Normal Presentation on Elliptic Ruled SurfacesNORMAL PRESENTATION 599 1, then L satisfies the...

Ž .JOURNAL OF ALGEBRA 186, 597]625 1996ARTICLE NO. 0388

Normal Presentation on Elliptic Ruled Surfaces

Francisco Javier Gallego*, †

Departamento de Algebra, Facultad de Matematicas, Uni ersidad Complutense de´Madrid, 28040 Madrid, Spain

and

B. P. Purnaprajna††

Department of Mathematics, Brandeis Uni ersity, Waltham, Massachusetts 02254-9110Department of Mathematics, 401 Mathematical Sciences Building,

Oklahoma State Uni ersity, Stillwater, Oklahoma 74078

Communicated by D. A. Buchsbaum

Received December 1, 1995

DEDICATED TO R. SRIDHARAN

CONTENTS

Introduction.

1. Background material.

2. General results on normal presentation.

3. Ampleness, base-point-freeness, and cohomology of line bundles on elliptic ruledsurfaces.

4. Normal presentation on elliptic ruled surfaces.

5. Koszul algebras.

References.

*E-mail: [email protected].† Partially supported by DGICYT Grant P1393-0440-C03-01.†† E-mail: [email protected].

597

0021-8693r96 $18.00Copyright Q 1996 by Academic Press, Inc.

All rights of reproduction in any form reserved.

GALLEGO AND PURNAPRAJNA598

INTRODUCTION

This article deals with the normal presentation of line bundles over anelliptic ruled surface. Let X be an irreducible projective variety and L avery ample line bundle on X, whose complete linear series defines

f : X ª P H 0 L .Ž .Ž .L

` m 0Ž . Ž . ` 0Ž mn.Let S s [ S H X, L and let R L s [ H X, L be thens0 ns0homogeneous coordinate ring associated to L. Then R is a finitelygenerated graded module over S, so it has a minimal graded free resolu-tion. We say that the line bundle L is normally generated if the naturalmaps

SmH 0 X , L ª H 0 X , LmmŽ . Ž .

are surjective for all m G 2. If L is normally generated, then we say that Lsatisfies property N , if the matrices in the free resolution of R over Sphave linear entries until the pth stage. In particular, property N says that1

Ž 0Ž ..the homogeneous ideal I of X in P H L is generated by quadrics. Aline bundle satisfying property N is also called normally presented.1

Let R s k [ R [ R [ . . . be a graded algebra over a field k. The1 2RŽ .algebra R is a Koszul ring iff Tor k, k has pure degree i for all i.i

Ž .In this article we determine exactly Theorem 4.2 which line bundles onŽelliptic ruled surface X are normally presented Yuko Homma has classi-

w xfied in Ho1, Ho2 all line bundles which are normally generated on an.elliptic ruled surface . In particular we see that numerical classes of

Žnormally presented divisors form a convex set. See Fig. 1 for the caseŽ . Ž .e X s y1; recall that Num X is generated by the class of a minimal

.section C and by the class of a fiber f and that C is ample. As a0 0corollary of the above result we show that Mukai’s conjecture is true forthe normal presentation of the adjoint linear series for an elliptic ruledsurface.

In Section 5 of this article, we show that if L is normally presented on Xthen the homogeneous coordinate ring associated to L is Koszul. We also

Ž .give a new proof of the following result due to Butler: if deg L G 2 g q 2on a curve X of genus g, then L embeds X with Koszul homogeneouscoordinate ring.

To put things in perspective, we would like to recall what is knownregarding these questions in the case of curves. A classical result of

Ž w x. Ž .Castelnuovo cf. C says that if deg L G 2 g q 1, L is normally gener-w x Ž .ated. St.-Donat and Fujita F, S-D proved that if deg L G 2 g q 2, then L

is normally presented. These theorems have been recently generalized toŽ w x. Ž .higher syzygies by Green see G , who proved that if deg L G 2 g q p q

NORMAL PRESENTATION 599

1, then L satisfies the property N . One way of generalizing the abovepresults to higher dimensions is to interpret them in terms of adjoint linearseries: let v be the canonical bundle of a curve X, and let A be anX

Ž Ž . .ample line bundle since X is a curve, A is ample iff deg A ) 0 . Ifm3 Ž mpq3.L s v m A respectively L s v m A , then Castelnuovo’s Theo-X X

Ž .rem respectively Green’s Theorem says that L is normally generatedŽ .respectively satisfies property N .p

ŽUnlike the case of curves, the landscape of surfaces not to speak of. Ž w x.higher dimensions is relatively uncharted. Recently Reider proved cf. R

that if X is a surface over the complex numbers, then v m Am4 is veryXample. Mukai has conjectured that v m Ampq4 satisfies property N .X p

w xSome work in this direction has been done by David Butler in B , wherehe studies the syzygies of adjoint linear series on ruled varieties. He provesthat if the dimension of X is n, then v m Am2 nq1 is normally generatedXand v m Am2 nq2 n p satisfies property N ; specializing to the case of ruledX p

surfaces, his result says that v m Am5 is normally generated and thatXv m Am8 is normally presented. In this article we consider not just theXadjunction bundle, but any very ample line bundle on an elliptic ruledsurface. In particular, we prove that v m Am5 is normally presentedXthereby proving Mukai’s conjecture for p s 1 in the case of an ellipticruled surface.

