fails. Hence, the result of Looijenga [L] on the freeness of the discrim-inant for stable germs defining an ICIS does not extend to nonisolated completeintersections.

However, for free complete intersections (V, 0), we establish (Theorem 4) thatprovided n < h(V ) (which can be smaller than hn(V )), KV has a Cohen–Macaulayreduction K∗

V . Thus, by Theorem 2, KV -discriminants are free* divisors for themodule of K∗

V -liftable vector fields. In the case of an ICIS (corresponding to V ={0} ⊂ Cp), the K∗

V and KV -liftable vector fields agree, and define the discriminantfor the versal unfolding with reduced structure, recovering the result of Looijenga[L]. However, Theorem 1 does not generally apply to this case because for Morse-type singularities for KV , vector fields are not K∗

V -liftable. We furthermore show inpart III that the question of CM-reduction and genericity of Morse-type singularitiesreappear for the relative case of a divisor on a nonisolated complete intersection.

Lastly, we give an answer to the third question we raised regarding the van-ishing topology of nonlinear sections of free* divisors. We show in Section 4 thatthe weaker properties of free* divisors are still sufficient to give formulas for thesingular Milnor numbers for nonlinear sections (Theorem 5) in terms of codimen-sions of appropriate normal spaces. Except now we must take into account “virtualsingularities” arising from the free* divisor structure. Because of the form for thenormal spaces, these codimensions are also given as Buchsbaum–Rim multiplicitiesof the normal spaces, leading to natural questions about whether the formulas canbe expressed in terms of Buchsbaum–Rim multiplicities of the original G-normalspaces.

I. Discriminants as Free or Free* Divisors and Their Vanishing Topology

1. Free* Divisors

Notation. We recall some of the notation used in Section 1 of Part I of this paper[D6]. If (V, 0) ⊂ (Cp, 0) is a germ of a variety, we let I(V ) denote the ideal ofgerms vanishing on V . We also let θp denote the module of vector fields on (Cp, 0)and consider the module of vector fields tangent to V .

Derlog(V ) = {ζ ∈ θp : ζ(I(V )) ⊆ I(V )}.Then, (following Saito [Sa]) if (V, 0) is a hypersurface, it is called a Free Divisorif Derlog(V ) is a free OCp,0-module. Its rank is then necessarily p. We furtherrecall (see [DM]) that H is a good defining equation for V if there is an “Euler-likevector field” e ∈ Derlog(V ) such that e(H) = H. This follows if V is weightedhomogeneous, and H can be chosen weighted homogeneous of non-zero weight.However, we can always find a good defining equation by replacing V by V × C


(see [DM, Section 3] and (1.4) below). This causes no changes in the properties; ifV is a free divisor then so is V × C. Also, by results in [D2], for the equivalenceof sections of germs of varieties, we can replace V by V × C without changing thedeformation theory. Finally if M is a submodule of θp generated by {ζ1, . . . , ζr},we let 〈M〉(y) be the subspace of TyCp spanned by {ζ1(y), . . . , ζr(y)}.

We now introduce a weakened form of the notion of free divisor.

Definition 1.1. For a hypersurface germ (V, 0) ⊂ (Cp, 0), and p′ = p+m ≥ p, letV ′ = V ×Cm ⊂ Cp′ . Then, a free* divisor structure for V defined on Cp′ , consistsof an O(Cp′ ,0)-submodule L ⊆ Derlog(V ′) which satisfies:

(1) L is a free O(Cp′ ,0)-module of rank p′; and(2) supp(θp′/L) = V ′.

Although a free* divisor structure for V is not unique, frequently L will be aclose approximation to Derlog(V ). Hence, we will denote L by Derlog∗(V ) andrefer to (V, 0) as a free* divisor defined by Derlog∗(V ).

Remark 1.2. This notion is really a “stable property” of V as the free* divisorstructure exists on V ′ = V × Cm rather than on V itself. However, as remarkedearlier, we can replace V by V ′ without changing either the deformation theoryor vanishing topology for nonlinear sections. Furthermore, many cases we considerinvolve discriminants and bifurcation sets for versal unfoldings, so we may replacethem by their products with some Cm. Hence, in what follows, to reduce notation,we shall frequently suppose that we have already replaced V by its product withthe appropriate Cm so that Derlog∗(V ) ⊆ Derlog(V ).

Remark 1.3. Unlike Derlog(V ), Derlog∗(V ) need not be a Lie algebra of vectorfields. However, frequently V will denote the discriminant (or bifurcation set) fora versal unfolding with respect to some geometric subgroup G of A or K [D5].Then, Derlog∗(V ) will be the extended tangent space to the group of G-liftablediffeomorphisms, which is a subgroup of DV , the diffeomorphisms preserving V .Hence, Derlog∗(V ) will naturally have a Lie algebra structure. It provides V witha singular foliation. Thus, when Derlog∗(V ) does have this additional Lie algebrastructure, we will specifically note it.

1.4. Properties of Free* Divisors. 1. As for free divisors, we say that h is agood defining equation for (V, 0) if h is reduced and there is an Euler-like vectorfield e ∈ Derlog∗(V ) satisfying e(h) = h. By replacing V by V ′ = V × C and h by

h1(x, t) = exp(t) · h, we obtain an Euler-like vector field e =∂

∂t. Again V ′, 0 has

a free*-divisor structure with

Derlog∗(V ′) = O(Cp+1,0)

{ζ1, . . . , ζp,

∂

∂t

}where {ζ1, . . . , ζp} generate Derlog∗(V ), with good defining equation h1. In allthat follows, we will see that this argument allows us to assume that the naturalmodules of vector fields defining free* divisors may always be assumed to containan Euler-like vector field. We will refer to the above as the standard constructionof an Euler-like vector field.

366 J. DAMON

2. For a free* divisor (V, 0) ⊂ Cp with good defining h and Euler-like vectorfield e we let

Derlog∗(h) = {ζ ∈ Derlog∗(V ) : ζ(h) = 0}

By the same argument as in [DM, Section 2], Derlog∗(h) is a free OCp,0-module ofrank p− 1 and

Derlog∗(V ) = Derlog∗(h)⊕OCp,0{e}

Finally, as for free divisors, we define Tlog∗(V )(y) = 〈Derlog∗(V )〉(y) and similarlyfor Tlog∗(h)(y).

3. Measuring Nontriviality for Free* Divisors. For any hypersurface (V, 0) ⊂Cp with reduced defining equation h, one free* divisor structure is defined usingDerlog∗(V ) = h · θp, which is trivial in the sense that for any y ∈ V , we haveTlog∗(V )(y) = (0).

Such trivial free* divisor structures provide no new useful infomation. The valueof a free* divisor structure depends on how closely it approximates Derlog(V ). Wenext define geometric measures of this closeness. For any OCp,0-module M ⊆θp we let Fq(θp/M) denote the q-th Fitting ideal of θp/M and let Fq(M) =supp(Fq(θp/M)). Then, Fp−q(M) = {y ∈ Cp : dim C〈M〉(y) < q}. If Fp−q−1(M) 6=Fp−q(M) and codim(Fp−q(M)) = s, then we say M has rank (at least) q off codi-mension s.

For specific modules of vector fields, we abbreviate Fq(Derlog∗(V )) by F ∗q (V ),

Fq(Derlog∗(h)) by F ∗q (h), and similarly for Fq(V ) using instead Derlog(V ), etc.

As Derlog∗(V ) ⊆ Derlog(V ), F ∗q (V ) ⊆ Fq(V ). In the case that Derlog∗(V ) is

a Lie subalgebra of Derlog(V ), Derlog∗(h) is a Lie ideal. Both Derlog∗(V ) andDerlog∗(h) define singular foliations of V . Then, F ∗

p−q(h) ⊆ F ∗p−q(V ) define the

analytic subsets where the leaves of the foliations have dimension < q.We shall see in Section 4 that to use the free* divisor structure to compute the

number of singular vanishing cycles for nonlinear sections f0 : (Cn, 0) → (Cp, 0) of(V, 0), we must have f0 transverse off 0 to the singular distribution of V definedby Derlog∗(h). For example, if n < p, then for each 0 ≤ q < p − n, this requiresn < codim(F ∗

p−q(h)).By the reduced rank of the free* structure at a point y ∈ V , we shall mean

dim CTlog∗(h)(y). Then, r = dim CTlog∗(h)(0) is the minimum reduced rank. Ifn < p − r, then a necessary condition for transversality off 0 is that the set ofpoints where the reduced rank = r has codimension < n. Thus, we seek free*divisor structures with as large a rank off as small a codimension as possible, withespecially F ∗

p−r(h) as small as possible.4. Exponents for Free* Divisors. Let {ζ1, . . . , ζp} generate Derlog∗(V ) with

ζi =∑j aij

∂

∂xi. We denote det(aij) by det(ζ1, . . . , ζp), or more simply det(ζi). It

is a generator of F0(θp/Derlog∗(V )). Also, let h be a reduced defining equationfor (V, 0) with h =

∏hi, where each hi defines an irreducible component Vi of

V . Then, by condition 2 of 1.1, det(aij) = u ·∏hmii , where u is a unit. The

exponents mi measure componentwise how much Derlog∗(V ) fails to define a freedivisor structure for V .


Remark 1.5. Many of the basic properties of free divisors given in [D3, Section 1–3]extend to free* divisors. To extend both the definitions and results to free* divisors,we simply replace TlogV and Tlog(h) by Tlog∗V and Tlog∗(h), and give the “*”analogues of algebraic transversality, algebraic general position, transverse union,almost free divisor, etc.

We illustrate the preceding with several examples.

Example 1.6. The Whitney umbrella (V, 0) is the image of the finite map germf0 : (C2, 0) → (C3, 0) with f0(x1, x2) = (x2

1, x1x2, x2) = (Y, Z, W ) and is definedby H(Y, Z, W ) = YW 2 − Z2 = 0. It is not a free divisor; in fact (see e. g. [D2,(1.1)], Derlog(V ) is generated by

η1 = 2Y∂

∂Y+ Z

∂

∂Z, η2 = 2Y

∂

∂Y−W

∂

∂W,

η3 = WY∂

∂Z+ Z

∂

∂W, and η4 = 2Z

∂

∂Y+W 2 ∂

∂Z.

(1.1)

However, it has several nontrivial free* divisor structures. One is given by D1

generated by {e = aη1 − bη2, η3, Zη4 −W 2η2}, where a 6= b, and a second by D2

generated by {e = aη1 − bη2, η4, Zη3 + WY η2}, where a 6= b, 0. D1 has reducedrank 1 off the “handle”, i. e. the Y -axis, while D2 has reduced rank 1 off the Y andW -axes. Both have exponent 2. We use D1 to compute the vanishing topology fornonlinear sections in Section 4.

Example 1.7. An arrangement of more than p hyperplanes through 0 ∈ Cp whichare in general position off 0 is not a free divisor for p > 2. However, such an arrange-ment is an almost free divisor, obtained as the pullback of a Boolean arrangement[D3, Section 5]. Consider the case of such an arrangement of four planes through0 ∈ C3. Any two such are equivalent by a linear transformation so we consider Adefined by xyz` = 0 where ` = x+y+z. Although it is not free, it is a free* divisorin C4 defined by Derlog∗(A) with generators

ζ1 =x(z+ `)∂

∂x−z(x+ `)

∂

∂z+

∂

∂w, ζ2 =−y(z+ `)

∂

∂y+z(y+ `)

∂

∂z+

∂

∂w,

ζ3 =−x(y+ `)∂

∂x+y(x+ `)

∂

∂y+

∂

∂w, and ζ0(= e) =x

∂

∂x+y

∂

∂y+z

∂

∂z.

