TC/96/28

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

RANK TWO VECTOR BUNDLESON ALGEBRAIC VARIETIES I

Shcng-Li Tan1

International Centre for Theoretical Physics. Trieste, Italy.2

MIRAMARE - TRIESTE

March 1996

Permanent address: East China Normal University, Shanghai, People's Republic ofChina.

2E-mail: tan@ictp.tricste.it

Introduction

In modern algebraic geometry, the classification of vector bundles hasreceived much attention. The purpose of this paper is to develop a new methodto classify rank two vector bundles.

As Hartshorne has noted, it is helpful to regard vector bundles as a specialkind of reflexive sheaves. A coherent sheaf T on an algebraic variety X is calledreflexive if the natural map of T to its double dual T** is an isomorphism. Inthis paper, all varieties are defined over an algebraically closed field k, itscharacteristics is denoted by char&. A variety is called factorial if all its localrings are unique factorization domains. We always identify vector bundles withlocally free sheaves.

First we recall the simple construction of reflexive sheaves from hypersur-faces. For i = 1, • • • , r, we let /; be a nonzero global section of an invertiblesheaf d on an integral normal algebraic variety X. We always assume thatthe hypersurfaces defined by the /, 's have no common components. Then/ = (/i, • • • , /r) defines a morphism, / : $%=l£j'1 —>• Ox, f is locally definedby / ( * ] , • • • , xr) = fix-i H \- frxr. The kernel of this map is a rank r — 1coherent reflexive sheaf. This sheaf is called the syzygy sheaf of f. A syzygysheaf is completely determined by its defining hypersurfaces / . The cotangentbundle of a projective space is just the syzygy sheaf of the hyperplans of coor-dinates (due to the Euler sequence). We shall show that any rank two reflexivesheaf can be constructed by this method.

THEOREM 1. Let T he a rank two reflexive sheaf on a projective factorialvariety X over k, and let C be an ample invertible sheaf If char k / 2,3, thenF®C~n is a syzygy sheaf for sufficiently large n. Furthermore, we can choosethree defining hypersurfaces of this syzygy sheaf such that any two of them haveno common components.

Therefore a rank two reflexive sheaf is completely determined by 3 hyper-surfaces. Then a natural problem is: when is a syzygy sheaf locally free? Dueto the well-known Hilbert-Burch Theorem, this problem ca,n be solved simply.We denote by Z(f) the common zero scheme of / . Then we have the followingsimple criterion.

THEOREM 2. Let F be the syzygy sheaf of / = (A, • • • , U) over a non-singular variety X. We assume that the hypersurfaces of f have no non-trivialcomponents in common. Let x 6 Z(f). Then the following are equivalent:

(i) T is locally free on x.(ii) Z(f) is locally Cohen-Macaulay and of pure codimension two at x.(iii) There exists a (r — 1) X r matrix A over Ox such that /; is the (r —

1) x (r — 1} minor of A by leaving out the i-th row.

The syzygy sheaf of / is locally free outside the zero locus Z(f). On anonsingular surface, reflexive sheaves are locally free, so syzygy sheaves arealways vector bundles.

In [Ser], Serre relates rank two vector bundles to locally complete intersec-tion subvarieties of codimension two — Serre construction. This constructionhas been proved very useful in the study of rank two vector bundles (see also[Hal] and [Sch]). Theorem 1 reduces the study of a rank two reflexive sheafto that of three hypersurfaces. Compared with Serre construction, Theorem1 has the advantage that the data of a syzygy bundle are quite simple, just 3hypersurfaces. The data depend essentially on the codimension one geometry.

For nonsingular surfaces, we do not need to worry about if it is locally free. Inparticular, the study of a rank two vector bundle on a projective space is re-duced to that of 3 homogeneous polynomials, the latter is elementary althoughit is also complicated.

As a direct application of Theorem 1, we consider the first coliomologymodules of rank two vector bundles T on a projective space 3Pn. We denote byS = k[x0> • • • ,xn] the homogeneous coordinate ring of Pn . The first cohomol-ogy module of T is a finite graded 5-module defined by

Similar to the Hartshorne-Rao modules of space curves, this module is animportant invariant of sheaves. In fact, Horrocks [Hor] proves that a ranktwo vector bundle on projective plan is completely determined, up to shiftsin degree, by its first cohomology module. For higher dimensional case, notall finite graded 5-modules can be the first coliomology modules of rank twovector bundles. A problem proposed by Hartshorne in ([Ha3], Problem 10) is:

Problem. Characterize those finite-length 5-modules M such that thereexists a rank two (resp. stable) vector bundle whose first cohomology moduleis isomorphic to M.

Let T be the syzygy sheaf of / = (/ lf /2 , / 3 ) , let / ( / ) be the ideal of Sgenerated by the / ; ' s , and let / ( / ) be the saturation of / ( / ) (cf. [Ha2], p.125,Ex. 5.10)). Then we can prove that the 5-module Hl{!F) is isomorphic to/ ( / ) / / ( / ) (cf. Theorem 5.2). By Theorem 1, we have

COROLLARY 3. A finite graded S-module M is the first cohomology mod-ule of a rank 2 reflexive sheaf if and only if M is isomorphic to I(f)/I(f) forsome f = (/i ,/2t/3)j where f is a triple of pairwise relatively prime homoge-neous polynomials.

Because the local freeness and the stability of a syzygy sheaf jFcan becharacterized by / (see Theorem 6.1), the above corollary gives also a charac-terization for the first cohomology modules of rank two (resp. stable) vectorbundles. Decker [Dec] has also given a different characterization of the first co-homology modules of rank two vector bundles on QP3 in terms of the minimalfree resolution of M.

Another important problem is the stability (in the sense of Mumford-Takemoto) of the syzygy sheaf T of / . In fact, we shall prove that the stabilityof T is determined by the minimal degree s(f) of the syzygies of / . Forsimplicity and applications, we pay our attention to the case of projectivespaces, r homogeneous polynomials g = (g\, • • • , gr) is call a syzygy of f =

If g = 0, it will be called a trivial or zero syzygy of / . The degree of a non-trivial syzygy g is defined as the degree of a nonzero component /,(/,•. Denoteby s(f) the minimal degree of the nonzero syzygies of / . If r = 3, then thestability of F is measured by s(f). Precisely, T is stable (resp. semistable) ifand only if s(f) > (deg A + deg/2 + deg/3) /2 (resp. « ( / ) > • • • ) • In fact,s(f) is exactly the the minimal number t such that i7°(Jr(i)) ^ 0. The impor-tance of this number has been noted by Hartshorne. He studied it for rank 2

reflexive sheaves on IP3 [Ha5]. Due to Theorem 1, some important work in clas-sical and modern algebraic geometry are equivalent to giving upper bounds ofs(f). For example, the important work of Hartshorne's in [Ha5]; Bogomolov'ainequality on stable vector bundles; the classical Cayley-Bacharach Theorem.(See Sect. 6 for details). Due to these relationships, in Sect. 6, we generalizeCayley-Bacharach Theorem to the most general case.

