manuscripta math. 84, 291 - 314 (1994) manuscri~ta mathematlca �9 Springer Verlag 1994

On graded Betti numbers and

geometrical properties of projective

varieties

Uwe Nagel 1 and Yves Pitteloud 2

In this paper we study graded Betti numbers of projective varieties. Using a spectral sequence argument, we establish an algebraic version of a duality Theorem proved first by Mark Green. Our approach doesn't require any smoothness or characteristic 0 assumption. We then study the graded Betti numbers of finite subschemes of a rational normal curve and apply these results to generalize another theorem of Mark Green, the Kp.1 theorem, to some non-reduced schemes. Our result applies for instance in the case of ribbons.

I n t r o d u c t i o n

Given a projective variety in F'~, one obtains a collection of invari- ants, the graded Betti numbers, by considering the minimal graded free resolution of the homogeneous coordinate ring of the variety. Recently a great deal of research has attempted to understand the interplay between the algebra and the geometry arising in such a situation.

This search was started in the pioneering paper [13], by M. Green. In this article new techniques were developed for the com-

1Supported by Deutsche Forschungsgemeinschaft 2Supported by the Swiss National Research Fund

292 NAGEL-PITTELOUD

putation of Betti numbers, making strong use of the Koszul com- plex, and various connections with geometrical questions were ex- plained. Most of the results of [13] are formulated in the setting of compact complex manifolds. The main object of the present paper is to examine how some of these results generalize to more general subschemes of 17'~.

The main computationals tools introduced in [13] are the fol- lowing:

the Vanishing Theorem (cf. Theorem 3.a.1), the Duality Theorem (cf. Theorem 2.c.6), the It'p,1 Theorem (cf. Theorem 3.c.1).

An algebraic version of the vanishing theorem has been proved by D. Eisenbud and J. Koh in [6] (cf. theorem 1.7) and hence we will concentrate our attention in this paper on the Duality Theorem and on the Kp,1 Theorem.

We will adopt a slightly different approach from that in [13], where graded SymH~ of the form |176 $ |174 are studied, with $, /: locally free sheaves on a variety X, a n d / : of rank one. Our projective varieties X will be already embedded in some I?" and we will consider arbitrary graded modules M over the polynomial ring R = k[Xo,..., x,~]; of course, in the most interesting case M is the homogeneous coordinate ring of X, or its canonical module. Our techniques will be mainly algebraic, although inspired by the techniques introduced in [13].

Section 1 deals with the Duality theorem. Roughly speaking the problem considered is the following. Given a Cohen-Macaulay graded R-module M (where R is the polynomial ring in n + 1 vari- ables), of dimension, say rn+ 1, one can dualize the minimal graded free resolution of M and obtains the minimal free resolution of Ext~- ' (M, R). In other words, if we denote by ~M the R-module Ext (M, R ) ( - n - 1) (the canonical module), we have an isomor- phism for all i and j

ToriR( M, k ),+j ~- Tor~_,,_,(WM, k )~+l_,_j.

We show, using a spectral sequence argument, how the Cohen- Macaulay hypothesis on M can be weakened and replaced by a suitable cohomological hypothesis on M, if one is just willing to

NAGEL-PITTELOUD 293

have the isomorphism above only for certain values of j (cf. the- orem 1.2). We explain how Green's Duality Theorem fits in the pattern of our result. Then we extend to singular curves a result of Green (cf. [13] Theorem 4.a.1)on the syzygies of a curve embedded by a very ample line bundle to singular curves (cf. theorem 1.6) generalizing a serie of results by Mumford, Saint-Donat and Fujita (cf. [20], [22] and [11]) about projective normality, and generation by quadrics of embedded curves.

I~ section 2 we consider the graded Betti numbers of a finite scheme on a rational normal curve. In certain cases we are able to compute all of them. This is related to results in [9] and will be used later.

In section 3 we consider the problem of bounding the length of the 2-linear part of the minimal resolution of the coordinate ring of a projective variety. If the variety lies on many quadrics, the bound provided by the Vanishing Theorem of Green/Eisenbud- Koh is much too big. Green however showed, in the so called Kp,a Theorem, that the codimension of the variety turns out to be a bound; moreover this bound is only reached in the case of a variety of minimal degree. We show how his results remain true under a much weaker hypothesis. We allow the schemes to be non-reduced and singular (cf. theorem 3.5). In particular we get a version of the Kv.1 theorem for ribbons, an interesting class of schemes introduced in [1].

Both the authors would like to warmly thank Tony Geramita for many interesting discussions during their stay at Queen's Uni- versity. The first author would also like to thank Mark Green for valuable discussions.

The authors are grateful to the referee for his helpful comments.

1 B e t t i n u m b e r s a n d d u a l i t y

Throughout the paper, we will denote by R the polynomial ring k[x0, . . . , x,~] over a field k. Gi'ven a graded R-module M, we denote by [M]j the homogeneous component of M of degree j; given an integer q, we write M(q) for the qth shift of M (defined by the formula [M(q)]j = Mq+j) and we write M v for the graded k-dual

294 N A G E L - P I T T E L O U D

of M. By the dimension of a graded R-module M we will mean the Krull dimension of the ring R/Ann(M).

The local cohomology modules of a graded R-module M, with respect to the irrelevant ideal ra = (x0,. �9 x,,), will be denoted by H~(M). Recall that they are graded R-modules. The Local Duality Theorem states that there is an isomorphism of graded R-modules (where Ext denotes the "graded Ext")

H~(M) v ~ Ext~+l-i(M,R)(-n- 1).

We write K(x__) for the Koszul complex of R associated to the variables x0 , . . . , x~. Given a graded R-module the Koszul homol- ogy modules Hp(x, M) are the homology modules of K(x__)| and the Koszul cohomology modules HP(x, M) are the homology mod- ules of Homn(K(x_),M) (where Horn denotes the "graded Horn"). The following properties of these Koszul modules are well known.