(1.2)

A computation shows det(ζi) = xyz`2, so the free* structure is reduced on allhyperplanes except ` = 0 and has reduced rank 2 up to codimension 3.

Third, consider general surface singularities in (V, 0) ⊂ (C3, 0). Surface sin-gularities which are free divisors always have nonisolated singularities. Isolatedsurface singularities in C3 are “almost free divisors” in the sense of [D3]. Weightedhomogeneous surface singularities with smooth singular set can be free or not, de-pending on special algebraic properties identified in [D8]. The next result showsthat all weighted homogeneous surface singularities are nontrivial free* divisors.

Proposition 1.8. If (V, 0) ⊂ C3 is a weighted homogeneous surface singularity ofnonzero weight, then (V, 0) has a natural “Pfaffian” free* divisor structure in C4

of exponent 2 and of reduced rank 2 off Sing(V ). In particular, if f has an isolatedsingularity, the free* divisor structure has reduced rank 2 off codimension 3.

368 J. DAMON

Proof. Let h be a weighted homogeneous defining equation for (V, 0). We supposewt(h) = d 6= 0, where wt(yi) = ai. Also, we denote the determinantal vector field

ηij =∂h

∂yi

∂

∂yj− ∂h

∂yj

∂

∂yi. For V ×C ⊂ C4, let w denote the coordinate for the last

factor. We define the generators for Derlog∗(V ) by

ζi = ηjk + aiyi∂

∂w(i = 1, 2, 3) and ζ0 = −de = −

∑aiyi

∂

∂yi.

where for i > 0, (ijk) is a cyclic permutation of (123).The matrix of coefficients for ζ1, ζ2, ζ3, ζ0 is skew-symmetric; hence, its determi-

nant is the square of the Pfaffian. A straightforward computation using the Eulerrelation for h shows the Pfaffian = d ·h, hence {ζi} defines a free* divisor structureof exponent 2. Furthermore, ζ1, ζ2, ζ3 generate Derlog∗(h) and their 2× 2 minorsgenerate an ideal containing J(h)2. Thus, the free* structure has reduced rank 2off Sing(V ). �

2. Cohen–Macaulay Properties of Geometric Subgroups

We now wish to extend the main Theorem 1 of Part I to more general situa-tions involving discriminants for versal unfoldings for general geometric subgroupsof A or K. These groups include all of the standard equivalences. Moreover theysatisfy the basic theorems of singularity theory such as the finite determinacy theo-rem, the versal unfolding theorem, and infinitesimal stability implies stability underdeformations (see [D5]).

We consider generally a geometric subgroup G of A or K for the category ofholomorphic germs. Recall this means that there is an action of G on F where F isan affine subspace of C(n, p), the space of holomorphic germs f : (Cn, 0) → (Cp, 0).There is also a corresponding action of the group of unfoldings Gun(q) on the spaceof unfoldings Fun(q) on q parameters, for all integer q ≥ 0. These actions satisfyfour conditions given in [D5]. For an unfolding F ∈ Fun(q), we have the orbit mapαF : Gun(q) → Fun(q) and the corresponding infinitesimal orbit map

dαF : TGun,e(q) → TFun,e(q).

The extended tangent space is the image dαF (TGun,e(q)) = TGun,e · F , and thenormal space is the quotient NGun,e ·F = TFun,e(q)/TGun,e ·F . In the case of germsf ∈ F , we denote the extended normal space by NGe ·f . These are analogues in theThom–Mather framework of “(relative) T 1”. A germ f has finite G-codimensionif dim CNGe · f < ∞. In this case it follows by an extension of the preparationtheorem for adequate systems of rings [D5, Corollary 6.16] that NGun,e · F is afinitely generated OCq,0-module.

We are interested in a special class of such subgroups G. Let F ∈ Fun(q) bean unfolding of a germ f having finite G-codimension. In what follows, we eitherrepresent F by F (x, u) = (F (x, u), u) or (Fu(x), u). We are interested in valuesu such that F (·, u) is not G-stable. Of course, in general G-equivalence does notnecessarily make sense for germs at points other then 0. Nonetheless we can define


the G-discriminant of the unfolding F by

DG(F ) = supp(NGun,e · F )

Following Teissier [Tei], DG(F ) has an analytic structure given by the 0-th Fittingideal of NGun,e · F (as an OCq,0-module).

Remark. There is ambiguity between what constitutes bifurcation sets versus dis-criminants for geometric subgroups. We should strictly speaking refer to DG(F )as the bifurcation set of F . However, by [D9], it often can be viewed as the dis-criminant of an associated group KV for appropriate V . For this reason we willthink of discriminants and bifurcation sets as being examples for different geometricsubgroups of this common notion of “discriminant”. Where we want to specificallyidentify the bifurcation aspect we will refer to it as the bifurcation set.

If DG(F ) geometrically captures where F (·, u) is not stable, this would implythat DG(F ) is preserved under G self-equivalences of F . Such equivalences are giveninfinitesimally by G-liftable vector fields.

Definition 2.1. For a geometric subgroup G of A or K, a G-liftable vector fieldfor F is a germ of a vector field η ∈ θq such that there is a germ of a vector fieldξ ∈ TGun,e satisfying:

dαF (η + ξ) = 0 (2.1)

If we integrate η + ξ and η, we obtain flows Φ and ϕ which commute with theprojection pr : Cn+p+q → Cq. If ϕ preserves DG(F ), then differentiating impliesthat η ∈ Derlog(DG(F )). We formalize this property as follows.

Definition 2.2. We say that a geometric subgroup G of A or K, has geometricallydefined discriminants if for any versal unfolding F of a germ f having finite G-codimension, if ξ is a G-liftable vector field for F , then ξ ∈ Derlog(DG(F )).

All geometric subgroups which we consider will have an appropriate geometriccharacterization of their discriminants and so have geometrically defined discrimi-nants in this sense.

Example 2.3. For a germ of a variety (V, 0) ⊂ Cp, the group KV has geometricallydefined discriminants.

To see this let F : (Cn+q, 0) → (Cp+q, 0) be a versal unfolding of a nonlinearsection f0 of V . Also, let DV (F ) denote the KV -discriminant of F . We will showthat if ζ is KV -liftable with local flow ϕt, then ϕt preserves DV (F ). This impliesζ ∈ Derlog(V ).

Then, ζ being KV -liftable means there are germs of vector fields

ξ ∈ O(Cn+q,0)

{∂

∂x1, . . . ,

∂

∂xn

}and η ∈ O(Cn+q,0){ζ1, . . . , ζp}

satisfying:(ξ + ζ)(F ) = η ◦ F . (2.2)

Let ϕt, ϕ1t, and ψt denote the flows induced by ζ, ξ, and η on respectively (Cq, 0),(Cn+q, 0) and (Cn+p+q, 0). Then, π◦ϕ1t = ϕt◦π, for π : Cn+q → Cq the projection.

370 J. DAMON

Also, we have the commutative diagram

(Cn+q, 0) F // (Cp+q+n, 0)

(Cn+q, 0)

ϕ1t

OO

F // (Cp+q+n, 0)

ψt

OO(2.3)

where F (x, u) = (F (x, u), u, x). Then, the algebraic transversality of F (·, u0) toV at x0 is equivalent to that of F to V = V ×Cq+n at (x0, u0). This latter conditionis equivalent to

dF (x0, u0)(TCn+q) + TlogVy0 = Ty0Cp+q+n. (2.4)

for y0 = F (x0, u0). However, as both ϕ1t and ψt are diffeomorphisms for fixed t,it follows from (2.3) that by applying dψt(y0), (2.4) is equivalent to

dF (x0, u0)(TCn+q) + dψt(y0)(TlogVy0) = Ty0Cp+q+n. (2.5)

Finally, ψt(V ) = V . Hence, let χ ∈ Derlog(V ) and let h define V . Then ψt∗(χ)(h) =χ(h◦ψt). Since h◦ψt vanishes on V , it has the form g(y, u, x) ·h. Thus, χ(h◦ψt) =g1(y, u, x) · h. Hence, ψt∗(χ) ∈ Derlog(V ); and so dψt(y0)(TlogVy0) = TlogVψt(y0).Thus, transversality, or failure thereof, is preserved under the action by (ϕ1t, ψt),and thus so is the KV -discriminant under ϕt. Thus, ζ is tangent to DV (F ) and sobelongs to Derlog(V ). This completes the verification.

Now we are ready to introduce the Cohen–Macaulay property for geometricsubgroups and state what we mean by a group having a reduction to such a group.

Definition 2.4. A geometric subgroup G of A or K will be said to be Cohen–Macaulay if it has geometrically defined discriminants; and if for each versal un-folding F ∈ Fun(q) of a germ f having finite G-codimension, NGun,e · F is Cohen–Macaulay as a OCq,0-module, with supp(NGun,e · F ) of dimension q − 1.

Example 2.5. For a free divisor (V, 0) ⊂ (Cp, 0) with n < hn(V ), the group KVacting on nonlinear sections f0 : (Cn, 0) → (Cp, 0) is Cohen–Macaulay. To see this,we note that by Example 2.3, KV has geometrically defined discriminants (thisdoes not depend on V being free). Also, as n < hn(V ), by the proof of Lemma3.4 of Part I [D6], NKV,un,e ·F is Cohen–Macaulay and by Proposition 2.4 of [D6],DV (F ) = supp(NKV,un,e · F ) has dimension q − 1.

For complete intersections, the situation becomes more complicated. However,the analogous group is close to being Cohen–Macaulay. This can be made preciseusing the following definition.

Definition 2.6. Given a geometric subgroup G of A or K which has geometri-cally defined discriminants, a Cohen–Macaulay reduction of G (abbreviated CM-reduction) consists of a geometric subgroup G∗ ⊂ G which still acts on F (and Fun)such that:

(1) G∗ is Cohen–Macaulay;(2) f ∈ F has finite G∗-codimension iff it has finite G-codimension;


(3) if F ∈ Fun(q) is an unfolding of a germ f which has finite G-codimension,then viewed as OCq,0-modules,

supp(NG∗un,e · F ) = supp(NGun,e · F ). (2.6)

Remark 2.7. It follows that the CM-reduction G∗ also has geometrically defineddiscriminants. As G∗ is a subgroup of G, G∗-liftable vector fields are also G-liftable.Hence, they belong to Derlog(DG(F )) = Derlog(DG∗(F )) by (2.3).

The second criterion in the “motto” stated in the introduction concerns thegenericity of Morse-type singularities. Because each group G would require itsown form of Morse-type singularity, we state the condition in a form which avoidsconsidering individual cases.

Definition 2.8. We say that a geometric subgroup G of A or K with a CM-reduction G∗, has generically G∗-liftable vector fields if for a G-versal unfolding F(on q parameters) of a germ f0 of finite G-codimension, there is a Zariski opensubset Z of DG(F )reg (intersecting each component in a neighborhood of 0) suchthat:

(1) if h is a reduced defining equation for DG(F ) then there is an inclusion ofsheaves restricted to Z

h ·Θq ⊂ Derlog∗(DG(F ));

where Derlog∗(DG(F )) denotes module of G∗-liftable vector fields; and(2) if u ∈ Z, then

Tlog∗(DG(F ))(u) = Tu(DG(F )).