By a simple computation on the Chern numbers, we find in Theorem 5.1a formula which can be used to compute the number of zeros of a syzygy ofthree plan curves. It seems that this kind of problem has not been studied inalgebraic geometry.

As the third application, syzygy sheaves can also be used to generalizeNoether's AF + BG Theorem which is viewed as the cornerstone of classicalalgebraic geometry.

Let F\, • • • , Fr be r fixed homogeneous polynomials and H an arbitraryhomogeneous polynomial in S ~ k[xQ, • • • , xn], A classical problem is:

Problem. Find the conditions imposed on H so that / / has an expressionas a linear combination

H = AiFi + A2F2 + • •- + ArFr, (Ai e S).

In classical algebraic geometry, it had already been noted that this prob-lem is one of fundamental importance, but also of great complexity and withvery wide ramification (cf. [SeK], p.247 and Appendix C). A necessary con-dition (called Noether's condition) is that the above expression holds locallyat the common zeros of F = (Fi, • • • , F r) . Noether's AF + BG Theorem saysthat this condition is also sufficient if n = r = 2. But in general it is not suf-ficient. In Sect. 7, we can see that this problem is equivalent to the vanishingof H1^^)). In particular, we have

THEOREM 4. Noether's Theorem holds for F (i.e., Noether's conditionis sufficient for any H) if and only if the ideal I(F) generated by F in S issaturated.

COROLLARY 5. Let F = (Fi,--- ,F r) be r curves on W2 with no commoncomponents. Then Noether's Theorem holds for F if and only if Fi, • • • ,Fr

are the (r — 1) x (r — 1) minors of a (r — 1) X r matrix whose elements arehomogeneous polynomials.

As the key point of the proof of Theorem 1, we show that the followingcategories are equivalent:

(A): Rank two coherent reflexive sheaves on X;(B): Triple coverings of X (or cubic extensions of the function field of X);(C): Triples of hypersurfaces (Fi, Fa, F3) with no common components.

For the subcategory of (A) of rank two vector bundles, the correspondingsubcategories of (B) a,nd (C) are respectively flat triple coverings and the setof those three hypersurfaces whose common zero set is a Cohen-Macaulaysubscheme of pure codimension 2.

If / : Y —• X is a (normal) triple covering, then the trace-free submoduleof / . (Oy) is a rank two reflexive sheaf. Our first step is to prove that any rank

two reflexive sheaf arises essentially from a triple covering (Theorem 3.2). It isSchwarzenberger who first considered this kind of problem. He proves in [Sew]that any rank two vector bundle on a nonsingular surface can be the directimage of a line bundle under a double covering. Hirschowitz and Narasimhan[HiN] generalize this result to higher rank vector bundles on any variety. Ourconsideration has the advantage that f*(Oy) is not only a module, but alsoan algebra. Therefore we can do more computations on triple coverings whichlead to (C).

Triple coverings are not well understood in algebraic geometry. It is Mi-randa [Mir] who first studied systematically flat triple coverings. He presentsthe data of flat triple coverings by rank two vector bundles with some mor-phisms. In Sect. 2, we show that the data of a triple covering (not necessarilyflat) can be simplified greatly as a triple (L,a,6), where L is a line bundle, aand b are respectively global sections of L2 and L3. The triple covering of thisdata is just defined as the normalization of the variety defined by z3+az+b = 0in L (see Sect. 2.2 for details). This construction is well-known. Our final (themost difficult) step of the proof is to compute the normalization of a cubicextension.

THEOREM 6. Let R be a Noetherian unique factorization domain suchthat 2 and 3 are units. Let p(s:) = x3 + ax + b be an irreducible polynomialover R, and let A = R[x]/(p(x)) = R[a], where a is a root of p(x). Lf B is theintegral closure of A in its fraction field, then B = R(& Bo, where Bo is thetrace-free R-submodule of B. Then there exist three patrwise relatively primenonzero elements f\, f% and / j in R, (3 and y in BQ such that

ft /3

where / j , fi, fs, ft and 7 are determined explicitly by a, b and a.

BQ is in fact isomorphic to the ij-module of all the syzygies of / =(/ii/2)/3)> which is computable in algebraic geometry (cf. [BaM]). If R isa principal ideal domain, then the computation of the syzygies is quit simple.Note that Theorem 6 is also of some interests in the number field case.

Due to the simple characterization (Theorem 2.4) and Theorem 6, a triplecovering is also determined by three hypersurfaces. In fact, because of this,the desingularization and the computation of the invariants of a triple coveringbecome easy (cf. Remark 4.1). We shall give some interesting applications ofthem in another paper.

In the second part of this series of articles, we shall investigate the non-existence of indecomposable rank two vector bundles on some projective spaces.

1. On syzygy sheaves

1.1. Invertible subsheaves of a syzygy sheaf Let R be a commutative ringwith an identity. Let / = ( A , ' " , / , ) € Rr. An element g = (gi,--- ,gr) € Rr

is called a syzygy of f if

Affl + • ' • + /rffr = 0.

The H-module of all syzygies of / is called the {first) syzygy module of f.

Syzygy sheaves are just the sheafization of syzygy modules.

As in the introduction, we let fi be a nonzero global section of an invertiblesheaf & on a variety X, let / = (/i, • • • i /r)> and let T be the syzygy sheaf of/ . If F{ is the hypersurface denned by /,-, then we shall also call J7 the syzygysheaf of the hypersurfaces F = (Fi, • • • , F r). If fact, one can prove that thesyzygy sheaf of F is independent of the choice of / . In this case £, = O{Fi).

i=\

Then the image of F is an idea! sheaf I generated by the sections /,-. LetZ = Z(F) be the subschemc of X defined by X. Then Z is just the intersectionvariety of the F;'s. Note that if F], • • • , F r have no components in common,then the codimension of Z in X is at least 2.

Now let £ = ©f-iCf—F,1). Tensoring the above sequence by an invertiblesheaf £, we get

0 _ > / f« (^ <g> £) — H°(f ® C) -^ H°{£).

A global section g = (#i, • • • , gr) € / /°(£ ® £) is called a C-syzygy (or simply

syzygy) of f if in / /°(£) we have

or equivalently, 5 e H°(T ® £). If ft- is nonzero for any J, and Gi is thehypersurface defined by gi, then we shall also say that the hypersurfaces G =(Gi, • • • , Gr) is a C-syzygy (or simply syzygy) of F . Obviously we have

LEMMA 1.1. An invertible sheaf £ is isomorphic to a subsheafofT if andonly if f has a nonzero £~x -syzygy.