R e m a r k 1.1 a) The Koszul complex K(x_) is a minimal graded free resolution of the R-module k. Consequently we have isomorphisms of graded modules

Hp(x_,M) ~- Tor~(k,M) and HP(x_,M) ~- Ext~(k,M).

b) Let M be graded R-module; if [M]i = 0 then

[Hp(x,M)]i+p = 0 and [H'(x_,M)]i_p = 0 for all p.

This follows at once from degree considerations. c) There are isomorphisms of graded modules

H'(z__,M) ~- Hn+l_p(X__,M)(n + 1), for every p.

This follows from the symmetry of the Koszul complex. d) There is a spectral sequence (hypercohomology) of graded

R-modules

E~'q= Hv(x_,H~(M)) :. Hv+q(x_,M),

where the differentials d~ : E~ ,q --, E~ +r,q-~+l are homogeneous maps of degree zero.

N A G E L - P I T T E L O U D 29s

The main result of this section is the following theorem, where given a graded R-module M of Krull dimension m + 1, we denote by WM the graded R-module Ezt~(m(M,R)(-n- 1).

T h e o r e m 1.2 Let M be a R-module of Krull dimension m + 1. Let j be an integer such that

[H~(M)]j_q = 0 and [H~(M)]j_q+I = 0 for q = O , . . . , m ,

and such that all the vector spaces [H~+I(M)]i are of finite type. There is then an isomorphism of k-vector spaces

[Tor~(M,k)]i+j ~ [Tor~_m_i(wM, k)],~+l_i_j, for all i.

R e m a r k 1.3 The hypothesis above includes, of course, the case where M is a Cohen-Macaulay module of finite type, and in this case the assertion is well known (cf., for example, [21], Lemma 2.1).

A special case of this theorem, Corollary 1.4, is a well known result about the Castelnuovo-Mumford regulari ty of a R-module M (cf. [5], Theorem 1.2). Recall that the Castelnuovo-Mumford

regular i ty of a R-module M is defined to be the largest integer j such tha t there is a q with H~(M)j_q ~ O.

C o r o l l a r y 1.4 Given a finitely generated graded R-module M, the following statements are equivalent:

a) [H~(M)]j_q = 0 for all q and all j > jo, b) [Tor~(M,k)]i+j = 0 for all i and all j > jo.

Proof of the Corollary: Suppose first tha t we have, for some integer jo, [Tori(M, k)]i+j = 0 for all j > j0. This means that in the minimal graded free resolution of M

0 --~ @R(-e , j ) ~ . . . ~ @R(-e i j ) --* . . . --* @R(-eoj) ---* M ~ 0

we have the inequali ty eij < i+jo. Now applying H o m R ( , R) to the resolution, we deduce that [Zxth( M, R)], is zero when l < - i - j0 and the conclusion follows by local duality.

Suppose now that [H~(M)]j_q = 0 for all j > j0; in part icular the hypothesis of Theoreml .2 is satisfied for those j , and hence we have the following isomorphism for all j > jo:

[Tori(M, k)]i+j ~- [Torn_m_,(wM, k)]n+l-i-j.

296 NAGEL-PITTELOUD

By local duality the module ~O M is isomorphic to H~+I (M) v and so the hypothesis on the local cohomology implies that [WM]l is zero for all l < m + 1 - j0; this in turn implies that [Torn-m-i(WM, k)]l is zero for all l < n + 1 - i - j0 which concludes the proof of the Corollary.

For the proof of Theorem 1.2 we will use the spectral sequence of Remark 1.1 d): The vanishing hypothesis in Theorem 1.2 will be just what is needed to have enough vanishing on the homogeneous part of the E2 term, in certain degrees.

Proof of Theorem 1.2: Let M and j be as in Theorem 1.2 and let s be an integer. We will look at the homogeneous part of degree j - s of the E2 term of the spectral sequence:

[E~'q]j_, = [HP(x_,H~(M))]j_~.

As the modules [H~(M)]j_q+, are zero for q = 0 , . . . , m, we obtain the following equalities by Remark 1.1 b):

[HP(x_,H~(M))]j_s=O, f o r p + q = s + l a n d q _ < m .

In other words [E~'q]j_, is zero for those values of p, q. On the other hand, E~ 'q is zero, in all degrees, for q > m + 1, as the Krull dimension of M is m + 1. These two facts imply the following isomorphism:

[ j~--~s-m- 1 , m + l ] . r,~ [ E s - m - l , m + l ] ~ ' 2 J 3 - s "-- t c ~ j j - s ,

We also have that the modules [H~(M)]j_q are zero for q = 0 , . . . , m, and this in turn implies the following equalities

[HP(x_,Hq~(M))]j_s=O, f o r p + q = s a n d q _ < m .

In other words, [E~]j_, is zero for those values of p, q. As E ; 'q when p > m § 1, the spectral sequence gives the isomorphism

[Z~o-m-l,m+l]j_s ,~ [gS(x_,M)]j_s.

= 0

Putt ing together these observations, we obtain an isomorphism

[H~(x_, M)]j_~ ~ [H~-'~-I(x_,H~+I(M))]j_~. (*)

NAGEL-PITTELOUD 29z

To conclude the proof of Theorem 1.2, it remains to identify the two vector spaces appearing in the isomorphism above.

By Remark 1.1, [He(x, M)]j_8 is isomorphic to the vector space [Tor,,+l_,(M, k)]j- ,+,+l, so setting i = n + 1 - s, this latter is precisely the left handside in the isomorphism of Theorem 1.2.