If G is itself Cohen–Macaulay, we say it has generically G-liftable vector fieldsprovided the preceeding conditions hold with G∗ = G.

Remark. In Part I, we showed that genericity of Morse type singularities for KV -equivalence for (V, 0) a free divisor implied the genericity ofKV -liftable vector fields.The crucial point is to be able to use the genericity of Morse-type singularities toestablish the genericity of G∗-liftable vector fields. We provide a strategy for doingthis and use this several times in Part III [D7].

We can now state the first basic result which implies that discriminants forvarious geometric subgroups G are free divisors.

Theorem 1. (i) Suppose that the geometric subgroup G is Cohen–Macaulay andgenerically has G-liftable vector fields, then DG(F ) is a free divisor with

Derlog(DG(F )) = module (Lie algebra) of G-liftable vector fields.

(ii) If instead G has a Cohen–Macaulay reduction G∗ which generically has G∗-liftable vector fields. Then, DG(F ) is a free divisor with

Derlog(DG(F )) = module (Lie algebra) of G∗-liftable vector fields.

As an indication of its applicability, we mention that besides including the resultsfrom Part I, this theorem also applies to: groups of equivalences for functions andalmost free divisors on an ICIS, allowing both to vary (by giving a CM-reduction),and to the group VK for “boundary singularities”, which is Cohen–Macaulay.

372 J. DAMON

In the absence of genericity of Morse-type singularities, we can still conclude

Theorem 2. (i) Suppose that the geometric subgroup G is Cohen–Macaulay. Then,DG(F ) is a free* divisor for

Derlog∗(DG(F )) = module (Lie algebra) of G-liftable vector fields.

(ii) If instead G has a Cohen–Macaulay reduction G∗. Then, DG(F ) is a free*divisor for

Derlog∗(DG(F )) = module (Lie algebra) of G∗-liftable vector fields.

Remark. We observe that despite the notation, G and F typically denote a collec-tion of groups and spaces which depend upon n, p, and (for unfoldings) q. WhetherG or G∗ are Cohen–Macaulay or whether there is genericity of G or G∗-liftable vectorfields may depend upon the dimensions n and p, so results may include conditionson dimensions.

It follows that despite the failure of genericity of Morse type singularities, wemay still conclude for discriminants that the module of liftable vector fields de-fines a free* divisor structure. As an example of this we consider the consequencefor KV equivalence for nonlinear sections f0 : (Cn, 0) → (Cp, 0) of a free divisor(V, 0). Even if n < hn(V ) (the holonomic codimension of V ), frequently V doesnot generically have Morse-type singularities in dimension n. Nonetheless, providedn < hn(V ), the discussion in Example 2.5 implies KV is Cohen–Macaulay. Hence,Theorem 2 allows us to conclude.

Theorem 3. Suppose that (V, 0) ⊂ (Cp, 0) is a free divisor and that F is aKV -versal unfolding of f0 : (Cn, 0) → (Cp, 0) with n < hn(V ). Then the KV -discriminant DV (F ) is a free* divisor for

Derlog∗(DV (F )) = module (Lie algebra) of KV -liftable vector fields.

As applications of this we give several corollaries.For finitely A-determined germs f0 : (Cn, 0) → (Cp, 0), we proved in Part I that

if f0 belongs to the “distinguished bifurcation class” of germs then the bifurcationset of the A-versal unfolding is a free divisor. This restriction excludes a numberof classes of germs occurring in the nice dimensions (in the sense of Mather [M]).We can extend this to all of the nice dimensions as follows.

Corollary 2.9. Suppose that f0 : (Cn, 0) → (Cp, 0) is a finitely A-determined germwith (n, p) in the nice dimensions. Then, the bifurcation set B(f) for the A-versalunfolding f of f0 is a free* divisor for

Derlog∗(B(f)) = the module of A-liftable vector fields.

In fact, by the argument for proving Theorem 3 of Part I, let f0 be obtained fromf by pull-back by a germ g0. Then, the bifurcation set B(f) = KD(f)-discriminantfor g, the KD(f)-versal unfolding of g0. By [D2] and [D6] the KB(f)-liftable vectorfields are the A-liftable vector fields. That (n, p) is in the nice dimensions thenimplies that n < hn(D(f)) so that Theorem 1 applies.

Second, we consider the case of the special A-equivalence of complete intersectionmap germs f0 on isolated complete intersections X, 0, where we allow both f0


and X to vary. In [MM], Mond and Montaldi show that deformations of suchgerms can be analyzed by the equivalence of sections of the discriminant of thecomplete intersection X0 = f−1

0 (0). For a limited range of dimensions given by[D6, Theorem 9.3], the bifurcation set for the versal unfolding will be a free divisor.More generally, we can relax the dimension restriction.

Corollary 2.10. Suppose that f0 : (X, 0) → (Cp, 0) is a complete intersection mapgerm defining an isolated singularity (X0, 0) on an isolated complete intersectionsingularity (X, 0). Let F : X → Cp+q be the versal deformation for this special Aequivalence (induced by the versal unfolding of the germ defining X0). If (X0, 0) isan ICIS defined by a simple germ, then the bifurcation set B(F ) is a free* divisorfor

Derlog∗(B(F )) = the module of KD(F )-liftable vector fields.

We can still use the argument of Mond–Montaldi [MM] to represent f0 as apullback of the versal unfolding F of the germ defining X0 by a section of D(F ).Then, D(F ) is free by [L] and the simplicity of the germ defining X0 implies thathn(D(F )) = ∞. Thus, Theorem 1 applies.

The third consequence is for nonlinear arrangements. Only Boolean arrange-ments (arrangements consisting of all coordinate hyperplanes) generically haveMorse-type singularities; nonetheless for general free hyperplane arrangements wecan conclude the following.

Corollary 2.11. Suppose that A ⊂ Cp is a free hyperplane arrangement. LetA0 = f−1

0 (A) be a (nonlinear) arrangement defined via f0 : (Cn, 0) → (Cp, 0).Then, the KA-discriminant of the KA-versal unfolding of f0 is a free* divisor for

Derlog∗(DA(F )) = the module of KA-liftable vector fields.

We now turn to the proofs of Theorems 1 and 2, although we prove them in thereverse order.

Proof of Theorem 2. The proof we give is a modification of an argument of Looi-jenga [L] but we use instead Saito’s criterion [Sa] for a free divisor. We give theproof in the case that G has a Cohen–Macaulay reduction G∗. For the case that Gitself is Cohen–Macaulay we just apply the argument given with G∗ = G.

We suppose f0 : (Cn, 0) → (Cp, 0) has finite G-codimension and that n and p aredimensions for which G has a Cohen–Macaulay reduction G∗. Then, by assumption,f0 has finite G∗-codimension. Suppose F ∈ Fun(q) is an unfolding of f0 and hasthe form F (x, u) = (F (x, u), u) with

ϕi = ∂iF =∂F

∂ui

∣∣∣u=0

, 1 ≤ i ≤ q. (2.7)

If {ϕ1, . . . , ϕq} spans NG∗e · f0, then since TG∗e · f0 ⊆ TGe · f0, it follows by theversality theorem (see [D5]) that F is both a G∗ and G-versal unfolding of f0.Moreover, suppose dim CNGe · f0 = q′ and that F1 is a restriction of F which isa G-versal unfolding on q′ parameters. Then, by the same versality theorem, Fis a G-trivial extension of F1 on q − q′ parameters. Hence, DG∗(F ) = DG(F ) 'DG(F1) × Cq−q′ . Hence, since a free* divisor structure is defined stably for some

374 J. DAMON

suspensionDG(F1)×C`, it is enough to consider F . If we take ϕq = 0, then∂F

∂uq= 0,

and∂

∂uqwill be a G∗-liftable vector field and (by the standard construction) an

Euler-like vector field.Next, we consider the G∗un,e normal space

NG∗un,e · F = TFun,e/TG∗un,e · F. (2.8)

It follows from the Preparation theorem (for adequate systems of DA algebras [D5,Section 6]) that NG∗un,e · F is a finitely generated OCq,0-module on the generators{ϕ1, . . . , ϕq}. Hence, we have an exact sequence

0 → L β−→ OCq,0

{∂

∂ui

}α−→ NG∗un,e · F → 0, (2.9)

where the map α sends∂

∂ui7→ ϕi and L denotes the kernel of α. Then, L will be

our Derlog∗(DG(F )).We first characterize L as the module of “G∗-liftable vector fields”. As

dαF (−ξ) ∈ TG∗un,e · F and dαF (η) = α(η),

(2.1) is equivalent to η ∈ L. Thus, at least L is the module of G∗-liftable vectorfields. Next we summarize the key properties of L by the following proposition

Proposition 2.12. Let F be the G∗ versal unfolding of f0 given in (2.7). For Lgiven in the exact sequence (2.9):

(1) L is the Lie algebra of G∗-liftable vector fields;(2) L contains an Euler-like vector field ;(3) L is a free OCq,0-module;(4) L is of rank q so β is given by a q × q-matrix, whose determinant (which

is the generator for the 0-th Fitting ideal of NG∗un,e · F ) defines the G∗-discriminant of F ; hence, supp(θq/L) = DG(F ).

Proof of Proposition 2.12. We have already seen that L is the set of G∗-liftablevector fields and contains an Euler-like vector field. Next, to see that L is a Liealgebra, we observe that it is the tangent space to the group of G∗-liftable diffeomor-phisms on Cq, 0, as described in remarks preceding Definition 2.2. By a standardargument, L is a Lie algebra.

To establish (3), we observe by (1) of the definition of CM-reduction that NG∗un,e ·F is Cohen–Macaulay as an OCq,0-module. Hence, it is a Cohen–Macaulay moduleover OD, where

D = supp(NG∗un,e · F ) (= DG(F )).

Since dimD = q−1, the Auslander–Buchsbaum formula implies that NG∗un,e ·F hasprojective dimension 1. Thus, it follows that in (2.9) that L is a free OCq,0-module.

Finally, it remains to prove (4). By (3), we know that L is a free OCq,0-module.As β is an inclusion, its rank is no greater than q, however, it cannot be less than qas supp(NG∗un,e ·F ) has dimension q− 1. Thus, L has rank q, and supp(NG∗un,e ·F )


is defined by the vanishing of det(β) (which is the generator of the 0-th Fittingideal of NG∗un,e · F ). This completes (4) and the proof of Proposition 2.12. �

To finish the proof of Theorem 2, we note that by (3) of the definition of CM-reduction,

supp(NG∗un,e · F ) = supp(NGun,e · F )

= DG(F ). (2.10)

Thus, supp(θq/L) = DG(F ).Finally, since G has geometrically defined discriminants, L ⊆ Derlog(DG(F )).

�

Proof of Theorem 1. We use the notation in the proof of Theorem 2. Again wegive the proof when G has a CM-reduction G∗.

If we extend the G-versal unfolding F1 to a G∗-versal unfolding F , then

DG∗(F ) = DG(F ) ' DG(F1)× Cq−q′. (2.11)

Thus, it is sufficient to prove that DG(F ) is a free divisor.The module of G∗-liftable vector fields is the free OCq,0-module on q generators

Derlog∗(DG(F )) ⊆ Derlog(DG(F )). We apply Saito’s criterion to a set of freegenerators of Derlog∗(DG(F )) = L given by Proposition 2.12. Thus, it is sufficientto show det(β), which is a generator for the 0-th Fitting ideal, is a reduced definingequation for DG(F ).