Therefore, the study of invertible subsheaves of a syzygy sheaf is equivalentto the study of the syzygies of / . The latter is a problem on the geometry ofr hypersurfaces. This reduction is very useful for the study of the stability ofsyzygy sheaves.

1.2. Reflexive sheaves. For the reader's convenience, we shall recall somebasic facts on reflexive sheaves which will be frequently used in this paper. Forthe details, see ([Ha4], Sect. 1).

Let X be a variety and let T be a coherent sheaf on X. We define thedual of T to be the coherent sheaf T* = Hom^, Ox). There is a naturemap of J7 to its double dual T**, the kernel of this natural map is the torsionsubsheaf of J7, and the cokernel is also a torsion sheaf. If the map T —+ J7** isan isomorphism, then J7 is called reflexive.

A coherent sheaf J7 on a, variety is called normal if for every open setU C X and every closed subset A C U of codimension > 2, the restrictionmap JFff/) —> T{U ~ A) is bijective.

PROPOSITION 1.2. If X is normal, then the following 4 conditions areequivalent.

(1) T is reflexive.(ii) T is torsion-free and normal.(iii) J7 is a subsheaf of a locally free sheaf with a torsion-free quotient(iv) J7 is torsion-free, and for each open U C X and each closed subset

A C U of codimension > 2, T\j = j^v\A> where j : U\A —• U is the inclusionmap.

Furthermore, assume that F is reflexive.(v) The singularity set of f (the set of points over which jF is not locally

free) is a closed subset of X of coditnension > 2, or > 3 if X is nonsingular.In particular, a reflexive sheaf on a nonsingular surface is locally free.

(vi) Iff : X —>Y is a proper dominant morphism of normal varieties withall fibers of the same dimension, then f±T is also a coherent reflexive sheaf.

(vii) If X is factorial, then any rank one reflexive sheaf is invertible.

COROLLARY 1.3. Any syzygy sheaf on a normal integral variety X is acoherent reflexive sheaf. In particular, any syzygy sheaf on a non-singularsurface is a syzygy bundle.

Finally, for a reflexive sheaf f, one can define its determinant invertiblesheaf det T. In particular, if X is factorial and T is of rank 2, then

Note that one can define the reflexive modules over a Noetherian integralring R. Similar to (Hi) of the previous proposition, we can see that asubmoduleof Rr with a torsion-free quotient is reflexive. In particular, we have

LEMMA 1.4. The syzygy module of f = (/i, • • • , / r ) G Rr is reflexive if Ris integral.

1.3. When a syzygy sheaf is locally free. Obviously, the singularity subsetof the syzygy sheaf T is contained in the intersection subscheme Z = Z(F).To find the singularity set is completely a local problem. Fortunately we canuse the well-known Hilbert-Burch Theorem (cf. [Bur], [Hil] or [Kap], p.148,Ex. 8).

LEMMA 1.5 (Hilbert-Burch Theorem). Let R be a commutative Noether-ian ring. If

o —• Rr~l -^RT MR —YR/I —„o

is a free resolution of a cyclic module Rjl, where I is an ideal of R andf = (/i, • • • , / r ) , then there exists a nonzero-divisor a £ R such that / ; = aA;,where A; is the (r - 1) x (r - 1) minor of the matrix A obtained by leaving outthe i-th row.

Note that the hypotheses of Hilbert-Burch Theorem are fulfilled if R isa regular local ring or a polynomial ring over a field, grad 7 = 2 and R/I isa Cohen-Macaulay ring (cf. [Sea]). In our case, the Fi's have no commoncomponents, so a is a unit in R.

Now we have

THEOREM 1.6. Let T be the syzygy sheaf of f = ( / j , • • • , / r ) over a non-singular variety X. We assume that the fi's have no non-trivial componentsin common. Let x € Z(F). Then the following are equivalent:

(i) T is locally free on x.(ii) Z(F) is locally Cohen-Macaulay and of pure codimension two at x.(iii) There exists a ( r - l ) x r matrix A over Ox such that fi is the (r —

1) x (r — 1) minor of A by leaving out the i-th row.

COROLLARY 1.7. F is locally free if and only if Z(F) is an empty subsetor a Cohen-Macaulay subscheme of pure codimension 2.

2. Structure of finite morphisms

2.1. Flatness of finite morphisms. Let / : Y —• X be a surjective finitemorphism of normal varieties. We denote by Sf the set of points i £ l overwhich / is not flat. We know that / is flat over x if and only if f+Oy is locallyfree at x. Thus Sj is nothing but the closed subset of the singularities of f*Oy.On the other hand, f*Oy is a reflexive coherent sheaf on X, as / is proper (cf.Proposition 1.2 (vi)). Thus the codimension of Sj in X is at least 2, and wehave (Proposition 1.2 (v))

LEMMA 2.1. If X is nonsingular, the codimension of S/ is at least 3. Inparticular, any finite covering of a nonsingular irreducible surface is flat.

We know that f*Oy is an C?x-algebra and the canonical map Ox —» f*Oyis defined by a global section of f*Oy which has no zeros on X. We denote by£ its quotient sheaf and call it the associated sheaf of / ,

(2.1) 0 —> Ox — f*Oy —> £ — 0.

By definition, £ is a coherent torsion-free sheaf. Since Ox and f*Oy arereflexive, we can prove easily that £ is also reflexive (cf. [OSS], p.154). Onthe other hand, we have a trace map tr. : f*Oy —> Ox, if we denote by £0 thetrace-free subsheaf of f*Oy (i.e., the kernel of tr.), by Proposition 1.2 (iii), £Qis reflexive.

Let d be the degree of / . If d is not divided by the characteristics of k,then ^ir. is a splitting of the sequence (2.1), so £Q coincides with E.

In particular, if / is a double covering of a nonsingular irreducible variety,then £ is a rank one reflexive coherent sheaf, and hence £ is invertible (Propo-sition 1.2 (vii)). Then we know f*Oy is locally free as (2.1) splits locally. Nowwe have

LEMMA 2.2. Any double covering of a nonsingular irreducible variety isflat.

2.2. A canonical construction of finite morphisms. We shall recall thewell-known construction of finite morphisms to a variety X,

Let C be an invertible sheaf on X, and a,- a global section of D for % =1, • • • , d. Denote by V(C) = Spec S(C) the associated line bundle of £, whereS(C) is the symmetric C)y-algebra of C, Let z be the global coordinate inthe fibers of V(£). Then z is a global section of p*£, where p is the bundleprojection of V(C). Thus we obtain a polynomial section of p*Cd

where a,- is viewed as a section of p*C\ Then the zero set of p(z) defines asubvariety E oiV(C). Let Y be the normalization of S. Then the compositionof the normalization with the bundle projection defines a finite morphism / :Y —• X of degree d. f will be called the finite morphism defined by p{z) inV{£).