Next, we have to identify the right handside of (.). By local duality, the module H 2 +~ (M)is isomorphic to the module w~ (here we use the fact that the [H~+X(M)]i are k-vector spaces of finite type). By Remark 1.1 a) and [24], Corollary 0.4.11, we have the following isomorphisms:

[HS-m-l(x, Hm+l~'M ~ jjjj-, ~_ [Ext~-~-l(k,w~)]j_8 ~- [Tor l(k ~OM)] j - - 8 - - ~ - - , S - -

and setting, as above, i = n + 1 - s, this latter is isomorphic to the [Tor,,_m_i(k, WM)]~+I-~-j. This concludes the proof of vector space n

Theorem 1.2.

R e m a r k 1.5 Given an m-dimensional locally Cohen-Macaulay equidimensional projective scheme over a field, an ample invertible sheaf L: on X with a base point free linear system W C H~ f~) of dimension n + 1 and a locally free sheaf g" on X, consider the R (= SymW)-graded modules

M = @qezH~ C | .C | and N = @qezH~ C* | K: X | ,E|

where )Ex denotes the dualizing sheaf and "*" denotes the Ox dual. Suppose moreover that the following equalities are satisfied, f o r q - 1 , . . . , m - l :

Hq(X,E | s | = 0 and Hq(X,E | s | = O.

There are then isomorphisms

[Tor~(M,k)]i+j ~- [Tor~_,~_,(N,k)],~+l_i_j for each i.

This assertion, similar to the Theorem 2.c.6 of [13], can be de- rived from our Theorem 1.2 as follows. By [14], III 2.1, there are isomorphisms

H~+'(M) ~- eq zH'(X,g | s174 for i _> 1 and H~ = H (M) = O,

298 NAGEL-PITTELOUD

where m denotes the irrelevant ideal in R. Hence, by Serre Duality, we have

N ~- @qezHomk(Hn(X,$ |176 k) = (eqezH':(X,E | s174 H~+I (M) v .

The assertion is now a direct consequence of Theorem 1.2.

Now let C be an integral curve and l e t / : be a very ample line bundle on C. Let S be the ring @H~ s174 considered as a graded module over the ring R = SymH~ f-). We say t h a t / : satisfies property (Np) if Toron(S,k) ~- k and Tor~n(S,k) ~- k ( - i - 1) ~ for 1 ~ i < p (cf. [4]). Let Ca(C) be the embedding of C into ~(H~ Then/2 satisfies (No) iff Ca(C) is projectively nor- mal. /: satisfies (N~) iff (No) holds and the homogeneous ideal I of Ca(C) is generated by quadrics. /:: satisfies (N2) iff (No),(N1) hold and the first syzygy module of I is generated by linear relations; and so on.

The following result was proved by Green in the case of a smooth curve over C (cf. [13], Theorem 4.a.1). It could also be derived using Theorem 3 of [7] instead of our Duality Theorem. Note also that there is a generalization of Green's result by Ein and Lazarsfeld to higher dimensional smooth varieties over the complex numbers (cf. [4]).

T h e o r e m 1.6 Let C be an integral curve of arithmetic genus g over an algebraically closed field. Let f~ be an invertible sheaf on C of degree > 2 g + l + p for some p > 1. Then L satisfies property (Np).

Proof. Let N be the R-module @H~ ]C | s174 where/C is the dualizing sheaf on C. Note tha t /2 is very ample and generated by global sections (see [11] Proposition 1.6). Hence, as in Remark 1.5 we have N -~ H~(S) v. From this we deduce that N is a torsion free R/p-module, where p denotes the ideal of eL(C) in R (cf. [21], Lemma 2.6). Now, rank [N]0 = g < rank [R]I = d - g + 1 where d = deg /:. Hence [N]j = 0 for all j < 0. Thus we obtain, by Remark 1.5,

[Tor~_g_,_,(S,k)]a_g+l_j ~- [Tor~(N,k)]j = 0 if j < i.

NAGEL-PITTELOUD 299

Moreover, Theorem 1.7 below gives

[Tor~_g_,_,(S,k)]d_g+l_, TM [Tor~(N,k)], = 0 for i > g,

proving the claim.

T h e o r e m 1.7 (cf. [61, Theorem 1.1) Let M be a graded module over the polynomial ring R = k[x0, . . . , x~] and suppose that M is torsion free over R / p for some absolutely irreducible prime ideal p. Then [Tor,(M,k)],+j = 0 for i > rank [M]j.

Theorem 1.7, which we have been using in the proof of Theorem 1.6, is the generalization to the non smooth case of the vanishing theo- rem of Green (cf. [13] Theorem 3.a.1) due to Eisenbud and Koh.

2 F i n i t e s c h e m e s o n a r a t i o n a l n o r m a l

c u r v e

Recently, J. Harris and D. Eisenbud [9] have extended the no- tion of linearly general position from sets of points to arbitrary 0-dimensional subschemes. A finite subscheme X C ?~ is said to be in linearly general position iff for every proper linear subspace L of P~ we have

deg X N L _ < d i m L + I .

Harris and Eisenbud could extend some results of Castelnuovo the- ory to schemes X in linearly general position. In particular, they have shown that X lies on a unique rational normal curve if deg X = n + 3 [9], Theorem 1 or deg X > 2n + 3 but X imposes only 2n + 1 independent conditions on quadrics [10], Theorem 2.2. In this sec- tion we want to study the graded Betti numbers of finite schemes on a rational normal curve. Note that these schemes are always in linearly general position. We need these results in the next section but they are also interesting in their own right.

For the remainder of the paper we will always assume that the ground field k is algebraically closed. The homogeneous ideal of a subscheme S C ~ will be denoted by Is. It is always assumed to be a saturated ideal.

aoo NAGEL-PITTELOUD

L e m m a 2.1 Let X be a finite scheme on a rational normal curve C ofF". Let p be an integer with 0 <_ p <_ n. Ifdeg X >_ 2n+ 1 - p then

[Tor~(k, Ix)],+~ = [Tor~(k, Ic)],+~ for all i >_ p.