We consider the Zariski open subset Z of DG(F )reg given in Definition 2.8. Ata point u ∈ Z, we may choose local coordinates u1 = h, u2, . . . , uq. Then, by

(2) of Definition 2.8, there are ζi ∈ Derlog∗(DG(F )) such that ζ1 = u1 ·∂

∂u1and

ζi(u) =∂

∂ui, i = 2, . . . , q. Hence, the determinant of the coefficients of the ζi

has the form unit · u1 which defines DG(F ) near u. Thus, the 0-th Fitting idealdefines DG(F ) with reduced structure near u. Hence, det(β) defines DG(F ) withreduced structure. By Saito’s criterion, DG(F ) is a free divisor with Derlog(DG(F ))generated by a set of generators of Derlog∗(DG(F )), so they agree. �

3. KV -Discriminants of Complete Intersections

In this section, we consider KV -Discriminants of versal unfoldings for sections ofnonisolated complete intersections (V, 0). We first identify a special class of “freecomplete intersection singularities” extending the class of free divisors.

Suppose the complete intersection (V, 0) ⊂ (Cp, 0) is defined by the equationH : (Cp, 0) → (Ck, 0). Just as for divisors, we define

Derlog(H) = {ζ ∈ θp : ζ(H) = 0}.

This is the module of vector fields tangent to the level sets of H. The associatedsheaf Derlog(H) is coherent.

376 J. DAMON

Definition 3.1. A complete intersection (V, 0) ⊂ Cp defined by H : (Cp, 0) →(Ck, 0) (p ≥ k) will be called an (H-) free complete intersection if Derlog(H) is afree OCp,0-module. Then, H will be called a free defining equation for V .

Remark. The coherence of Derlog(H) implies that if V is H-free, then Derlog(H)will have rank p− k as an OCp,0-module.

We will frequently abuse terminology and call V a free complete intersection.For example, a product of free divisors is a free complete intersection (see Section 5for properties and examples). For a free divisor a good defining equation is a freedefining equation (but not conversely, see [D3, Example 2.11d]).

In general, the properties of free complete intersections follow those of free di-visors very closely. Nonlinear sections have a singular Milnor fiber and one cancompute the singular Milnor number and higher multiplicities as for nonlinear sec-tions of free divisors (see [D3, Parts 2, 3]). One might then expect that just as forisolated singularities and free divisors, the KV -discriminants of versal deformationsof nonlinear sections of a free complete intersection (V, 0) will also be free, providedV generically has Morse-type singularities in the appropriate sense. We next con-sider a simple example for which this is not the case. This example was worked outwith Anne Fruhbis-Kruger, using a package she developed for the computer algebrasystem Singular [Sg]. We make use of results from Sections 5 and 7.

Example 3.2. We consider the free complete intersection V = V1×V2 ⊂ C5, whereV1 = {(x, y, z) ∈ C3 : h1(x, y, z) = xyz = 0} and V2 = {(w, v) ∈ C2 : h2(w, v) =wv = 0} are both Boolean arrangements, and hence free divisors. Also, by Lemma7.7 of Part I, each Vi generically has Morse type singularities in all dimensions.Hence, by Proposition 7.5 below, the product V = V1 × V2 also genericially hasMorse-type singularities in all dimensions.

Next, we define a linear section f0 : (C2, 0) → (C5, 0) by f0(s, t) = (s+2t, s+ t,t−s, s−3t, t−5s). By choosing the weights of all variables to equal one, f0 and Vare both weighted homogeneous. By Proposition 5.6 below, Derlog(V ) is generatedby: {

x∂

∂x, y

∂

∂y, z

∂

∂z, w

∂

∂w, v

∂

∂v, h2

∂

∂x, h2

∂

∂y, h2

∂

∂z, h1

∂

∂w, h1

∂

∂v

}.

Using these we can compute NKV · f0 and verify by the infinitesimal criterionfor versality for KV -versality (Theorem 1 of [D1]) that a KV -versal unfoldingF (s, t, u) = (F (s, t, u), u) of f0 where u = (u1, . . . , u5) is given by

F (s, t, u) = f0(s, t) + (0, 0, u1, u2t+ u3, u4t+ u5).

For F to be weighted homogeneous, we must assign weights wt(u1, u2, u3, u4, u5) =(1, 0, 1, 0, 1). Thus, although the module of KV -liftable vector fields is graded,the graded pieces are not finite dimensional, but are modules over the ring ofgerms on (u2, u4). However, the package of Anne Fruhbis-Kruger, KVequiv.libfor the program Singular, enables one to find generators of fixed weight and lowdegree in (u2, u4). From among low weight KV -liftable vector fields, were identifiedζi, i = 0, . . . , 5 of weights 0, 1, 1, 2, 2, 3, such that their (u2, u4) components afterreducing mod (u2, u4) were respectively: (0, 0), (u1, ∗), (u5, ∗), (u2

3, ∗), (0, u25+g1),


and (0, u33 + g2). Here ζ0 is the Euler vector field and has u5 component equal to

u5; and g1 does not contain as a nonzero term u23. Also, there is no element with

(u2, u4) component (0, u5); nor do the terms obtained from ζ1, ζ2, and ζ3 with u4

component 0 contain terms with pure powers u25 or u3

3 in the u2 components. Thus,it follows that all 6 of the ζi are needed as generators so the number of generators≥ 6, and the module of KV -liftable vector fields is not a free module.

Thus, for sections of free complete intersections, the best one can hope for ingeneral is that discriminants are free* divisors (see the discussion in Section 8).

We shall establish this by giving a Cohen–Macaulay reduction K∗V for KV and ap-

plying Theorem 2. To define K∗V , we recall the subgroup KH of KV which preserves

the level sets of H (see e. g. [DM, Section 3]). The subgroup of diffeomorphismsK∗V lies between KH and KV . It consists of elements of KV (i. e. diffeomorphisms

of (Cp × Cn, 0)) which when restricted to V × Cn are induced by restrictions ofdiffeomorphisms from KH (see Section 6).

We must replace the holonomic codimension hn(V ), used for free divisors, byh(V ). This is the codimension of the set of y for which Tlog(h)(y) 6= TySi, whereSi the canonical Whitney stratum of V containing y. For example, by an argu-ment in [D3, Section 7], for the free complete intersection V =

∏ri=1 Vi, h(V ) =

min{h(Vi)}+ r − 1.

Theorem 4. Let (V, 0) ⊂ (Cp, 0) be a free complete intersection of codimensionk. If k ≤ n < h(V ), then for nonlinear sections f0 : (Cn, 0) → (Cp, 0), K∗

V is aCohen–Macaulay reduction for KV . Hence, for a KV -versal unfolding F of f0, theKV -discriminant DV (F ) is a free* divisor for

Derlog∗(DV (F )) = module (Lie algebra) of K∗V -liftable vector fields.

In light of Theorem 1, we ask when we have the genericity of K∗V -liftable vector

fields. In fact this almost never happens. It is not due to the failure of genericity ofMorse-type singularities for free complete intersections. We shall see in Section 7that this does often hold. Rather, it is due to KV -liftable vector fields which are notK∗V -liftable. The exception is for the case of smooth V . As a result, we recover the

result of Looijenga [L] that for isolated complete intersection singularities (whichcorrespond to V = {0}) the discriminant D(F ) for a versal unfolding F is a freedivisor, with Derlog(D(F )) the module of KV -liftable vector fields.

4. Vanishing Topology for Sections of Free* Divisors

Given the abundance of free* divisors following from the results in the precedingsections, we now consider the vanishing topology of nonlinear sections of a free*divisor (V, 0). As (V, 0) is a hypersurface, nonlinear sections have a singular Milnorfiber and singular Milnor number. We shall give a formula for the singular Milnornumber as the length of a determinantal module as in [DM], except that we willfind it necessary to subtract certain multiplicities of “virtual singularities”.

Let (V, 0) ⊂ (Cp, 0) be a free* divisor. We suppose, if necessary, that we havereplaced V by an appropriate suspension. We also suppose H is a good definingequation for V with Euler-like vector field e. Let f0 : (Cn, 0) → (Cp, 0) be a non-linear section of (V, 0) which has finite KV -codimension. We recall [DM, Section 4]

378 J. DAMON

that a topological stabilization of f0 consists of a family ft : U → Cp, where U isneighborhood of 0 in Cn, satisfying: i) when t = 0, ft is a representative of the germf0 which is geometrically transverse to V on U \ {0}, and ii) ft is geometricallytransverse to V on U for all sufficiently small t 6= 0. Then, by [DM, Section 4], fora sufficiently small ball Bε about 0 of radius ε > 0, f−1

t (V ) ∩ Bε is, up to home-omorphism, independent of t and is homotopy equivalent to a bouquet of spheresof dimension n− 1. This is the singular Milnor fiber, and the number of spheres isthe singular Milnor number, denoted µV (f0).

To compute this number for free* divisors, we note Derlog∗(V ) defines a singular,possibly nonintegrable, distribution on V . We must take into account the “virtualsingularities” where ft fails to be algebraically transverse to the distribution.

Definition 4.1. For a stabilization ft, a virtual singularity for a given t is a pointx ∈ U such that ft(x) ∈ V but

dft(x)(Cn) + Tlog∗(H)(ft(x)) 6= T(ft(x))Cp.

By [DM], the codimension of the module TKH,e ·f0 computes the singular Milnornumber in the case of a free divisor V . We introduce an analogue for the case of afree*-divisor V .

TK∗H,e · f0

def= OCn,0

{∂f0∂x1

, . . . ,∂f0∂xn

, ζ1 ◦ f0, . . . , ζp−1 ◦ f0},

where Derlog∗(H) is generated by ζ1, . . . , ζp−1. The normal space is given by

NK∗H,e · f0 = OpCn,0/TK

∗H,e · f0. (4.1)

Then, K∗H,e-codim(f0, x) = dim CNK∗

H,e · f0.

Remark. Although TK∗H,e ·f0 is not always an extended tangent space for a group

action, it is in the case that V = DG(F ) is a discriminant with G Cohen–Macaulayor having CM-reduction G∗. Then, K∗

H is a subgroup of diffeomorphisms KV definedby replacing diffeomorphisms preserving V by G (resp. G∗) liftable diffeomorphismspreserving the level sets of H (see Section 6).

For a virtual singularity x of ft, we let K∗H,e-codim(ft, x) denote the K∗

H,e-codimension of the germ ft : (Cn, x) → (Cp, ft(x)). Then, we obtain the formulafor the singular Milnor number.

Theorem 5. Suppose (V, 0) ⊂ (Cp, 0) is a free* divisor with good defining equationH. Let f0 : (Cn, 0) → (Cp, 0) be a nonlinear section of (V, 0) which has finite K∗

H-codimension. For ft a stabilization of f0,

µV (f0) = K∗H,e-codim(f0)−

∑K∗H,e-codim(ft, xi), (4.2)

where the sum is over the finite number of virtual singular points xi for a givensufficiently small value of t.

For t sufficiently small, all of the virtual singular points will occur in a sufficientlysmall neighborhood of 0.


Proof of Theorem 5. We outline the argument which follows along the lines of [DM,Section 5] except using the reduction. The module NK∗

H,e · f0 is a determinantalmodule. If we sheafify the corresponding module for the stabilization ft we obtaina sheaf N on a neighborhood U of 0 which is the quotient sheaf of OpU by thesubsheaf which is an OU -module generated by{

∂ft∂x1

, . . . ,∂ft∂xn

, ζ1 ◦ ft, . . . , ζp−1 ◦ ft}.