We always assume that Y is integral or equivalently, p(z) is irreducibleover the function field of X.

2.3. Structure of triple coverings. In what follows, a triple coveringis defined as a surjective finite morphism of degree 3 between two normalvarieties. Two triple coverings fx : Yy —• X and /2 : Y2 —- X are said to beisomorphic if Yi and Y2 are isomorphic over X.

Let U be an open subvariety of X. Then any triple covering / : Y -* Xinduces a new triple covering fu • /

LEMMA 2.3. Let A be a closed subset of X of codimension at least two,and let U = X\A. If the induced triple coverings fm and f2U ore isomorphic,so are fi and /2-

Proof. Since F; is isomorphic to Spec fi*Oyt over X, we only need to showthat f\*Oyx

a n d f2*Oy2 are isomorphic Ox -algebras. We have known thatthey are both reflexive. By hypothesis, fuOy^ \u and f2*Oy2 \u are isomorphicOJJ-algebras. Now from Proposition 1.2 (iv), we know f\^Oyl is isomorphic tofi*Oy2 as O^-modules. Obviously, this isomorphism preserves the C^-algebrastructures. This completes the proof. D

THEOREM 2.4. Let f :Y —• X be a triple covering of a factorial variety.If charfc ^ 3, then there exist an invertible sheaf C and two sections a 6H°(C2), b G H°{C3) such that f is defined in V(C) by

z3 + az + b = Q.

Proof. Since char A; / 3, the trace map tr. : f*Qy —* Ox is nonzero,and the trace-free subsheaf £Q of f+Oy is a rank 2 reflexive sheaf. We haveknown that in this case the trace map splits the sequence (2.1) as f*Qy ~Ox © £, where £ = £o is the associated sheaf of / . Let C C £ be a maximalrank one subsheaf with a torsion-free quotient. Then C is reflexive and henceinvertible. Obviously, the quotient sheaf £/C is isomorphic to XA ® M-, whereM = (£/£)** is an invertible sheaf and XA is the ideal sheaf of a subscheme Aof codimension at least two. Let U ~ X \A, and let { f/,-1 i 6 / } be an affineopen covering of U such that C and £ are trivial over each Ui. Let Zi and{ZJ, Wi} be respectively the local bases of £ and £ on Ui. Then on UiDUj, wehave

(2-2) v̂ J = l% » w w !

where l{j and niij are respectively the transition functions of C and M. Since{1,Zi,wi} is a base of the C[/t-algebra f*Oy\vi as an £9[;t-module. Hence wehave

(2.3) z? = CiWt + c\zi + c'!,

Zi can be viewed as a rational function on V, but not a rational function onX. Hence the minimal polynomial of zi over Kx is of degree 3. This means C{is nonzero for each i.

Substituting Zi and wi in (2.3) by (2.2), we have

*j = mijt^CiWj + ( c^ 1 + SijiT^Zi + l^c'l.

Thus, on Ui n Ujt we havec3 = tjf

hence {c;} is a section c of M ® C 2 over U. Let C be the divisor of c on U.Since 2; can be viewed as an endomorphism of f+Oylu, over Ou, by multi-

plication, by choice, the trace of Z{ is zero. Hence the characteristic polynomialof zi over Ojjt is of the form:

zf + aiZi + bt = 0.

With the same proof as above, we can see that {a,} and {&;} are respectivelytwo sections of C2 and C3 over U. Because X is factorial and A has codimensionat least two, these two sections can be extended respectively to two globalsections a <E H°(C2) and b € H°{C3}.

Now we can construct a new triple covering / ' : Y' —> X by

z3 + az + b = 0.

Note that on U{ \ C , z; generates / * O K since u/,- G Oj / ^cM by (2.3). In fact,f*Oy\ut is exatly the normalization of Oj/Jz;] in the function held Ky, as aj;is integral over 0{js[

zi\- Hence / . ( O y ) ! ^ = f*(Oy)\u- T n a t is t o s a y f'u a n c ]

/[/ are isomorphic triple coverings of U. By Lemma 2.3, / is nothing but / ' ,which is what we wanted. •

Remark 2.1. One may obtain a similar characterization for separablefinite coverings of higher degrees by using the primitive elements of the fieldextensions. If the degree of the covering is a prime, and if the characteristicof the field k is bigger than the degree, the above proof can be used withoutchanges.

3. Rank 2 reflexive sheaves and triple coverings

The aim of this section is to find the relationships between rank 2 reflexivecoherent sheaves and triple coverings.

LEMMA 3.1. Let A be a codimension > 2 subscheme of a factorial varietyX. Then any triple covering of X \ A can be extended uniquely to a triplecovering of X.

Proof. Let U = X \ A, and let fu : V —- U be a triple covering. By Theo-rem 2.4, we can find an invertible sheaf CJJ on U and sections a £ H°(U, Cfj),b e H°{U,£u)i s u c h t h a t fu is defined by

z3 + az + b - 0.

Since X is factorial and A has codimension > 2, Pic(X) = Pic(t/). Thus CJJis the restriction of an invertible sheaf C on X, and the sections a and b canbe extended respectively to global sections of C2 and C3 on X. From thesesections and C, we can construct a triple covering of X, which is obviously theextension of fu.

The uniqueness has been proved in Lemma 2.3. •

THEOREM 3.2. Up to tensoring an invertible sheaf, any rank 2 reflexivesheaf on a projective factorial variety is isomorphic to the associated sheaf ofa triple covering.

Proof. Let £ be a rank 2 reflexive sheaf on a projective factorial varietyX, and let A be the singularity locus of £ and X. Then A has codimension atleast 2. Let V = X \ A. Then U is nonsingular and £' = Z\j is locally free.

In order to construct the desired triple covering of X, we consider theprotective line bundle

p :P = $>(£')—*U.

We have an invertible sheaf O(l) on P.We can choose a very ample invertible sheaf £ on U as X is projective. It

is well-known that M = O(3) ® p*Cn is a very ample invertible sheaf on P ifn is sufficiently large ([Ha2], p.172, Ex. 7.14 (b)). Since P is nonsingular, byBertini Theorem, we can find a generic nonsingular irreducible divisor V in thelinear system \M\ of M. Let / ' : V —*• U be the projection. We can assumethat / ' is surjective. We know that the restriction of 0{V) to a generic fiberof p is C r i(3), hence / ' is a generically finite morphism of degree 3. BecauseV and U are nonsingular, we can find a closed subset A' of U of codimension> 2 such that / ' is a flat finite morphism over U' = U \ A'. Without lossof generality, we assume U = U', namely we assume that / ' : V —* U is aOat triple covering. By Lemma 3.1, / ' can be extended uniquely to a triplecovering / : Y —*• X.