Proof: The exact sequence

0 ~ Ic ' Ix ~ I x / I c ~ 0

gives an exact sequence

[Tor51(k, Ix/Ic)],+~ -* [Tor~(k, Ic)],+~ ~ [Tor~(k, Ix)],+2-*

[roC(k, Since Ix does not contain a linear form we get [Tor~,(k, Ix/Io)],+~ = O.

By [9], Theorem 3.2 we have for the Hilbert function of X, hx(2) > 2 n + 1 - p . Therefore we get rank [Ix/Ic]2 <_ 2 n + 1 - p - (2n + 1) = p. Since I x / I c is torsion free as an R/Ix-module , the Vanishing Theorem 1.7 gives [Tor~(k, Ix/Ic)],+2 = 0 for all i > p. The claim is now a consequence of the above exact sequence.

It is easy to compute the Betti numbers of a rational normal curve. Recall that a subscheme V of F" is called variety of minimal degree if V is integral, nondegenerate and deg V = codim V + 1. Note that this is the minimal possible degree of an integral nondegenerate subscheme of F'~. For example, a curve of minimal degree is precisely a rational normal curve.

By a famous classification theorem of Bertini (cf. [8]), a variety of minimal degree is either a rational normal scroll or a cone over the Veronese surface in Fh. In the first case the homogeneous ideal of the scroll is generated by the maximal minors of an 1-generic matr ix of linear forms (see [8]). The resolution of such an ideal is given by the Eagon-Northcott complex; as the Bett i numbers of a Veronese surface are the same as the Bett i numbers of a surface scroll in Fs, we have thus the following result.

L e m m a 2.2 Let V C ~ be a variety of minimal degree and codi- mension c. Then V has a minimal free resolution of the following shape:

0 --* e R ( - c - 1) '~ - ~ . . . -~ e R ( - 2 ) '~ --, Iv --. 0

NAGEL-PITTELOUD am

/< 1) where fl, = rank [Torin(k, Sv)]i+2 = (i + 1 ,+2 , ",

Now we are able to compute the graded Betti numbers of cer- tain subschemes of a rational normal curve. This generalizes [2], Proposit ion 3.1 and [3], Theorem 6.

P r o p o s i t i o n 2.3 Let X be a finite subscheme of ~ ~ of degree 2 n + l - p where 0 <_ p < n. Suppose that X is a subscheme of a rational normal curve. Then the minimal graded free resolution of I x has the following form:

0 ~ R ( - n - 2) n"-I ~ R ( - n - 1) z"-~ @ R ( - n ) ""-2 ~ . . .

.--+ R ( - p - 3)/~" @ R ( - p - 2)"" ---.>, R ( - p - 1)"p- ' -.->,

---+ R(-p)<~'>-~... --+ R ( - 2 ) "~ ---+ Ix ---+ 0

where n+2

a i : (i + 1)(~+2) i f p < i < n - 2

and

n n - t - 2 n

Proof: X has Hilbert function 1, n + 1, deg X , . . . by [9], Theorem 3.2. Thus we get (cf., for example, [21], Remark 2.5(v))

[Tor~(k, Ix)]i = 0 for all j r i + 2, i + 3.

Hence, using the addit ivi ty of the rank function on exact sequences (applied here to the minimal free resolution of R / I x ) , as well as the above description of the Hilbert function of X, we obtain, as in [2], Proposit ion 1.6

rank [mor~(k, I x ) ] , + 2 - rank [Tor~_l(k, Ix)],+2 =

n+2 - ( ; )

But, from Lemma 2.1 and Lemma 2.2, after some computations, we get

rank [Tor~(k, Ix)],+~ = ( p + 1)(,+2) n+2 X. = (p + 1 ) ( ,+2 ) - ( ;) deg

302 NAGEL-PITTELOUD

It follows that [Torn l(k, Ix)]p+2 = 0. Using, for example, our Duality Theorem we get

[Tor~(k, Ix)],+3 = 0 f o r a l l i < p - 1.

Thus the claim follows by Lemma 2.2, Lemma 2.1 and [2], Propo- sition 1.6.

After writing this paper, the preprint of K. Yanagowa [25] came to our attention. He obtains the above result (cf. [25], Corollary 3.4) by completely different means.

3 T h e l e n g t h of t h e l inear par t o f a r e s o l u t i o n

One of the most beautiful results in M. Green's paper [13] is the so-called Kp.1 Theorem. The aim of this section is to show that some of the assumptions for this result in [13] can be relaxed.

In this section we will consider hyperplane sections of projective schemes, so we need to make this notion clear. Let I C R = k[xo,. . . ,z,~] be an homogeneous ideal and let S = Proj(R/I). We denote, as before, the homogeneous ideal of S by Is. Let H = Proj(R/lR) be a general hyperplane. The geometric hyperplane section SfqH is Proj(R/Is + lR) and will always be considered as a subscheme of Proj(R/lR). Its homogeneous ideal IsnH is thus the saturation of the ideal (Is + lR)/lR. This latter ideal is called the algebraic hyperplane section of S.

Let I C R be a homogeneous ideal with initial degree t = min {s E 7/11 : [I]s ~ 0}. Then a nonzero element of [Torln(k,I)]i+t corresponds to a/-l inear syzygy in the sense of [6]. If I is saturated and S = Proj(n/I) then we call the /-linear syzygies of I also /-linear syzygies of S. A linear syzygy of S is an /-linear syzygy for some i. The linear syzygies correspond to a subcomplex of the minimal free resolution. This subcomplex is called the t-linear or simply linear part of the resolution (for more details, cf. [6]).

The following facts show that linear syzygies are easier to study than arbitrary syzygies.