The support of N is one-dimensional and is then Cohen–Macaulay by results ofMacaulay [Mc] and Northcott [No]. For the projection π : Cn+1 → C, its restrictionto supp(N ) is finite to one so π∗(N ) is Cohen–Macaulay (and in fact free) over C.Thus,

dim CNK∗H,e · f0 = dim C(π∗(N )/t · π∗(N ))

= dim C(π∗(N )/(t− t0) · π∗(N ))

=∑

dim CNK∗H,e · (ft, xi). (4.3)

Here the sum in (4.3) is over points (xi, t0) ∈ supp(N ). This sum splits intotwo parts, a sum over points in V × {t0} and a sum over points in the comple-ment. For (ft0(xi), t0) = (yi, t0) /∈ V × {t0}, by assumption, Tlog∗(V )(yi) = Tyi

Cpso the generators {ζ1, . . . , ζp} evaluated at yi are linearly independent. Hence,{ζ1, . . . , ζp−1} span the tangent space to the level set of H at yi = ft0(xi). Hence,the argument in Lemma 5.6 of [DM] implies that dim CNK∗

H,e · (ft, xi) equals theMilnor number of H ◦ ft0 at xi. Using complex Morse theory as in [DM, Section 4]yields that the sum over (xi, t0) ∈ supp(N ) \ V equals the singular Milnor numberof f0. The remaining terms correspond to points in supp(N )∩V which are exactlythe virtual singularities each contributing K∗

H,e-codim(ft, xi). �

Ultimately, we would like to reexpress the terms on the RHS of (4.2) in terms ofinvariants of NKV,e · f0 (or possibly NKH,e · f0); however the dimensions of thesemodules do not behave well under deformation. Another invariant of finite lengthOCp,0-modules N is the Buchsbaum–Rim multiplicity [BRm], which we denote bymBR(N). For determinantal modules such as N = NK∗

H,e · f0, mBR(N) = dim CN .Hence, we may restate Equation (4.3) for computing the singular Milnor numbersolely in terms of Buchsbaum–Rim multiplicities

µV (f0) = mBR(NK∗H,e(f0))−

∑mBR(NK∗

H,e(ft, xi)) (4.4)

with the sum is over the finite number of virtual singular points xi for a givensufficiently small value of t. This leads us to the basic question.

Question. Can we replace the terms on the RHS of (4.4) by expressions involvingthe Buchsbaum–Rim multiplicities of the modulesNKV,e·f0 (or possiblyNKH,e·f0),and allowing certain virtual singularities?

Remark. In the weighted homogeneous case, for a Cohen–Macaulay reduction G∗of a group G, we give in [D9] a formula for the number of vanishing cycles in astabilization of an unfolding f of f0 in terms of the normal space of G∗.

380 J. DAMON

Next, we seek circumstances when we can simplify the expression representingthe sum over the virtual singularities. In the first special case, we suppose that forthe section f0, all stabilizations ft have virtual singularities on a curve C. For eachirreducible branch Ci of C, there is a Zariski open subset of jets of germs of sectionsg : (Cn, 0) → (Cp, y) with y ∈ Ci, with NK∗

(H,y),e ·g having minimum dimension ci.Then, we may perturb any stabilization so that at any virtual singular point x withft(x) = y ∈ Ci we have K∗

H,e-codim(ft, x) = ci. Then, we can compute the singularMilnor number as follows. Let (D(f0), Ci) denote the intersection multiplicity ofthe discriminant D(f0) and Ci (where D(f0) has nonreduced structure so we countintersection points y ∈ D(f0) with multiplicity given by the number of criticalpoints mapping to y).

Corollary 6. Suppose (V, 0) ⊂ (Cp, 0) is a free* divisor with good defining equa-tion H. Let f0 : (Cn, 0) → (Cp, 0) be a nonlinear section of (V, 0) of finite K∗

H-codimension with the property that any stabilization of f0 has its virtual singularitieson the curve C. Then,

µV (f0) = K∗H,e-codim(f0)−

∑i

ci · (D(f0), Ci) (4.5)

Proof. We may first perturb f0 by an isotopy so that ft is transverse (i. e. D(ft) istransverse) to C. If ft is not one-to-one on its critical set, we further perturb it so itdoes become so while still remaining transverse to C. Then, on each branch Ci therewill be (D(f0), Ci) transverse intersection points. The number of such transversepoints will remain fixed under an additional sufficiently small perturbation. Wemay further slightly perturb ft if necessary to a stabilization ft which only hasvirtual singularities x with ft(x) ∈ Ci of K∗

H,e-codim(ft, x) = ci. Hence, the sumon the RHS of (4.3) becomes exactly the sum on the right hand side of (4.5). �

Example 4.2 (“Twisted Whitney Umbrellas”). As a first example, consider asection of the Whitney umbrella V in (1.6) with the free* divisor structure withDerlog∗(V ) (given by D1 for (a, b) = (1, 0)) generated by {e = η1, η3, Zη4 −W 2η2}. Consider a section of the form (Y, Z, W ) = g0(X, y, z) = (y, z, p(X, y))where X = (x1, . . . , xn−1) so g0 : (Cn+1, 0) → (C3, 0). If we view V as the imageof F = (y2, uy, u), then the pullback of F by g0 yields the germ f0(X, y) =(y2, yp(X, y2), X) with f0 : (Cn, 0) → (Cn+1, 0). In the case n = 1 this givesa family of germs appearing in Mond’s classification [Mo1]. For f0 to have finiteA-codimension, we must have that (pX , ypy) = (px1 , . . . , pxn−1 , ypy) generates anideal of finite codimension. The singular Milnor number µV (g0) equals the numberof vanishing cycles for the image of a generic perturbation of f0. In fact, for thecase of germs f0 : (C2, 0) → (C3, 0), Mond [Mo2], de Jong–Van Straten [JVS], etc.have directly related this to Ae-codim(f0).

We compute the singular Milnor number for general n. In carrying out compu-tations, to reduce notation, we represent the ring O(Cn+1,0) by the coordinates forthe particular Cn+1, e. g. OX,y,z. From

∂g0∂xi

= (0, 0, pxi),∂g0∂y

= (1, 0, py),∂g0∂z

= (0, 1, 0) (4.6)


we obtain by projection along∂g0∂y

and∂g0∂z

onto the third component

NK∗H,e · g0 ' OX,y/(pX , p2(p− 2ypy))

Thus,

dim C(NK∗H,e · g0) = 2dim COX,y/(pX , p) + dim C(OX,y/(pX , p− 2ypy) (4.7)

A straightforward calculation shows that the virtual singularities only occuralong the “handle” of the Whitney umbrella, i. e. the Y -axis; and that a genericvirtual singularity has K∗

H,e-codim = 2. The intersection number of D(g0) with theY -axis is given by the number of points where z = p = pX = 0. Hence, it equalsdimOX,y/(pX , p). Thus, by Corollary 6 and (4.7)

µV (g0) = dim COX,y/(pX , p− 2ypy)

If p is weighted homogeneous, this is exactly Ae-codim(g0), see [Mo1] for the caseof n = 2.

Example 4.3. The second type of example we consider is a weighted homogeneoussurface singularity which is a free* divisor by Proposition 1.8. Let (V, 0) ⊂ (C3, 0)be a weighted homogeneous surface singularity and let V ′ = V × C, with lastcoordinate w. Given a nonlinear section of V , f1 : (Cn, 0) → (C3, 0) we constructa section of V ′, f : (Cn, 0) → (C4, 0) defined using coordinates x = (x1, . . . , xn)for Cn and y = (y1, y2, y3) by f(x) = (y, w) = (f1(x), f2(x)). If f1t denotes astabilization of f1 as a section of V , then ft = (f1t, f2t) is a stabilization of f as asection of V ′. Moreover, their singular Milnor fibers are homeomorphic. It sufficesto consider f and ft and NK∗

He.As an example, consider a Pham–Brieskorn singularity (V, 0) ⊂ (C3, 0) defined

by h(y) = yp1 + ym2 + y`3 = 0, and let wt(yi) = ai. We consider an “Ak−1” typesection of V × C

f(x) = (f1(x), f2(x)) = (xk1 +n∑i=4

x2i , x2, x3, x1),

and consider a deformation where we only deform f1.To apply Theorem 5, we recall from the proof of Proposition 1.8 the determinan-

tal vector fields ηij =∂h

∂yi

∂

∂yj− ∂h

∂yj

∂

∂yi, and the generators for Derlog∗(h) given

by

ζi = ηjk + aiyi∂

∂w(i = 1, 2, 3) and ζ0 = −de = −

∑aiyi

∂

∂yi. (4.8)

where for i > 0, (ijk) is a cyclic permutation of (123).

∂f

∂x1=(∂f1∂x1

, 0, 0, 1),

∂f

∂x2= (0, 1, 0, 0)

∂f

∂x3= (0, 0, 1, 0)

and∂f

∂xj= (2xj , 0, 0, 0) for j ≥ 4.

382 J. DAMON

If we project NK∗He ·f off the free submodule generated by fxi

, i = 1, 2, 3 onto thefirst component and take the quotient by fxi

, i ≥ 4, we obtain from the generatorsin (4.8)

NK∗He · f ' OC3

/(−a1g1

∂g1∂x1

,∂h

∂y3◦ g − a2x2

∂g1∂x1

, − ∂h

∂y2◦ g − a3x3

∂g1∂x1

)(4.9)

with g : (C3, 0) → (C4, 0) obtained from f by setting xi = 0, i ≥ 4. Consideringdegrees in (4.9) we obtain dim CNK∗

He · f = (2k − 1)(`− 1)(m− 1).Second, to determine the contribution from virtual singularities, we have xi = 0

for i ≥ 4, h = 0, and the generators in (4.9), with g replaced by gt, vanish. From

the first generator, either g1 t, or∂g1 t∂x1

= 0. If∂g1 t∂x1

= 0, setting the second and

third generators equal to 0 implies that y2 = y3 = 0. We only consider pointswhere h ◦ ft = 0; thus y1 = 0 so g1 t = 0. However, generically we can assume both

are not 0, so we may assume∂g1 t∂x1

6= 0. Then, there are k solutions to g1 t = 0

and (`− 1)(m− 1) solutions to the second and third generators = 0. At each suchvirtual singular point, they generate the maximal ideal so the corresponding localalgebra as in (4.9) has dimension 1. Hence, Theorem 5 gives

µV (f1) = µV ′(f) = (2k − 1)(`− 1)(m− 1)− k(`− 1)(m− 1)

= (k − 1)(`− 1)(m− 1). (4.10)

This is also the Milnor number of the Pham–Brieskorn singularity xk1 + xm2 + x`3 +∑ni=4 x

2i = 0. However, the singular Milnor fiber for a stabilization such as (xk1 +∑n

i=4 x2i + t)p + xm2 + x`3 = 0 must have singularities along xk1 +

∑ni=4 x

2i + t =

x2 = x3 = 0, and is superficially quite different from the Milnor fiber of this Pham–Brieskorn singularity. Is there an intrinsic way of seeing the relation between theseapparently quite distinct objects?