Since OP(-V) S O(-3) ®p*C~n, by standard formulas (cf. [Ha2], p.254,Ex. 8.4), we have

R1P.O{-V) <*{£®det£® £n)*\u.

From the exact sequence

0 _ OP(-V) -^OP^OV^O,

we obtain

0 —, Ou —^ KOv ~^(£®det£® tn)*\u ~~> 0.

If T is the associated sheaf of / , then T\j is the associated sheaf of / ' . Hence

Because both sheaves are reflexive, we have

By the formula of Sect. 1.2., £* 3* £ ® (detf )" 1 . The theorem has beenproved. •

COROLLARY 3.3. Let C be an ample invertible sheaf on X. Then thereexists a nature number no such that for any n > n0, £ <g> £~n is the associatedsheaf of a triple covering of X.

Proof. We only need to know that there exists a nQ such that 0(3) ®p*!(det £')~2 ® p*Xfj is very ample for all n > n0. n

10

4. Integral closure of a cubic extension

In this section, we shall prove Theorem 6. By Corollary 3.3, Theorem 6is a local version of Theorem 1.

Let R be a Noetherian unique factorization domain (UFD) such that 2and 3 are units. Let p(x) = x3 + ax + b be an irreducible polynomial over R,and let A = R[x]/(j)(x)). Then A = R[a] is an integral domain, where

a3 + act + b = 0.

Our purpose of this section is to find out the integral closure B of A in itsfraction field K,

THEOREM 4.1. Assume that Bo is the trace-free R-submodule of B. Wehave B = R © BQ. If a / 0, then there exist three patrwise relatively primenonzero elements fi, fi and fy in R, 0 and j in B such that

/? + R and f3 | flU + f2v/3 h

where fi, f2, fs, fi and 7 are determined explicitly by a, b and a.If a = 0 and b = m3s2t, where s and t are square-free, then B has a base

O: Q?

m' m2s

Note that BQ is isomorphic to the syzygy module of (fx, f2, / j ) .In fact, the second part of the previous theorem can easily be presented as

the first part (fi = 1 for all i). Because the proof of the second part is trivial,in what follows, we always assume that a / 0.

We shall say that A is non-normal (resp. singular) over a prime p of R ifthere is a prime ideal q of A over p such that Aq is non-normal (resp. singular).We also say that p is contained in the non-normal (resp. singular) locus of A.Miranda classified in ([Mir], Lemma 5.1) the codimension one singular locusof A:

LEMMA 4.2. Let p be a prime in R. Then A is singular over p if and only

if(i) p\ a andp2 \ b, or(ii) p \ a but p2 j S (hence p \ b),

where S = 4a3 + 27b2 is the discriminant of p(x).

Proof. Reduce to the localization of R at the prime ideal (p) and then use([Mir], Lemma 5.1). Q

Note that the codimension one singular locus p of A is exactly the non-normal locus of A, because A is flat over R.

We shall denote by xv the highest power of p in x, and by [s/t] the maximalinteger < s/t.

Without loss of generality, we can assume that a and 6 have no primedivisors p such that p2\ a and p3\ b. Otherwise, we let £ = min{[ap/2], [and let

a = p~2ta, b = p~3ib, a = p~la.

11

Thena-3 + aa + b = 0,

and R[a] has the same integral closure as R[a].Now we can see that the non-normal locus p £ R of A can be divided into

the following types (I) to (III).(O) ap > 1, bp = 1, (5P = 2);(I) ap = l,bp> 2, («„ = 3);

(II) a p > 2 , 6P = 2, («p = 4);(III) ap=bp = 0, 5P > 2.Note that the primes of type O are not contained in the non-normal locus

of A.We let

m3= Yl P1"^1' TO; = I I P' for* = 0,1,2.p is of type III p is of type i

Then we have

a = momim^ft, & = momfm^bt (a, b) = 1.

Now we can define the /; 's in Theorem 4.1 as follows,

/ i = o S m 2, h =b, f3 = m3.

We can see easily that /; and /,• are relatively prime if i / j .Let A C A' be the it!-submodule generated by 1 and the elements of the

form

(4.1) -fa+-£-/3, u.ueii , /3|«A + u/2,

where /3 = (3a2 + 2a)/3mim2 .

LEMMA 4.3. The R-module A is a reflexive Noetherian ring integral overA.

Proof, Let Ao be the ij-submodule of A generated by the elements of theform (4.1). Then we can see easily that Ao is isomorphic to the syzygy moduleof / = (/j , /2 , / s) . Thus we have an exact sequence

0 —• Ao —-+ R3 -^ R .

By Hilbert Basis Theorem, Ao is a Noetherian .R-module, and hence A =i J 0 A o is also Noetherian. Therefore A is a finite i?-module. By Lemma 1.4,Ao is reflexive, so is A. If we can prove that A is a subring of K, then it iswell-known that A, containing A, is integral over A.

In order to prove that A is a ring, we only need to prove that the productof any two elements of the form (4.1) is still of this form.

By some computations, we have

a2 —

aft = --

12

Thus if xi = (uia+ vi0)/f3, i = 1,2, satisfy (4.1), i.e, there are two elementsW\ and w-2 in R such that

(4.2) Aiii + M + /3u>; = 0, t = 1, 2.

then x\x2 = {ua + v/3 + t» ) / / | , where

M = - -m o v 2 (Awi + / )

+

First we have to prove that /3 divides « and u, and / | divides «?.

Indeed, from (4.2) u is obviously divided by /3 . On the other hand, bydefinition, / | divides 5 — 27mom^m|5, where

5

so / | divides 5. From (4.2), we have

thus

It implies that fs divides v.Similarly, we can prove that w is divided by / | by considering f$w.Now we have to check that ufi + v/2 is divided by / | . In fact, this can

also be proved similarly by considering A(«A +^/2}- Thus we have completedthe proof. D

In order to prove Theorem 4.1, we only need to prove that A is the integralclosure of A in its fraction Jield K. Then by Serre's criterion for normality, itis sufficient to prove that A satisfies the following two conditions.

(Ri) A is nonsingular in codimension one. ^(S2) Every ideal / of codimension two contains a regular sequence on A

with two elements.

LEMMA 4.4. A satisfies Serre's condition 82-

Proof. Since A is reflexive, A satisfies S2 (cf. [Vas]). •

LEMMA 4.5. A satisfies Serre's condition R\.

Proof. In order to prove that A is nonsingular in codimension one, we onlyneed to prove that A is the codimension one normalization of A. Equivalentlywe have to show that for any p of type I, II and III, we can find an element zin A such that p is not contained in the non-normal jocus of R[z] (cf. [Sto]). Infact, by Proposition 4.2, it is sufficient to find a z in A such that its discriminantS(z) is not divided by p2.