N A G E L - P I T T E L O U D 303

R e m a r k 3.1 a) If I is a saturated ideal and l is a linear form whose residue class modulo I is a non-zero divisor in R/ I , then it is well-known that all the syzygies of (I + lR)/1R are restrictions of the syzygies of I. In particular, the Betti numbers of I as an R-module are the same as the Betti numbers of (I + lR)/IR as an R/IR-module.

b) If t is the initial degree of I then it is not hard to see (cf. [2]) tha t

[ToriR(~.,I)]i+t ~ (Ai[R]I @ [I]t) n I~i

where we have denoted by Ii'~ the kernel of the Koszul differential Ai[R]I | [R]t ' Ai-I[R]I | [R]t+l.

c) It follows from b) that if J1 C -12 are homogeneous ideals with the same initial degree t, then every linear syzygy of J1 is also a linear syzygy of J2; in other words the canonical morphism [Tor~(k, Jx)],+t--+ [Tor~(k, J2)],+t is injective for all i.

d) Let S be a subscheme of IP '~ with defining ideal I. Given a general hyperplane H, with corresponding linear form l, we set J1 = ( I+ lR) / lR and J2 = Isng. Applying c) to these ideals in the ring R/IR, and then applying a), we see tha t the linear syzygies of S restrict to linear syzygies of S n H, provided tha t Is and ,[SAg have the same initial degree.

For arbi t rary subschemes of ~'~ there is the following bound for the length of the linear part of the minimal resolution.

L e m m a 3.2 Let S C I? n be a closed subscheme of codimension c and let t be the initial degree of its homogeneous ideal Is. Let L C IP" be a general linear subspace of dimension c and suppose that ISo L has initial degree t too. Then

[Tor~(k, Is)]i+t = 0 for all i >_ c.

Proof: Suppose [ror~(k, Is)],+t # 0 for some i > c. Let L = Proj(B). Then we get from Remark 3.1 d) tha t Tor~(k, IsnL) ~ O. But this contradicts the theorem of Auslander and Buchsbaum since B/IsnL is a 1-dimensional Cohen-Macaulay ring.

There are conditions ensuring that the assumption above is sat- isfied. If t = 2 and S has a top-dimensional component S' such

a04 N A G E L - P I T T E L O U D

that Sr d is nondegenerate then IsnL has also initial degree 2. This follows by induction because Is~redN H is nondegenerate in H for a general hyperplane H by Bertini 's Theorem.

If S is an integral curve in pa then the assumption of Lemma 3.2 is satisfied if deg(S) > t 2 + 1 (Laudal 's lemma). For some general- izations of this result, we refer to [19] and [23].

We need some more notation. For a finite set J we write I J] for its cardinality. If J = { j l , . . . , j , } C { 0 , . . . , n } we set ej = xjl A . . . A xj~. Thus the ej form a basis of A~[R]I, where J varies in the set of i-tuples { j l , . . . , j i } C { 0 , . . . , n } with j~ < . . . < ji. Let 0 # a e [Torin(k,I)]i+t for some homogeneous ideal I with initial degree t. By Remark 3.1 b), we can write

e j o F j JJl=i

with forms F j E [I]t. Let I(a) denote the ideal generated by the F j , and call it the ideal asssociated to the (linear) syzygy a. By definition, we have tha t I(a) C I. Moreover, we set V(a) = Proj (R/ I (a) ) and call it the subscheme associated to a. Note tha t we do not know a priori if I(a) is saturated. Note, more- over, tha t I(a) does not depend on the choice of a basis for Ai[R]I.

In case t = 2,. the Kpa-Theorem in [13] gives a much bet ter description of the length of the linear par t of the resolution than Lemma 3.2. The basis for this improvement is the so-called Strong Castelnuovo Lemma. Using Green's ideas this has been refined in [3] and can be s ta ted as follows.

P r o p o s i t i o n 3.3 Let S C ~'~ be a closed subscheme such that S contains a finite subscheme Y of degree at least n + 3. Then S is contained in a rational normal curve C of ]P'~ if and only if [Tor~_2(k, Is)],~ # 0 and Y is in linearly general position.

Moreover, if 0 ~ (x E [Tor~_2(k, Is)],~ and if Y is in linear general position, then I(a) is prime and V(a) is the unique rational normal curve of ~'~ containing S.

Note tha t we do not assume tha t S is zero dimensional. Since only a slightly weaker and less precise version of Proposi t ion 3.3 is proved in [3], we include a proof using the results of [3].

NAGEL-PITTELOUD a05

Proof." If S C C then we get by Remark 3.1 c) and Lemma 2.2, that 0 7~ [Tor~_~(k, Ic)],~ ~ [Tor~_2(k, Is)],, Moreover, Y is in linearly general position because every finite scheme of degree at least n on a rational normal curve is in linearly general position. This proves one implication.

To prove the other implication let 0 -# a C [Tor~_2(k, Is)],~. Since Y is in linearly general position it lies on an unique rational normal curve C by Theorem 1 in [9]. Thus, using Remark 3.1 c) and Lemma 2.1 we get a e [Tor~_2(k, Iy)],~ ~- [Tor~_2(k, Ic)],~. Therefore we obtain, by Remark 3.1 b), that I(o~) C Iv. Now our Lemma 2.2 and Proposition 4 in [3] give that

Since Iv is generated by quadrics it follows that I (a) = Ic com- pleting the proof.

The result above applies, in particular, to an irreducible, nondegen- erate subscheme S of positive dimension because on such a scheme S we can find a set of t points in linearly general position for any given positive integer t. Here we use the notion nondegenerate in the following sense.

Def in i t ion 3.4 A closed subscheme S C P'~ is said to be nonde- generate if STud is not contained in any hyperplane of ]P".