Example 4.4. Corollary 7 also applies to free divisors when n ≥ hn(V ). As alast example, consider the surface singularity (V, 0) ⊂ C3 defined by F (x, y, z) =f(x, y) + zg(x, y), where both f and g are weighted homogeneous with f definingan isolated curve singularity and wt(g) ≥ wt(f). Let J(f) be the Jacobian idealof f . Suppose (J(f) : g) is a complete intersection ideal with generators {h1, h2}satisfying the numerical condition

wt(h1) + wt(h2) + wt(g) = wt(x) + wt(y) + wt(H)

where H is the Hessian of F . Then V is a free divisor but with nonholonomicstratum the z-axis [D8]. A generic virtual singularity is given by a Morse-typesingularity for K-equivalence (Example 7.7) of the form

ϕ(x1, . . . , xn) = (x1, x2, z0 +n∑j=3

x2j )

whose discriminant is transverse to the z-axis. A straightforward computationshows

dimNK∗H,e · ϕ = dim COx,y/(J(f) : g),


the colength of the ideal (J(f) : g). Thus, the singular Milnor number for anonlinear section ϕ : (Cn, 0) → (C3, 0) is given by

µV (ϕ) = K∗H,e-codim(ϕ)− col(J(f) : g) · (D(ϕ), z-axis). (4.11)

II. Discriminants of Sections of Free Complete Intersections

5. Free Complete Intersections

We establish in this section several basic properties of free complete intersections.Suppose V is a free complete intersection with free defining equation h, we firstremark that V ×Cq is a free complete intersection with free defining equation h ◦πwhere π : Cp+q → Cp denotes projection. This follows from the following simpleLemma whose proof is straightforward.

Lemma 5.1. Let u = (u1, . . . , uq) denote local coordinates for Cq, and let ζi ∈ θp,1 ≤ i ≤ r. Then,

Derlog(h) = OCp,0{ζ1, . . . , ζr}iff

Derlog(h ◦ π) = O(Cp+q,0)

{ζ1, . . . , ζr,

∂

∂u1, . . . ,

∂

∂uq

}.

Next, we give a simple criterion for establishing the freeness of complete inter-sections.

Proposition 5.2. Suppose (Vi, 0) ⊂ (Cp, 0), i = 1, . . . , r are divisors (not neces-sarily irreducible) with reduced defining equations hi : (Cp, 0) → (C, 0) such that:

(1) V ′ = (⋃Vi, 0) ⊂ (Cp, 0) is a free divisor (defined by

∏hi);

(2) there exist Euler-like vector fields ei for hi such that ei(hj) = δijhj ;(3) V = (

⋂Vi, 0) is a complete intersection defined by h = (h1, . . . , hr).

Then, (V, 0) is a free complete intersection with free defining equation h.

Proof. We observe that Derlog(h) ⊂ Derlog(V ′), and Derlog(V ′) is a free OCp,0-module. It is sufficient to prove Derlog(h) is a direct summand of Derlog(V ′) forthen it is a finitely generated projective OCp,0-module, hence free. We define aprojection

ψ : Derlog(V ′) → Derlog(h), ξ 7→ ξ −r∑i=1

ξ(hi)hi

ei. (5.1)

To see it is well-defined, let ξ ∈ Derlog(V ′). Then, ξ is tangent to Vi,reg \⋃j 6=i Vj .

Thus, ξ(hi) vanishes on Vi,reg\⋃j 6=i Vj , and hence its closure Vi. Hence, ξ(hi) = ϕihi

andξ(hi)hi

= ϕi. At least ψ(ξ) ∈ θp.Moreover, it is easily checked that ψ is OCp,0-linear. It remains to check that

ψ is projection onto Derlog(h). However, it is straightforward to check that: ifξ ∈ Derlog(V ′), then ψ(ξ)(hj) = 0 all j, and if ξ ∈ Derlog(h), then ψ(ξ) = ξ. Thiscompletes the proof. �

384 J. DAMON

We list several consequences.

Corollary 5.3. The product (V, 0) = (∏Vi, 0) ⊂ Cp of free divisors (Vi, 0) ⊂

(Cpi , 0), i = 1, . . . , r, with good defining equations hi is a free complete intersectionwith free defining equation h = (h1 ◦ π1, . . . , hr ◦ πr), where πj : Cp → Cpj denotesprojection.

Proof. Let V ′i = Vi ×

∏j 6=i Cpj , which is still free with defining equation hi ◦ πi.

Then, V ′ =⋃V ′i is the production union of the Vi and hence also free. Also, the

Euler-like vector fields ei for each hi can be trivially extended to Cp and satisfy (2)of Proposition 5.2. Finally, V =

∏Vi =

⋂V ′i . Thus, by Proposition 5.2, (V, 0) =

(∏Vi, 0) is a free complete intersection with free defining equation h = (h1 ◦ π1,

. . . , hr ◦ πr). �

As a consequence of Corollary 5.3, we deduce that products of any of the freedivisors listed in Part I, such as discriminants of versal unfoldings, Coxeter arrange-ments, etc. yield free complete intersections. These are the main examples. Thesimplest such is the product of {0} ⊂ C yielding the free complete intersection{0} ⊂ Cp. The product of r discriminants of versal unfoldings yields a free com-plete intersection which is part of another discriminant of a versal unfolding whereat least r singular multigerms appear.

Remark. Products of free divisors form a very special class of nonisolated completeintersection singularities. However, a nonlinear section of such a product is theintersection of the pullbacks of the individual free divisors. For a nonlinear sectionalgebraically transverse in a punctured neighborhood, such pullbacks are “almostfree complete intersections” [D3] which naturally extend the class of ICIS.

A second consequence is a weaker form of Proposition 5.2.

Corollary 5.4. Suppose (Vi, 0) ⊂ (Cp, 0), i = 1, . . . , r, are divisors with reduceddefining equations hi : (Cp, 0) → (C, 0) such that :

(1) V ′ =⋃Vi, 0 ⊂ (Cp, 0) is a free divisor (defined by

∏hi);

(2) V = (⋂Vi, 0) is a complete intersection defined by h = (h1, . . . , hr).

Let Cr have coordinates (t1, . . . , tr). Then, (V ×Cr, 0) is a free complete intersec-tion with free defining equation h′ = (et1h1 ◦π, . . . , etrhr ◦π), where π : Cp+r → Cpdenotes projection.

Proof. We replace each Vi by V ′i = Vi × Cr, and replace hi by h′i = etihi ◦ π.

Then, ei =∂

∂tiis an Euler-like vector field for h′i and ei(hj) = 0 if i 6= j. Then,

V ′′ = V ′ × Cr =⋃V ′i is still a free divisor (defined by

∏h′i), and V × Cr =

⋂V ′i .

Thus, Proposition 5.2 applies, yielding the result. �

That products of free divisors are free complete intersections extends to productsof free complete intersections.

Corollary 5.5. The product of free complete intersections (Vi, 0) ⊂ (Cpi , 0), i =1, . . . , r, with free defining equations hi is free with free defining equation h =(h1 ◦ π1, . . . , hr ◦ πr), where again πi : Cp → Cpi denotes projection.


Proof. The proof is by induction on r and reduces to verifying the result for r = 2.Denote h = (h1 ◦ π1, h2 ◦ π1), and let x = (x1, . . . , xp1) and y = (y1, . . . , yp2)denote local coordinates for Cp1 and Cp2 . Also, let

Derlog(hi) = O(Cpi ,0){ζ(i)1 , . . . , ζ(i)

ri} (5.2)

where ri = pi − ki. Finally let p = p1 + p2. Then,

Derlog(h) = Derlog(h1 ◦ π1) ∩Derlog(h2 ◦ π2) (5.3)

(still πi : Cp → Cpi denotes projection). By Lemma 5.1

Derlog(h1 ◦ π1) = OCp,0{ζ(1)1 , . . . , ζ(1)

r1 } ⊕ OCp,0

{∂

∂y1, . . . ,

∂

∂yp2

}(5.4)

with an analogous formula for Derlog(h2 ◦π2). Each summand respects the decom-position

θp = OCp,0

{∂

∂x1, . . . ,

∂

∂xp1

}⊕OCp,0

{∂

∂y1, . . . ,

∂

∂yp2

}.

Hence, by (5.3) and (5.4)

Derlog(h) = OCp,0{ζ(1)1 , . . . , ζ(1)

r1 } ⊕ OCp,0{ζ(2)1 , . . . , ζ(2)

r2 }. (5.5)�

By contrast, the module structure Derlog(V1×V2) becomes increasingly compli-cated. We use the above notation, except now the Vi are free divisors so ri = pi.

Proposition 5.6. Suppose (Vi, 0) ⊂ Cpi are free divisors with good defining equa-tions hi : (Cpi , 0) → (C, 0) for i = 1, 2. Let p = p1 + p2. Then

Derlog(V1 × V2) = OCp,0

{ζ(1)i , 1 ≤ i ≤ p1; ζ

(2)i , 1 ≤ i ≤ p2;

h2∂

∂xi, 1 ≤ i ≤ p1; h1

∂

∂yi, 1 ≤ i ≤ p2

}(5.6)

In particular, even if we restrict to V = V1×V2 which is a complete intersectionof dimension p− 2, we obtain

Derlog(V )|V = OV,0{ζ(1)i , 1 ≤ i ≤ p1; ζ

(2)i , 1 ≤ i ≤ p2} ⊂ O(p)

V,0 (5.7)

and Derlog(V )|V has p generators which will fail to give it good algebraic propertiesfor either discriminants or for computing the vanishing topology of nonlinear sec-tions. This observation motivates the introduction of Cohen–Macaulay reductionfor free complete intersections.

Proof of Proposition 5.6. We have Euler-like vector fields ei for hi so that ei(hj) =δijhj . LetM denote the module on the RHS of (5.6). We easily seeM ⊆ Derlog(V ).

For the reverse inclusion, let ζ ∈ Derlog(V ). We may write ζ = ζ1 + ζ2 withζi ∈ θ(πi), for πi the projection onto Cpi . Hence, ζ(hi) = ζi(hi). We claim ζi ∈M ,i = 1, 2.

For example, we may write ζ(h1)(= ζ1(h1)) = α1h1 + α2h2. Then,

(ζ1 − α1e1)(h1) = (ζ − α1e1)(h1) = α2h2.

386 J. DAMON

As ζ1−α1e1 ∈ θ(π1), α2h2 ∈(∂h1

∂x1, . . . ,

∂h1

∂xp1

), the ideal in OCp,0 generated by the

∂h1

∂xi. However,

∂h1

∂xi∈ O(Cp1 ,0) and h2 ∈ O(Cp2 ,0). Hence, h2 is not a zero divisor

in OCp,0

/(∂h1

∂x1, . . . ,

∂h1

∂xp1

). Thus, α2 ∈

(∂h1

∂x1, . . . ,

∂h1

∂xp1

). If α2 =

∑bi∂h1

∂xi, let

ξ = ζ1 − α1e1 −∑

bih2∂

∂xi.

Then ξ(h1) = 0 and ξ ∈ θ(π1). Hence, by Lemma 5.1, ξ =∑λjζ

(1)j . Thus,

ζ1 = α1e1 +∑

bih2∂

∂xi+∑

λjζ(1)j ,

showing ζ1 ∈M . �

Remark. The conclusion of Proposition 5.6 extends in a straightforward fashionto finite products.