For this purpose, we consider a general element z in Ao, i.e., z =v(5)j'/3, and there is a w in R such that

faw = 0.

13

Note that if M is the representation matrix of (l,z,z^) under the base(1, a, a2), then 8(z) = (det M)2S. By a straightforward computation, we have

(4.3) 5 { z ) = , , .m m J

Now let p be a prime in R contained in the non-normal locus of A.Firstly, if p is of type I with bp > 3, then we take 7' = ex + j3, i.e.,

u = v = fa. Note that p divides mi and / ; , but does not divide mo/i- Then,from the above formula we have £(7% = 1 since Sp = 3.

If p is of type II, or of type I with bp = 2, we consider /?, (u = 0, v = p6).By the above formula, we can see that 6{0)p = 1.

Now we assume that p is of type III and 5P is even. Let r = (/2« — fift)jh-Then

Hence 5(r)p = 0.Finally, we consider the remaining case when p is of type III and 5P is odd.

In this case, replacing v in (4.3) by -(fiu + hw)/f2, we get

t e r m s ^

Note that in this case 5P = 2/3P 4-1- If P dose not divide MW2, then we can seethat S(z)p = 1. So we need to find a syzygy (u, u, ^ ) of (/1, /3 , /s) such that(p, MUJ) = 1.

It is obvious that such a syzygy exists, because for generic (1,^1*3 € R,we have a syzygy

(£1/2 - £2/3, £3/3 - hhi hh ~ £3/2),

which satisfies our requirement.Up to now, we have completed the proof of this lemma •

Therefore Theorem 4.1 has been proved.

Proof of Theorem 1. By Corollary 3.3, we only need to prove that theassociated sheaf of a triple covering is a syzygy sheaf.

Let f : 7 -> X be a triple covering determined by (C, a, b) as in Theorem2.4. As in the local case, we can assume that a ^ 0 and that any commoncomponent T of A = div(a) and B = div(6) satisfies Hr{A) < 2 or )J,r(B) < 3,where /^r(-4) and /ir(JS) a r e respectively the multiplicities of V in A and B.Now we can define globally mi, / j , a and J5 such that their restrictions to thelocal ring R = Ox,x coincide with the local definitions above. Then we cansee easily that the trace-free submodule of TT*(OY) is nothing but the syzygysheaf of ( / i , / 3 , / 3 ) . Q

Remark 4.1. If we try to resolve the singularities of a triple covering bythe embedded resolution of its branch locus, then the most important stepis the computation of the normalization (Theorem 4.1). The invariants of a

14

syzygy sheaf can be computed easily (see the next section), so do those of atriple covering. For example, if x : Y —• X is a triple covering of a nonsingularprojective surface X, and if the trace-free subsheaf of ff*(C?y) is the syzygysheaf of F = (Ft , F?, F3), then we have

If X = P2, one can prove easily that the irregularity of Y is at most 1.

5. Invariants of syzygy sheaves

5.1. The number of zeros of a syzygy. Let T be the syzygy sheaf ofthe hypersurfaces F = (Fu--- ,Fr), and let Z = Z(F). Assume that thesehypersurfaces have no common components. From the exact sequence

(5.1) 0 —» T — © 0 ( - F t ) — Xz —• 0,

we have

Hence the Chern classes of T can be computed easily. For example,- E-=i ft, and c2(f) = Y:(<j FiFj - c2(Iz).

Now we give an application of the above computation. Let X be a non-singular projective surface over k. Let F = ( F j , ^ , F3) be 3 curves on X) andlet G = (Gi, G2, G3) be a syzygy of F, We denote by Z(G) the scheme of thecommon zeros of Gi,C?2 and G3.

THEOREM 5.1. I-Vie assume that both F and G have no components incommon. Then we have

deg Z(G) = FXF2 + F2F3 + FXF3 - {Fx + F2 + F3)C + C2 - deg Z{F),

where C = .F,- + Gi for each i.

Proof. Let T be the syzygy sheaf of F,

0 —> ? -^ 0 C?(-F;) —> ^ ( F ) —^ 0.

Then we have

ci{?) = -Fl - F2 - F3,

c2(Jr) = ^1*2 + F2F3 + F1F3 - degZ(F).

On the other hand, G is a global section of F ® C, hence

deg^(G) = c2{F®C) = c2(jf) +

Now we can obtain easily the desired formula.

IS

For higher dimensional variety X, we can obtain the same formula aboutthe Chern classes C2{Zz(F)) a Rd C2(2z{G))-

On P2, if degF = (n,n,n), and degG = (m,rn,m), then

deg Z(G) = n2 + m2 -nm- deg Z{F).

For example, three cubics must have 7 points in common if they are a syzygyof three lines in general position. The converse is also true (cf. Corollary 6.5).

5.2. The first cohomology module of F. Let X = IF", and let T bedefined by / = (/i, • • • , fr)- As in the introduction, the invariant of the firstcohomology module of J- is defined as

This is a finite graded S-module (cf. [Ha4], Theorem 2.5). From the long exactsequence of (5.1), we can obtain

M — 0,

where a; is the degree of /,-. The image of / is the ideal / ( / ) of S = k[xQ, • • • , xn]generated by the / , ' s . H®(2z) is the ideal of S generated by homogeneouspolynomials vanishing on Z, hence it is nothing but the saturation / ( / ) of/ ( / ) in S. Thus we have

THEOREM 5.2. With the notations as above, the first cohomology moduleof F is isomorphic to / ( / ) / / ( / ) .

Now we can obtain Corollary 3 by Theorem 1.

6. Stability of rank two syzygy sheaves and applications

6.1. Stability of rank two syzygy sheaves. In this section, we shallconsider the stability of a, ranktwosyzygy sheaf on an-dimensional nonsingularprojective variety X with respect to a fixed ample divisor / / . For any coherents h e a f s , we denote by degj 7 — CiiJ7)}!11^1 the degree of T with respect to H.A ra,nk two torsion-free sheaf is stable if for every rank one coherent subsheafL C T, the i nequality deg C < deg T /2 holds. If the weaker inequality deg C <deg T ji holds, we say F is semistable.

Let F = (Fi, F2)F'i) be 3 hypersurfaces on X without component in com-mon. Let T be its syzygy sheaf. We denote by a,- = degF, the degree of F;. IfG is a £-syzygy of F, then we shall call also deg£ the degree of G. First wedefine an invariant s[F) of F which measures the stability of F.

s(F) := min{deg£ | F has a non-trivial £-syzygy }.

In fact s(F) is the minimal degree of the syzygies of F .

THEOREM 6.1. With the notations above, we have(i) T is semistable if and only if s(F) > (oi + a2 + a$)f2.(ii) J7 is stable if and only if s(F) > («i + a2 + a3)/2.