This definition is a bit unusual but useful for our purposes. We are now ready to state the algebraic analog to Green's Kp.1-

Theorem.

T h e o r e m 3.5 Let S = P r o j ( n / I s ) C IP~ be a closed subscheme of dimension m > 0 and codimension c >_ 2. Suppose that S has a nondegenerate m-dimensional component S'. Then we have:

(i) [ToriR(k, Is)],+2 = 0 for all i >_ c; (ii) [Tor~_l(k, Is)]c+l 7~ 0 if and only if S is a variety of minimal

degree; (iii) if deg S~ d >_ c+4 and ei therchar(k) = 0 or S~ d issmooth,

then

306 NAGEL-PITTELOUD

[TorY_2(k, Is)]c # 0 if and only if S is contained in a variety of minimal degree having dimension m + 1;

5v) The claim in 5ii) is also true if S is a nondegenerate, in- tegral subscheme of degree c + 3 and either char(k) = 0 or S is smooth.

Note that [Tor~_l(k, Is)], . l ~ [Tor~(k,R/Is)],+l and this cor- responds in Green's notation to the Koszul homology group K~.I (from which the name of the theorem is derived).

The assumption that S has a nondegenerate top-dimensional component is necessary for Theorem 3.5 (i) and (ii). One need only consider the example of two skew lines in p3. Moreover, the claim in (iii) fails in general for integral schemes with deg S = codim S + 2 as the del Pezzo surface of degree 9 in p9 shows. This example is due to Green [13].

Part (iii) of the above theorem is the most difficult to prove. We will deduce it from the following more general result.

P r o p o s i t i o n 3.6 Let S = P r o j ( R / I s ) C P= be a closed subscheme of codimension c >_ 2. Suppose that for a general linear sub- space L C P'~ of dimension c, S N L contains a finite scheme in linearly general position in L of degree at least c + 4. Then [Tory_~.(k, Is)]~ ~ 0 if and only if S lies on a variety of minimal degree having codimension c - 1.

In particular: If 0 # a E [Tor~_2(k, Is)]c then I (a) is prime and V(~) is a variety of minimal degree having codimension c - 1 and containing S.

Proof: If S lies on a variety V of minimal degree having codi- mansion c - 1 then, by Lemma 2.2,

rank [Tor~_2(k, Is)]c > rank [Tory_2(k, Iy)]c = c - 1.

For the other implication we induct on the dimension m = n - c of S. If m = 0 then Proposition 3.3 gives the claim. Suppose m > 0. The assumption on S N L implies in particular that Isn L has initial degree 2. Therefore Remark 3.1 d) applies to all successive general hyperplane sections of S.

NAGEL-PITTELOUD ao7

Let o # ~ = ~ ~j | F j e [To~_~(k,*s)]o.

IJ I - -c-2

Let H be a general hyperplane and let l be its defining equation. We choose a basis {10 = l, ll,...,l,~} of [R]I and we denote by lj the product ljl A . . . A lj, where J = { j l , . . . , j s ) . Then the set {lj : IJI = c - 2 , J C { 0 , . . . , n } with j l < " " < j c - 2 } is abas i s for Ac-2[R]I and we can write a as

= ~ Ij | a~ + ~ l A l, | Q, e A~ | [ I + . I JJ=c-2 , o~J IIl=c-3, oqH

Let F denote the image of a form F E R under the canonical map onto /~ = R/lR; we see that the restriction of a to the general hyperplane H is

0 # a l H = ~ I~oG~e[To~_~(~,I~)]o. IJl=c-2, 0~J

S ince H is general we have dim S = dim S f3 H. Hence we may apply the induction hypothesis and get that I(a[H) is a prime ideal defining a variety W of minimal degree having codimension c - 1 in H ~ ~,-1. In particular we have I(aIH) - Iw C IsnH.

We now want to show that (~x E /(a{H) for all Q1 occuring in (~. Since a is mapped onto zero under the Koszul map we see that

~ = ~ lI| Qx e [Tor~-a(k, Is)]~-~. Ill--c-3, 0~I

Suppose fl # 0. Then we get as above

0 # Z I g = Z lI|176 IsnH)]c-l"

Let L = L' Ig H = Proj(R') be a general linear subspace of di- mension c. Since we consider only linear syzygies we get (cf. Re- mark 3.1)

R' R' rank [Torr Isnn)]~_~ = rank [Tor~_a(k, Isnn,ng)]~_~ > rank [Tor~_a(k, IsnH)]~_~ > rank [Tor~_a(k, Iw)]c_~ =

R I rank [Tor~_3( k, Iwnn,)]~-l.

ao8 NAGEL-PITTELOUD

The last equality is true because W, as a variety of minimal degree, is arithmetically Cohen-Macaulay. Since S fq L lies on the rational normal curve WN L ~, Lemma 2.1 applies, and we conclude that the inequalities above are in fact equalities. Thus we get, in particular,

r a n k [To@_3(k, ISNH)]c--1 = r a n k [Tor~_3(k , 1W)]c-1.

Hence Iw C IsnH implies

EL,I=o-3 t, | Q, E [Tor~-dk, r-sn,-,)]c-, = [Tor~_~(k, Iw)]c_~ C Ac-a[R]I ~ [ffw]2.

Therefore we have that Q1 E I(alu). It follows that

I(a) + lR/lR = I(alH) = Iw C R.

Since l is a general linear form we get deg I(a) = deg lw = c. Moreover, since Iw is perfect, I ( a ) is perfect too. Thus I ( a ) i s equidimensional and has only nondegenerate components because 1w is nondegenerate. Since any nondegenerate component of l ( a ) has degree at least c = deg I (a) , I (a ) must be prime. This con- cludes the proof of Proposition 3.6.

We are now ready for the proof of our version of the Kp.1 theo- rem.