6. CM-reduction for KV -Equivalence for Free CompleteIntersections

Before we begin the proof of Theorem 3, we give more detailed information aboutthe group K∗

V . We recall that it is the subgroup of diffeomorphisms in KV whichwhen restricted to V ×Cn are induced by restrictions of diffeomorphisms from KH .Here KH -equivalence is defined by the subgroup of KV given by

KH = {Ψ ∈ K : H ◦Ψ = H}.

for H the composition of H with projection onto Cp. The extended tangent spaceis given by

TKH,e · f0 = OCn,0

{∂f0∂x1

, . . . ,∂f0∂xn

, ζ1 ◦ f0, . . . , ζr ◦ f0}

(6.1)

where Derlog(H) is generated by ζ1, . . . , ζr.We have the inclusions

KH ⊂ K∗V ⊂ KV .

It is easily checked that K∗V is a geometrically defined subgroup of A or K. To give

the extended tangent spaces, we note that I(V ) · θp ⊂ Derlog(V ) and

TK∗V,e · f0 = OCn,0

{∂f0∂x1

, . . . ,∂f0∂xn

, ζ1 ◦ f0, . . . , ζr ◦ f0}

+ I(V ) · θ(f0) (6.2)

where again ζ1, . . . , ζr generate Derlog(H). If V is a free complete intersection,then r = p− k.

In addition, consider an unfolding F : (Cn+q, 0) → (Cp+q, 0) of f0. We writeF (x, u) = (F (x, u), u) where we use local coordinates u for Cq. There are definedthe associated tangent spaces for the unfolding groups. Let {ζ1, . . . , ζm} be a set


of generators for Derlog(V ). Then, the extended tangent spaces for the unfoldinggroups are given by

TKV,un,e · F = O(Cn+q,0)

{∂F

∂x1, . . . ,

∂F

∂xn, ζ1 ◦ F , . . . , ζm ◦ F

}and

TK∗V,un,e ·F = O(Cn+q,0)

{∂F

∂x1, . . . ,

∂F

∂xn, ζ1 ◦ F , . . . , ζr ◦ F

}+ I(V ) · θ(F ). (6.3)

We note that the normal space for K∗V can be written

NK∗V,un,e · F = θ(F )/TK∗

V,un,e · F

= OX ,0

{∂

∂yi

}/OX ,0

{∂F

∂x1. . . ,

∂F

∂xn, ζ1 ◦ F , . . . , ζr ◦ F

}(6.4)

where X = F−1(V ).To begin the proof of Theorem 3 that K∗

V is a CM-reduction of KV , we firstestablish condition (2) of 2.6. The codimensions are related by the next proposition.

Proposition 6.1. Suppose (V, 0) ⊂ (Cp, 0) is a free complete intersection with freedefining equation H : (Cp, 0) → (Ck, 0). If n < h(V ), then f0 : (Cn, 0) → (Cp, 0)has finite KV -codimension iff it has finite K∗

V -codimension.

Proof. Let T KV,e · f0, T K∗V,e · f0, NKV,e · f0, and NK∗

V,e · f0 denote the associ-ated sheaves to the tangent and normal spaces TKV,e · f0, TK∗

V,e · f0, NKV,e · f0,and NK∗

V,e · f0. Then, by the Nullstellensatz for coherent analytic sheaves, finiteKV -codimension is equivalent to supp(NKV · f0) = {0}, and similarly for K∗

V -equivalence.

We claim that the condition n < hn(V ) implies that

supp(NKV,e · f0) = supp(NK∗V,e · f0). (6.5)

In fact, this follows from a more general result for unfoldings which we now consider.�

Associated to each of these normal spaces for an unfolding F of f0, we havesheaves denoted by NKV,un,e · F , and NK∗

V,un,e · F which are sheaves of OCn+q -modules. The relation between the supports of the normal sheaves is given by thefollowing.

Proposition 6.2. Suppose (V, 0) ⊂ (Cp, 0) is a free complete intersection withfree defining equation H : (Cp, 0) → (Cr, 0), with n < h(V ). Let F be an unfoldingof f0 : (Cn, 0) → (Cp, 0). Then, as sheaves of OCn+q -modules,

supp(NKV,un,e · F ) = supp(NK∗V,un,e · F ). (6.6)

Proof of Proposition 6.2. For the assertion we note that

T K∗V,un,e · F ⊆ T KV,un,e · F,

so we have the inclusion ⊆ in (6.6). For the reverse inclusion, we examine wherethey differ. Because of the sheaf inclusion

I(V ) ·Θ(F ) ⊂ T K∗V,un,e · F,

388 J. DAMON

a point (x, u) ∈ supp(NK∗V,un,e · F ) implies F (x, u) ∈ V . If F (x, u) ∈ V , then

(x, u) /∈ supp(NK∗V,un,e · F ) is equivalent to

TlogH(x) +DF (x, u)(T(x,u)Cn) = TF (x,u)Cp. (6.7)

However, the assumption n < h(V ) implies that

TlogH(x) = TlogV(x) = TxSi,

where Si is a stratum of the canonical Whitney stratification containing x. Thus,(6.7) implies that F is algebraically transverse to V at (x, u) (see [D3, Section 2]).Thus, (x, u) ∈ supp(NK∗

V,un,e · F ) implies (x, u) ∈ supp(NKV,un,e · F ) . Thus, wehave equality in (6.6). �

Finally, applying Proposition 6.2 to f0 viewed as an unfolding on 0 parameters,we obtain (6.5), completing the proof of Proposition 6.1.

Now we turn to considering the KV -critical set and discriminant. These weredefined in Part I in the case that (V, 0) was a divisor. However, the definitionfor (V, 0) a complete intersection is the same. Briefly, given (V, 0) ⊂ (Cp, 0)and f0 : (Cn, 0) → (Cp, 0) which has finite KV -codimension, let F : (Cn+q, 0) →(Cp+q, 0) be a KV -versal unfolding of f0. We write F (x, u) = (F (x, u), u) wherewe use local coordinates u for Cq. Also, we denote the projection π : (Cn+q, 0) →(Cq, 0).

Definition 6.3. We define the KV -critical set of F to be

CV (F ) = supp(NKV,un,e · F ).

Analogously we define the K∗V -critical set of F to be C∗

V (F ) = supp(NK∗V,un,xe ·F ).

Remark 6.4. It follows from the definition of CV (F ), that it consists of points(x0, u0), with y0 = F (x0, u0), such that the germ F (·, u0) : (Cn, x0) → (Cp, y0) isnot algebraicially transverse to V at x0. As n < hn(V ), this is equivalent to therestriction of the projection π : (Cn+q, 0) → (Cq, 0) to X (= F−1(V )) not beingthe germ of a submersion at (x0, u0).

Suppose f0 : (Cn, 0) → (Cp, 0) has finite KV -codimension and n < h(V ). ByPropositions 6.5 and 6.7, if F is an unfolding of the finite KV -codimension germ f0,the critical sets CV (F ) for KV and C∗

V (F ) for K∗V agree. Also, π|CV (F ) is finite to

one. By Grauert’s theorem the direct image of sheaves NKV ·Fdef= π∗(NKV,un,e ·F )

and NK∗V · F def= π∗(NK∗

V,un,e · F ) are coherent. However, these sheaves havesupport exactly the KV and K∗

V -discriminants of F . Hence, the KV discriminantDV (F ) = π(CV (F )) is an analytic subset of the same dimension as CV (F ) andsimilarly for D∗

V (F ) = π(C∗V (F )). Moreover, by Proposition 6.7 their images under

the projection π agree, so

DV (F ) = π(CV (F )) = π(C∗V (F )) = D∗

V (F ).

Thus, condition (3) of Definition 2.6 is satisfied. It remains to show that K∗V is

Cohen–Macaulay, i. e. that NK∗V ·F is Cohen–Macaulay with support of dimension

q − 1.


Proposition 6.5. Suppose that (V, 0) ⊂ (Cp, 0) is a free complete intersectionof codimension k, with k ≤ n < h(V ). Then, NK∗

V · F is Cohen–Macaulay withsupp(NK∗

V · F ) = D∗V (F ) of dimension q − 1.

Proof. Let f0 : (Cn, 0)→ (Cp, 0) have finite KV -codimension and let F : (Cn+q, 0) →(Cp+q, 0) be a KV -versal unfolding of f0. As F is a KV -versal unfolding of f0,F : (Cn+q, 0) → (Cp, 0), viewed as a map, is algebraically transverse to V at 0.Thus, If we write V = V0×Cs with TlogV0 = {0}, then X = F−1(V ) is diffeomorphicto V0 × Cs′ (and so is also a complete intersection of dimension n+ q − k where kis codimension of V ). In particular, it is Cohen–Macaulay.

As V is a free complete intersection with free defining equation H, Derlog(H) isfreely generated by say ζ1, . . . , ζp−k. By (6.4) NK∗

V,un,e ·F is an OX -module whichis the quotient of OpX by the OX -submodule generated by the n+ p− k generators

{∂F

∂x1(x, u), . . . ,

∂F

∂xn(x, u), ζ1 ◦ F (x, u), . . . , ζp−k ◦ F (x, u)

}.

It has support C∗V (F ). Since n ≥ k, n + p − k ≥ p, we apply results of Eagon–

Northcott [EN] extending those of Macaulay–Northcott [Mc], [No], to concludethat C∗

V (F ) is a subvariety of X of codimension ≤ n+ p− k − (p− 1) = n− k + 1in X so it has dimension ≥ n + q − k − (n − k + 1) = q − 1. Moreover, if thecodimension is exactly n− k + 1 (i. e. dimension exactly q − 1), then NK∗

V,un,e · Fis a Cohen–Macaulay OX -module and C∗

V (F ) is Cohen–Macaulay.If dimC∗

V (F ) = q, then so would dimπ(CV (F )) = dimπ(C∗V (F )) = q, implying

π(CV (F )) is locally onto near 0. However, we can now use the definition of CV (F ) assupp(NKV,un,e). By the parametrized transversality theorem, there are points u0 ∈Cq arbitrarily close to 0 for which F (·, u0) is geometrically transverse to V . As n <h(V ) ≤ hn(V ),geometric and algebraic transversality agree (see [D2, Section 3]);thus, DV (F ) = π(CV (F )) can not have dimension q. Thus, the dimension is atmost q − 1.

By the above, dimC∗V (F ) = dimCV (F ) ≤ q − 1 so it is exactly q − 1. Thus,

the critical set C∗V (F ) is Cohen–Macaulay of dimension q − 1 and NK∗

V,un,e · F isa Cohen–Macaulay OX -module. As π|CV (F ) is finite to one, we conclude (e. g.using [Se]) that the direct image under a finite map NK∗

V · F = π∗(NK∗V,un,e · F )

is Cohen–Macaulay as a OD∗V (F )-module and hence as a OCq -module. Its support

is D∗V (F ) = π(C∗

V (F )) which is, hence, Cohen–Macaulay of dimension q − 1. Thiscompletes the proof of Theorem 4. �

Remark 6.6. We observe that the only two properties of F which were used inProposition 6.10 were: that F : (Cn+q, 0) → (Cp, 0), viewed as a map, is alge-braically transverse to V , and that generically F (·, u0) is tranverse to V . Thus,for any unfolding F with these properties, both C∗

V (F ) and D∗V (F ) will be Cohen–

Macaulay of dimension q − 1. Hence, this is true for CV (F ) and DV (F ) using theinduced structure.

390 J. DAMON

7. Morse-type Singularities for Sections of General Varieties

We extend the notion of Morse-type singularity defined for sections of divisorsto arbitrary analytic germs (V, 0). We also will see that specific properties whichhold for Morse-type singularities for divisors continue to hold for free completeintersections.