16

Proof. T is semistable if and only if for any invertible subsheaf £ C / , wehave CH < ^c\{T)H = ~(ai + a2 + ag)/2. By Lemma 1.1, this is equivalentto the fact that for any non-trivial £-1-sy2ygy of F, — CH > (ai + «2 + «3)/2.By definition of s(F), it is equivalent to s(F) > (ai + 02 + «3)/2. This proves(i). The proof of (ii) is the same if we change > to >. •

Therefore the stability of JF is completely determined by the syzygy in-variant s(F). lit what follows, we shall try to understand the behavior of s(F).Sometimes it is more convenient to consider

syz (F) := s(F) — max{o;}.

6.2. Syzygies on surfaces. By Lemma 1.1, s(F) is the minimal degree ofinvertible sheaves C such that HQ{!F®C) / 0. Because most of our applicationsare for P2, for simplicity, we only consider the case X — P2. In fact, one cangeneralize easily the results to the general case.

For this purpose, let F = (Cm,Cn, Cm +"-7) , where Cl is a curve of degreei. Without lose of generality, we assume that m<n<m + n~ 7.

Note first that T is stable (resp. semistable) if and only if syz (F) > 7/2(resp. > 7/2).

THEOREM 6.2. We assume that the three curves of F have no commoncomponents, and let Z = Z(F). Then we have syz (F) < 7. Furthermore,

(i) ifj<0, thensyz(F) = y,(ii) if 7 > 0 and deg Z > mn — y2/4, then

(iii) if 7 > 0 and deg Z < mn - y2 /4, then

7 / ~2 f !0 < syz (F) < —}- y nin — deg Z H .

Proof. We still denote by C1 the defining polynomial of the curve C \We can see that (Cn, - C m , 0) is a non-trivial syzygy of F, hence s(F) <i, i.e., syz (F) < 7. If 7 < 0, then this syzygy is of minimal degree. Hence

syz (F) = 7.Case (ii) is due to the Bogomolov-Schwarzenberger inequality. In fact the

condition degZ > mn — j2/4 is equivalent to c\{T) > 4c2(Jr). Hence T isnot stable, and there exists a maximal invertible sheaf L~l C ^ , such thatdeg£ - 1 > ^deg-/7, i.e., I = deg£ < m + n - y/2. By Lemma 1,1, there existsa £-syzygy G of F such that G has at most finite number of zeros. By Theorem5.1, we have

£2 - (2m + 2n - 2)C + mn + (m + n)2 - (m + n)y + deg Z(F) + deg Z(G) = 0.

Because t < ra + n — 7/2, we have

n - l - Jdeg Z{F) + deg Z(G) + y - mn .

Note that syz (F) < £ — m — ra + 7 and deg Z(G) > 0, hence we have the desiredinequality of (ii).

For case (iii), we have to use a different method. Note that if H°0, then s(F) < L By Riemann-Roch Theorem,

17

X (^i)) = \ («5(^W) + 3ci(^W)) - cs(^W) + 2

v2

Hence if

£ >- ' - • • - 2 • y — o - • 4 4

then x(F{t)) > 0, which implies that at least one of H°(F(£)) and H2{F{£))is non-vanishing. By Serre duality, h2 {?(£)) = /io(/"(2m + 2ra - 7 - 3 - ^)).But £ > 2m + 2ra - 7 - 3 - £, hence if°(^(^)) + 0 and s(F) < L Thussyz (f1) < £ — m~n + y. Then we can obtain the desired inequality by choosinga minimal £. •

Now we recall the classical remarkable Cayley-Bacharach Theorem:(A) Every curve Cm+n~3 which passes through mn — 1 of the points of

Cm • C n , assumed distinct, passes through them all.(B) Every curve C m + n ~ 7 (7 > 3) passes through mn - \ (7 - 1) (7 - 2) of

the points Cm • Cn passes through the remainder, except when these remaining^(7 — 1)(7 — 2) points lie on a C7~3 .

Originally, Cayley stated the above two theorems in a single form whichtook no account of the exceptional case. The necessary modification is due toBacharach (see [SeR], p.98, for the history).

In fact, this is a kind of problem on bounding syz (F). Due to Theorem6.2, we have

COROLLARY 6.3. / / c m + " - > passes through mn- 7 ( 7 + l)/2 + S {5 > 0)points of Cm • Cn, then the remaining 7(7 + 1) — & points He on a curve ofdegree syz (F) < 7.

Proof By assumption and the previous theorem, we can see easily thatsyz (F) < 7. Let {G\, G2, G3) be a syzygy of F with degree s(F) < m + n.Then G3 / 0 and degG'3 = syz(F). Because the mn points Cm • Cn lie onG3 and Cm+n~'y, the remaining points must lie on G3. This completes theproof. •

Now we can generalize Cayley-Bacharach Theorem as follows.

COROLLARY 6.4. Assume 7 > 3. }fQm+n~'y passes through at least mn-7 + 2 points of Cm • Cn, then it passes through all of them.

Proof In our case, we can use Theorem 6.2 (ii) to prove that

Hence syz (F) = 0. By Corollary 6.3, F has a syzygy (Gi,G2 ,G3) such thatG3 / 0 is of degree 0, so G3 is a nonzero constant. Thus Cm+n~"' is a combi-nation of C m and Cn. This implies that Cm+n~'1 passes through all the pointsCm-Cn. •

If degZ = mn - £(7 - 1)(7 - 2), then by Theorem 6.2 (ii) for 7 < 5and (iii) for 7 > 5, we have syz (F) < 7 — 2, hence syz (F) < 7 - 3. This isCayley-Bacharach Theorem B.

18

COROLLARY 6.5. IfCm+n 7 passes through at least mn—2j+5 ( or mn —2 if f — 3) of the points of Cm -Cn, then the remaining points lie on a line.

Proof, The condition implies syz (F) < 1. •

6.3. Syzygies on P3. Let F = (Fm,Fn, Fm+n~^ be 3 surfaces in P3

with degrees m, n and m -f n — 7 respectively. Let Z = Z(F). Assumethat m < n < m + n + 7- The following theorem is due to Bogomolov andHartshorne.

THEOREM 6.6. (char& = 0). We assume that the three surfaces of F haveno common components, and let Z — Z(F). Then syz (F) < 7. Furthermore,

0) iff <0, then syz (F) = 7;(ii) (/ 7 > 0 and c2{Jz) > mn - j2/4, then syz (F) < 7/2;

(iii) iff > 0 and 02(^2) < mn - 72/4, then

z (F) < I + ^3mn - ^72 - 3c2(Iz) + 1 -1 .

Proof, (i) is trivial, (ii) follows from the Bogomolov inequality, (iii) isjust a translation of the main theorem of [Ha5]. D

Note that the inequality of (iii) is the best possible one (cf. [Ha6]).