Proof of Theorem 2.5: By the remarks immediately after the proof of Lemma 3.2, the existence of S ~ allows us to apply Re- mark 3.1 and Lemma 3.2. The latter implies claim (i).

By Lemma 2.2 the conditions in (i i)- (iv) are sufficient for the non-vanishing of the appropriate Tot groups. Now we show the reverse implication beginning with (ii).

Let L = Proj(R') C ~'~ be a general linear subspace of di- mension c + 1. Suppose that we have a # 0 in [Tor~_l(k, Is)]c+a. Then we get, by Remark 3.1, that 0 ~ alL E [Torcn'_x(k, IsnL)]c+l. Furthermore, S fq L has a 1-dimensional nondegenerate component coming from SL Hence Proposition 3.3 applies to S N L and we obtain that I(a[L) = I0, i. e., the 1-dimensional scheme S fq L is contained in a rational normal curve C. It follows that S fq L = C and deg S = deg C = c + 1. On the other hand, deg S ~ _> c + 1

NAGEL-PITTELOUD 309

because S' is nondegenerate. It follows that S has exactly one top- dimensional component, namely a variety V of minimal degree. It remains to exclude the possibility that S has lower-dimensional components. Assume Is = Iv N J where J is an ideal of dimension < m. We have, by Lemma 2.2:

( c + 1 ) = r a n k [ I v ] 2 > 2 _ rank[Is]2.

On the other hand we have Ic = I(a[L) C I(a)R'. This implies

rank[Is]2>_ rank[I(a)]2>_ rank [Ic]2 = (c + l ) " 2

Putt ing both estimates together we see, in particular, that [Iv]2 = [Is]2. It follows that [Iv]2 C J. But Iv is generated by quadrics, a fact that implies Iv C J, a contradiction. Hence S = V and (ii) is shown.

To prove (iii) we first note that the assumptions "char(k) = 0" or "Sired smooth" ensure that, for a general linear subspace L of dimension c, Sr d n L is a set of deg S~, d points in linearly general position in L (see [15], p. 197 and [18], Proposition 5). Hence Proposition 3.6 applies and proves (iii).

The proof of (iv) follows an argument of M. Green. We sketch it here for the sake of completeness. Let 0 7~ a E [TorcR_2(k, Is)]c. Let H0 , . . . , H,~ be general hyperplanes. Then we want to show

Claim: V(C4Hon...n,O,n...nH,~) n Hi = V(C~IHon...nH,,,) for all i = 0 , . . . m where H0 n . . . N Hi n . . . N/arm means that Hi is omitted in the intersection.

Proof of the claim: As in (iii) we have for all i that the intersec- tion SNHoN. . . nHin . . . nHm is a set of c+3 points in linearly general position. Thus V(O~]Hon...nB, n...nU,,,) is a rational normal curve due to Proposition 3.3 and V(O~]Hon...nft, n...nH,,) N Hi C V(a]Hon...nH,~) is a set of c independent points in a (c - 1)-plane. Then it follows by [12] that

On the other hand the ideal of c independent points is generated by (~) quadrics. Thus our claim follows from the containment relation above.

3~0 NAGEL-PITTELOUD

Now we choose pencils of general hyperplanes {HA, : Ai E ~1} (i ---- 1 , . . .m) . Then there exist open subsets U~ C ~1 such that S n H~ 1 n . . . n Hx~, is in linearly general position for all Ai E Ui. We put

T = U V(aIH~ln...nH~,,) AI 6UI ,.,.,AM6U.,

where- denotes the closure in the Zarisky topology. We want to show that T is the required variety of minimal degree containing S. To clarify ideas we assume Ui = ~1 for all i. The general case follows by using relations like T n L = Ux~eu~ ..... X~eVm V(aiH~,n.. .nHx~) n L for a general hyperplane L.

Under our assumption we obtain

T = U,~,et~ ..... ;,,~e~ V ( a l g ~ n . . . n g ~ ) D_

Let now L1, . . . ,Lm be be further general hyperplanes. Then we get by using the claim

TN L1 n . . . n Lm - - U,k16~. l ..... A,,61*l V(OtIH~ln...nHx m) N L1 N . . . N Lm

= U ~ e ~ ..... ~ e ~ V(alg~n.. .nH~.~_,)nL ~ n L~ N . . . n Lm-~ n H~m

~-- UA16:~1 ..... A rn_16~ l V(OtlHAln...nH~,n_l)nL,n n L1 n . . . n Lm-1 0 . .

=

By Proposition 3.3 V(OL]Lln...nLm ) is a rational normal curve. Then it follows as in the proof of (ii) that T is a variety of minimal degree of dimension m + 1 concluding the proof of Theorem 3.5.

In [1] D. Bayer and D. Eisenbud have introduced ribbons. A ribbon is a subscheme S C ~" such that the ideal sheaf 2" of S ~ in S has square 0 and the conormal sheaf Z/2" 2 is a line bundle on S. It is easy to see that the general hyperplane section of a ribbon of dimension > 1 is again a ribbon. Moreover, by [9], Theorem 3.1, the general hyperplane section of an one-dimensional irreducible ribbon is in linearly general position. Thus we get, from Theorem 3.5(ii) and Proposition 3.6, the following corollary.

C o r o l l a r y 3.7 Let S C 1P'~ be a nondegenerate irreducible ribbon

over an algebraically closed f ield k o f characteris t ic zero. Le t c = codim S . Then we have

NAGEL-PITTELOUD 311

(i) [TorRc_l(k, Is)]r = O; (ii) [Tory_2(k, Is)]~ 7~ 0 if and only if S is contained in a variety

of minimal degree and of codimension c - 1.

Another application of Theorem 3.5 is concerned with a problem which arises, for example, in the study of varieties with maximal geometric genus. One wants to know if a scheme is contained in a variety of minimal degree, knowing that its general hyperplane section is contained in such a variety.