Definition 7.1. Given (V, 0) ⊂ (Cp, 0) and an integer n > 0, a germ g : (Cn, 0) →(Cp, 0) is a Morse-type singularity in dimension n if g is KV -equivalent to a germf0 which has KV,e-codim = 1 and for a common choice of local coordinates, bothf0 and V are weighted homogeneous.

Furthermore, we say V has a Morse-type singularity in dimension n at x if thereis a germ g : (Cn, 0) → (Cp, x) which is a Morse-type singularity in dimension n(for this, we use K(V,x)-equivalence).

Definition 7.2. V is said to generically having Morse-type singularities in dimen-sion n if: all points on the canonical Whitney stata of V of codimension ≤ n + 1have Morse-type singularities of nonzero exceptional weight type; and any stratumof codimension > n+ 1 lies in the closure of a stratum of codimension = n+ 1.

We first observe that the normal form established in part I for Morse type sin-gularities holds for an arbitrary (V, 0) not just a free divisor.

Lemma 7.3 (Local Normal Form). Let f0 : (Cn, 0) → (Cp, 0) be a Morse-typesingularity for (V, 0) ⊂ (Cp, 0). Then, up to KV -equivalence, we may assume(V, 0) = (Cr × V0, 0) for (V0, 0) ⊂ (Cp′ , 0), and with respect to coordinates forwhich (V0, 0) is weighted homogeneous, f0 has the form

f0(x1, . . . , xn) =

(0, . . . , 0, x1, . . . , xp′−1,

n∑j=p′

x2j

).

Proof. The proof given in part I does not specifically refer to V being a free divisorexcept at one point and there the reference can easily be removed as follows. Theinitial reduction makes no reference to special properties of V . It allows us toreduce to the case p′ = p and show that f0 can be put in the form

f0(x1, . . . , xn) = (x1, . . . , xp−1, f0(x1, . . . , xn)) (7.1)

with df0(0) = 0. Then, in (4.15) of [D6], we explicitly give the generators forTKV,e · f0 and make use of the generators of Derlog(V ). Suppose instead that wehave a set of generators {ζi, i = 0, . . . , `} for Derlog(V ) with ` ≥ p and with ζ0denoting the Euler vector field. By the form of f0 in (7.1)

TKV,e · f0 ⊆ OCn,0

{∂

∂y1, . . . ,

∂

∂yp−1

}⊕mn

{∂

∂yp

}. (7.2)

As KV,e-codim(f0) = 1, we have equality in (7.2). Thus,

∂

∂yi∈ TKV,e · f0 for i = 1, . . . , p− 1.


Also, the projection of TKV,e · f0 onto OCn,0

{∂

∂yp

}' OCn,0 is mn

{∂

∂yp

}. The

Euler relation and (7.1) imply for the Euler vector field ζ0ζ0(yp) ◦ f0 = bpf0 ∈ m2

n

where bp = wt(yp). Hence,{ζi(yp) ◦ f0, i = 1, . . . , `;

∂f0∂xj

, j = p, . . . , n

}generate mn. (7.3)

Also, by (7.1), ζi ∈ mpθp for i = 1, . . . , `. Thus,

ζi(yp) ◦ f0 ∈ mp−1OCn,0 mod (m2n) (7.4)

and, by (7.3), must generate this ideal. Thus,

{ζi(yp), i = 1, . . . , `} generate mp−1OCp,0 (mod (yp) + m2p). (7.5)

After this point, the number of vector fields in Derlog(V ) is no longer of significanceand the rest of the proof remains the same. �

Having established the normal form, the arguments given in part I allow us todeduce Corollaries 4.20, 4.22, and 4.24 of [D6] for Morse-type singularities for anyV . These three results are summarized as follows.

Corollary 7.4. (1) If (V, 0) ⊂ (Cp, 0) has a Morse type singularity in dimensionn, then it has Morse-type singularities in the “allowable dimensions” m ≥ p′ − 1where p′ = p− r and r = dim (Tlog(V )0).

(2) Suppose (V, 0) = (Cr×V0, 0) is weighted homogeneous with (V0, 0) ⊂ (Cp′ , 0)and (Tlog(V0)(0)) = (0). Then V has Morse type singularities in all allowable di-mensions iff there is a weighted hyperplane in Cp′ which is transverse to the orbitsof Aut1(V0) in a punctured neighborhood of 0. Here Aut1(V0) denotes the group oflinearized automorphisms of V0.

(3) If (V, 0) ⊂ (Cp, 0) has Morse type singularities, then there is a Zariski opendense subset of Σ1 consisting of jets of Morse-type singularities (here Σ1 denotesthe 2-jets of germs which are not algebraically transverse to V at 0).

Also, just as in the hypersurface case, we say that V with Morse type singular-ities, has “exceptional weight type” positive, negative, or zero type if V satisfies(7.5) with wt(yp) having the corresponding positive or negative sign or = 0. Then,the next result ensures that having Morse-type singularities is preserved underproducts.

Proposition 7.5. If (Vi, 0) ⊂ (Cpi , 0) for i = 1, 2 have Morse type singularitiesof the same non-zero exceptional weight type, then the product V = V1 × V2 hasMorse type singularities of the same exceptional weight type.

Proof. The proof we give for products is actually very close to the proof of Propo-sition 4.29 given in [D6] for an analogous statement for product-unions.

We let p = p1 + p2. We may independently make weighted homogeneous changeof coordinates to factor (Vi, 0) = (Cri × Vi0, 0). Then, the product factors

V1 × V2 = Cr1+r2 × (V10 × V20).

392 J. DAMON

Hence, we may suppose that TlogVi(0) = (0). Furthermore, we suppose that Cpi−1

are the weighted subspaces given by (2) of Corollary 7.4. Then, as each Vi hasthe same exceptional weight type, we may multiply weights if necessary so thatwt(y(1)

p1 ) = wt(y(2)p2 ) where each Vi has coordinates (y(j)

i ) . Then, we consider theweighted homogeneous subspace

M = (Cp1−1 × {0}) + ({0} × Cp2−1) +

⟨(∂

∂y(1)p1

,∂

∂y(2)p2

)⟩.

Now, we claim that M satisfies the conditions of (2) of Corollary 7.4 so that V hasMorse type singularities with the same exceptional weight type. The verificationfollows exactly the proof of Proposition 4.29 of Part I, since the vector fields usedthere which generate Derlog(V1 ∪× V2) also belong to Derlog(V ). �

Remark 7.6. The products of discriminants of versal unfoldings of simple germs orthe products of Boolean arrangements are examples of free complete intersectionswhich genericially have Morse-type singularities (of positive exceptional weight) inall dimensions and at any point.

Example 7.7. {0} ⊂ Cp is a free complete intersection which is a product of thefree divisors {0} ⊂ C. It has Morse-type singularities of all dimensions, which are(K equivalent to) the standard Σn−p+1,0 germs

f0(x1, . . . , xn) =

(x1, . . . , xp−1,

n∑j=p

x2j

).

The final property of Morse type singularities that carry over to general V is

the KV -liftability of u∂

∂ufor the versal unfolding F of a Morse-type singularity in

normal form f0 from Lemma 7.3

F (x, u) = (F (x, u), u) with F (x, u) = f0(x) + (0, . . . , 0, u). (7.6)

Lemma 7.8. For the KV -versal unfolding F of a Morse-type singularity in (7.6)with nonzero exceptional weight, the discriminant DV (F ) (defined by u = 0) is

reduced and u∂

∂uis a KV -liftable vector field.

The proof is exactly as given in Lemma 4.10 of [D6].

8. KV -Discriminants as Free and Free* Divisors

We conclude by briefly discussing the consequences of our results for KV -dis-criminants DV (F ) of versal unfoldings F for free complete intersections (V, 0).When V is a free divisor which generically has Morse-type singularities, the KV -discriminants are free divisors. Reexpressed in terms of Theorems 1 and 2, thisfollows because genericity of Morse-type singularities for V implies the genericityof KV -liftable vector fields. This together with KV being Cohen–Macaulay (for Va free divisor) implies that the module of KV -liftable vector fields is free and equalsDerlog(DV (F )).


Free complete intersections (V, 0) still generically have Morse-type singularitiesin many cases (e. g. by Proposition 7.5). This still implies the genericity of KV -liftable vector fields by the same arguments given in Part I using instead Corollary7.4 and Lemma 7.8. However, by Example 3.2, we see that the module of KV -liftablevector fields need not be free.

Suppose instead we use the CM-reduction K∗V , and V generically has Morse-

type singularities (for KV -equivalence). A sufficient condition for the genericity ofK∗V -liftable vector fields is that for a Morse-type singularity at a point y ∈ V in a

stratum of codimension ≤ n + 1, with the normal form in Lemma 7.3, the vector

field u∂

∂uis K∗

V -liftable. This cannot be true in general by Example 3.2, otherwiseby Theorem 1, the KV -discriminant would be free using the module of KV -liftablevector fields.

The question reduces to deciding when for a Morse-type singularity in normal

form Lemma 7.3, u∂

∂uis K∗

V -liftable.We consider the two highest dimensional strata: the smooth stratum, and the

strata which up to diffeomorphism have the form V1 × V2 where V1 ⊂ Cp1 is a freedivisor and V2 is smooth. In the second case, we may assume V2 = Cp2−k ⊂ Cp2 .However, V2×Cp2−k ⊂ Cp1+p2−k is still a free divisor, so we can reduce to the casewhere V2 = {0}. The situation for these cases is given by the next proposition,whose proof is given in Part III [D7].

Proposition 8.1. (i) Suppose (V, 0) ⊂ (Cp, 0) is smooth. Then, for any unfoldingF , TK∗

V,un,e · F = TKV,un,e · F . Hence, any KV -versal unfolding F is K∗V -versal,

and all KV -liftable vector fields are also K∗V -liftable.

(ii) Suppose V = V1 × {0} ⊂ Cp, where (V1, 0) ⊂ (Cp1 , 0) is a free divisor andp = p1 + p2 with p2 > 0. Then, for a versal unfolding of a Morse-type singularity

in normal form (Lemma 7.3), u∂

∂uis not K∗

V -liftable.

When (V, 0) is smooth, the KV -discriminant is exactly the K∗V -discriminant by

Proposition 6.2. By Proposition 8.1, a Morse-type singularity at any point of V

also has u∂

∂uK∗V e-liftable. Thus, by the argument of Part I, we have genericity

of K∗V e-liftable vector fields. Hence, Theorem 1 implies the KV -discriminant of a

versal unfolding is a free divisor, recovering the result of Looijenga.

Corollary 8.2 (Looijenga [L]). If V is smooth, the KV -discriminant for the versalunfolding is a free divisor. In particular, the discriminant for the versal unfoldingof an ICIS is a free divisor.

We note the second part of Proposition 8.1 essentially implies that Example 3.2is the typical state of affairs; and we will not generally have freeness of KV -discrim-inants. Except for ICIS singularities, we must settle for free* divisors.

Acknowledgments. The author is very grateful to Anne Fruhbis-Kruger for de-veloping a package for Singular [Sg] and for working with him to use it for verifyinga basic counterexample, and also to David Mond and the referee for a number ofvery useful comments.

394 J. DAMON

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Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA

E-mail address: [email protected]



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on the legacy of free divisors ii: free* divisors and ... · that the simple boundary singularities...

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