7. On Noether ' s Theorem

In this section, we shall consider the general case of Noether's AF + BGTheorem.

Let F i , - - - ,Fr be r fixed homogeneous polynomials and H an arbitraryhomogeneous polynomial in S = k[xo, ••• ,xn]. We shall try to find the condi-tions imposed on H so that H has an expression as a linear combination

(7.1) / / = 4 , F i + A2F2 + ••• + ArFr, (Ai € S).

A necessary condition (called Noether's condition) is that the expression(7.1) holds locally at the common zeros of F = (Fi, • - • , Fr). Noether's AF +BG Theorem says that this condition is also sufficient if n = r = 2. But ingeneral it is not sufficient.

We shall say that Noether's Theorem holds for F in degree k if for every Hof degree k satisfying the above local condition must admit a global expression(7.1). Furthermore, if for every k > 0, Noether's Theorem holds for F in degreek, then we shall say that Noether's Theorem holds for F.

Now we want to find the condition on F so that Noether's Theorem holdsfor it. Assume that n > 2 and r > 2.

We still use F; or H to denote the corresponding hypersurfaces in IP". Inmodern language, we let J7 be the syzygy sheaf of F. Let degF,- = a,-, anddeg/ / = k. From the long exact sequence of

r

0 _> T{k) - , : 0 O(k - a{) -^ Iz(F)(fc) — 0,I=I

we have

19

(7.2)

The local condition for H is just that H e H°(lZ(F)(k)). Noether'sTheorem holds for F in degree k means that the map F in (7.2) is surjective,or equivalently IIl{F{1t)) = 0. Thus we have

THEOREM 7.1. Noether's Theorem holds for F if and only if the idealI(F) generated by F in S is saturated.

COROLLARY 7.2. Let F = (Fi, • • • , Fr) be r curves on P2 with no commoncomponents. Then the following conditions are equivalent:

(i) Noether's Theorem holds for F.(ii) The syzygy sheaf of F splits as a sum of r — 1 line bundles.(iii) Ft, •' • , Fr are the (r — 1) x (r — 1) minors of a (r — 1) x r matrix

whose elements are homogeneous polynomials.

Proof. The splitting criterion of Horrocks for vector bundles implies theequivalence of (i) and (ii). The equivalence of (ii) and (iii) follows from thehomogeneous case of Hilbert-Burch Theorem. •

COROLLARY 7.3. Let F = (Fi, F2, F3) be 3 curves on F2 without commoncomponent, and of degrees («i,a2jG3)- Then Noether's Theorem holds for Fin degree k for all k > a^ + a2 + 03 - 3.

Proof. From the long exact sequence (7.2), we can see that Hl(!F(k)) = 0for k < 0. On the other hand T* = ^{ai + a2 + a$), by Serre duality, ifk > a% + a2 + «3 — 3, then we have

= H1 {7{al + a2 + a3 - 3 - k)) = 0.

Thus Noether's Theorem holds for F in degree k. •

AcknowledgementsThe author would like to thank the International Centre for Theoretical Physics, Tricste; for

hospitality and financial support during this research. This research was partially supported bythe K.C. Wong Education Foundation and the National Natural Science Foundation of China.He would also like to thank Professor G. Xiao who introduced him to Rick Miranda's work ontriple coverings several years ago while he was a student. Finally, he expresses his appreciationto Professors. S. Lang, M. S. Narasimhan and M. Schneider for their encouragement and help.

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REFERENCES

[BaM] D. BAYER, D, MUMFORD, What can be computed in algebraic geometry? in Compu-tational algebraic geometry and commutative algebra, Sympos. Math,, XXXIV, Cam-bridge Univ. Press, Cambridge, 1993, pp.1-48.

[Bur] L. BURCH, On ideals of finite homological dimension in local rings, Proc. CambridgePhilos. Soc, 64 (1968), 941-948.

[Dec] W. DECKER, Monads and cohomology modules of rank 2 vector bundles, CompositioMath. 70 (1990), 7-17.

[Hal] R. HAOTSHORNE, Varieties of small codimension in projective spaces, Bull. Arner. Math.Soc, 80(1974), 1017-1032.

[Ha2] , Algebraic Geometry, Graduate Texts in Math., 52, Springer-Verlag, NewYork, 1977.

[Ha3] ——, Algebraic vector bundles on projective space: a problem list, Topology, 18(1979), 117-128.

[Ha4] , Stable reflexive sheaves, Math. Ann., 254 (1980), 121-176.[Ha5] , Stable reflexive sheaves II, Invent. Math., 66 (1982), 165-190.[Ha6] , Stable reflexive sheaves III, Math. Ann., 279 (1988), 517-534.

[Hil] D. HILBERT, Uber die Theorie der algebraisdien Fornieii, Math. Ann., 36 (1890), 473-534.

[HiN] A. HIRSCHOWFTZ, M, S, NARASIMHAN, Vector bundles as direct images of line bundles,Proc. Indian Acad. Sci. (Math. Sri.), 104 (1994), 191-200.

[Hor] G. HOFIROCKS Vector bundles on the punctured spectrum of a local ring, Proc. LondonMath. Soc. 14(1964), 689-713.

[Kap] I. KAPLANSKY, Commutative Algebra, W. A. Benjamin Inc., New York, 1970.[Mir] R. MIRANDA, Triple covers in algebraic geometry, Amer. J. Math., 107 (1985), 1123-

1158.[OSS] C. OKONEK, M. SCHNEIDER, H. SPINDLER, Vector Bundles on Projective Spaces, Progress

in Math., 3, Birkauser, Boston, 1980.[Sea] M. SCHAPS, Deformations of Cohen-Macauley schemes of codimension 2 and non-

singular deformations of space curves, Amer. J. Math., 99 (1977), 669-685.[Sch] M. SCHNEIDER, Holomorphic vector bundles on IP™, Sem. Bourbaki, 1978/1979, expose

530.[Sew] R. L. E. SCHWARZENBERGER, Vector bundles on the projective plan, Proc. London Math.

Soc, 11 (1961), 623-640.[SeK] J. G. SEMPLE, G. T. KNEBBONE, Algebraic curves, Clarendon Press, Oxford, 1959.[SeR] J. G. SEMPLE, L. ROTH, Introduction to Algebraic Geometry, Clarendon Press, Oxford,

1949.[Ser] J. P. SERRB, Sur les modules projectifs, Sem, Dubrieul-Pisot 1060/1961, expose 2.[Sto] G. STOLZENBBRG, Constructive normalization of an algebraic variety, Bull. Amer.

Math. Soc, 74 (1968), 595-599.[Vas] W, V. VASCONCELOS, Computing the integral closure of an affine domain, Proc. Amer.

Math, Soc, 113 (1991), 633-638.

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