C o r o l l a r y 3.8 Let V C F~ be a nondegenerate integral and lin- early normal subscheme of codimension c and degree at least c + 3. Suppose that either char(K) = 0 or V is smooth. If the general hyperplane section of V is contained in a variety of minimal degree having codimension c - 1 in the hyperplane, then V is contained in a variety of minimal degree having codimension c - 1.

This result is well-known if deg V > 2c + 1. In this case V lies on the variety of minimal degree cut out by all quadrics containing V (cf. [16], p. 45). This fails if deg V <_ 2c. Of course, the assumption linearly normal ensures that for all hyperplanes H, all quadrics containing V ~ H lift to quadrics containing V, but the variety cut out by these lifted quadrics depends on H. Thus we do not know a priori its general hyperplane section.

Proof: Let H = Proj(R ' ) be a general hyperplane. As V fq H lies on a variety of minimal degree and is nondegenerate thanks to Bertini's Theorem we obtain, using Remark 3.1 c) as well as

R I Lemma 2.2, that [Torc_2(k, IvnH)]c # O. By the linear normality of V, every quadric containing V f3 H lifts to a quadric containing V. Therefore every linear syzygy of V f3 H lifts to syzygy of V. If follows that [Torn~_2(k, Iv)]c # O. Thus Theorem 3.5 shows the claim.

If the characteristic of the ground field is zero and V C ~n is smooth of dimension > ~ - ~ then V is linearly normal according to Zak's solution of the first part of Hartshorne's conjecture on varieties of low codimension.

312 NAGEL-PITTELOUD

R e f e r e n c e s

[1] D. Bayer, D. Eisenbud: Ribbons, in preparation

[2] M. P. Cavaliere, M. E. P~ossi, G. Valla: On the resolution of certain graded algebras, Trans. Amer. Math. Soc. 337, 389 - 409 (1993)

[3] M. P. Cavaliere, M. E. Rossi, G. Valla: The strong Castelnuovo lemma for zerodimensional subschemes, Preprint, University of Genova, 1993

[4] L. Ein, R. Lazarsfeld: Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension, Invent. Math. 111, 51 - 67 (1993)

[5] D. Eisenbud, S. Goto: Linear free resolutions and minimal multiplicitiy, J. Algebra 88, 89 - 133 (1964)

[6] D.Eisenbud, J. Koh: Some linear syzygy conjectures, Adv. Math. 90, 47 - 76 (1991)

[7] D. Eisenbud, J. Koh, M. Stillman: Determinantal equations for curves of high degree, Amer. J. Math. 110, 513 - 539 (1988)

[8] D. Eisenbud, J. Harris: On varieties of minimal degree (A cen- tennial account), Algebraic Geometry, Bowdoin, 1985, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc. Providence RI (1987), 3- 13

[9] D. Eisenbud, J. Harris: Finite projective schemes in linearly general position, J. Alg. Geom. 1, 15 - 30 (1992)

[10] D. Eisenbud, J. Harris: An intersection bound for rank 1 loci, with applications to Castelnuovo and Clifford theory, J. Alg. Geom. 1, 31 - 60 (1992)

[11] T. Fujita: Defining equations for certain types of polarized variety, in: Complex Analysis and Algebraic Geometry, W.L. Baily, Jr. and T. Shioda, eds. Cambridge Univ. Press (1977), 165- 173

NAGEL-PITTELOUD 313

[12]

[13]

[14]

[15]

[16]

M. Green: The canonical ring of a variety of general type, Duke Math. J. 49, 1087 - 1113 (1982)

M. Green: Koszul cohomology and the geometry of projective varieties, J. Diff. Geom. 19, 125 - 171 (1984)

A. Grothendieck: E1fments de g~om6trie alg6brique III, Publ. Math. IHES 11 (1961)

J. Harris: The genus of space curves, Math. Ann. 249, 191 - 2O4 (1980)

J. Harris: A bound on the geometric genus of projective va- rieties, Ann. Scuola Norm. Sup. Pisa C1. Sci. (4) 8, 36 - 68 (1981)

[17] R. Hartshorne: Algebraic Geometry, Springer-Verlag, New- York 1977

[18]

[19]

[20]

[21]

[22]

[23]

[241

[25]

D. Laksov: Indecomposability of restricted tangent bundles, in: Tableaux de Young et functeurs de Schur en algebre et geometry, Asterisque 87 - 88, 207 - 219 (1981)

E. Mezzetti: Differential-geometric methods for the lifting problem and linear systems on plane curves, Preprint 1993

D. Mumford: Varieties defined by quadratic equations, C.I.M.E. (1969)-III, 29 -100

U. Nagel: On the equations defining arithmetically Cohen- Macaulay varieties in arbitrary characteristic, Preprint, 1993

B. Saint-Donat: Sur les ~quations d~finissant une courbe algfibrique, C.R. Acad. Sc. Paris 274, 324 - 327 (1972)

R. Strano: On generalized Laudal's lemma, Projective Com- plex Geometry, Cambridge University Press (to appear)

P. Stfickrad, W. Vogel: Buchsbaum Rings and Applications, Springer-Verlag, Berlin, 1986

K. Yamagawa: Some extensions of Castelnuovo's lemma on zero-dimensional schemes, Preprint, 1993

314 NAGEL-PITTELOUD

Uwe Nagel FB Mathematil-Informatik / 17 Universit~it-GH Paderborn D-33095 Paderborn Germany e-marl: uwen@uni-paderborn.de

Yves Pitteloud Department of Mathematics and Statistics Queen's University Kingston, Ont, K7L3N6 Canada e-maih pittelou~hilda.mast.queensu.ca

(Received August 22, 1993; in revised form March 22, 1994)