In a sequel to this article we generalize our results on normal presenta-tion to higher syzygies. We there show the following: let L s B m ??? m1B be a line bundle on X, where each B is base-point-free and ample.pq1 iThen L satisfies property N . As a corollary we show that v m Am2 pq3

P Xsatisfies property N .p

1. BACKGROUND MATERIAL

CONVENTION. Throughout this paper we work over an algebraic closedfield k.

We state in this section some results we will use later. The first one isthis beautiful cohomological characterization by Green of the property N .pLet L be a globally generated line bundle. We define the vector bundleM as follows:L

0 ª M ª H 0 L m OO ª L ª 0. 1.1Ž . Ž .L X

Ž .In fact, the exact sequence 1.1 makes sense for any variety X and anyvector bundle L as long as L is globally generated.

LEMMA 1.2. Let L be a normally generated line bundle on a ¨ariety XiŽ m2yi.such that H L s 0 for all i G 1. Then, L satisfies the property N iffp

1Ž p9q1 .H n M m L ¨anishes for all 1 F p9 F p.L


w xProof. The lemma is a corollary of GL, Lemma 1.10 .

Ž . Ž . 1Ž 21.2.1 If the char k / 2, we can obtain the vanishing of H n ML. 1Ž m2 . 2m L by showing the vanishings of H M m L , because n M m L isL L

in this case a direct summand of Mm2 m L.LThe other main tool we will use is a generalization by Mumford of a

lemma of Castelnuovo:

THEOREM 1.3. Let L be a base-point-free line bundle on a ¨ariety X andiŽ yi .let FF be a coherent sheaf on X. If H FF m L s 0 for all i G 1, then the

multiplication map

H 0 FF m Lmi m H 0 L ª H 0 FF m Lmiq1Ž . Ž . Ž .

is surjecti e for all i G 0.

w xProof. Mu, p. 41, Theorem 2 . Note that the assumption made thereof L being ample is unnecessary.

It will be useful to have the following characterization of projectivenormality:

2Ž .LEMMA 1.4. Let X be a surface with geometric genus h OO s 0 and letX1Ž .L be an ample, base-point-free line bundle. If H L s 0, then L is normally

1Ž .generated iff H M m L s 0.L

Proof. The line bundle L is normally generated iff the map

am 0 0 mmS H X , L ª H X , LŽ . Ž .

is surjective. The map a fits in the following commutative diagram:

bmm0 m 06

H L S H LŽ . Ž .

6g1

mm y20 m2 0H L m H LŽ . Ž .

6 6g a2

...

6gmy2

gmy10 mmy1 0 0 mm6

H L m H L H LŽ . Ž . Ž .


The map b is surjective. From this fact it follows that the surjectivity ofa is equivalent to the surjectivity of g ( ??? (g . Theorem 1.3 impliesmy 1 1the surjectivity of g , . . . , g . Hence the surjectivity of g implies the2 my1 1surjectivity of a . On the other hand, if m s 2 the surjectivity of a implies

Ž .the surjectivity of g . Finally from 1.1 we obtain1

H 0 L m H 0 L ª H 0 Lm2 ª H 1 M m L ª H 0 L m H 1 L .Ž . Ž . Ž . Ž . Ž . Ž .L

1Ž .Therefore the vanishing of H L implies that the surjectivity of g is11Ž .equivalent to the vanishing of H M m L .L

2. GENERAL RESULTS ON NORMAL PRESENTATION

As mentioned in the Introduction, according to our philosophy, theŽtensor product of two base-point-free line bundles B and B provided it1 2.is ample and that certain higher cohomology groups vanish should be

normally presented. This philosophy is made concrete in the following

PROPOSITION 2.1. Let X be a surface with geometric genus 0 and B and11Ž . 1Ž . 2Ž U .B base-point-free line bundles such that H B s H B s H B m B2 1 2 2 1

2Ž U . 1Ž mpq1 .s H B m B s 0 and let L s B m B . Then H M m L s 0 for1 2 1 2 LŽ .p s 0, 1. In particular, if L is ample and char k / 2, then L is normally

presented.

We will prove a more general version of Proposition 2.1 in Section 5.From this proposition we will obtain corollaries for Enriques surfaces

Ž . Ž .Corollary 2.8 and for elliptic ruled surfaces Theorem 4.1 . To proveProposition 2.1 we will need several lemmas and observations:

OBSERVATION 2.2. Let X be a surface with geometric genus 0, let P be an< <effecti e line bundle, and let B be a line bundle such that, for some p g P ,

B m OO is tri ial or has a global section ¨anishing at finite subscheme of ppŽ . 1Ž . 1Ž . 1Ž .e. g., let B be base-point-free . If H P s H B s 0, then H B m P s 0.

OBSERVATION 2.3. Let X be a surface, let P be an effecti e line bundle,2Ž . 2Ž .and L a coherent sheaf. If H L s 0, then H L m P s 0.

LEMMA 2.4. Let X be a surface, let B be a globally generated line bundle1Ž . Ž .such that H B s 0, and let Y be a cur e in X such that B m OO Y isY

Ž .globally generated. Then B m OO Y is also globally generated.X


Proof. The result is the surjectivity of the middle vertical arrow in thefollowing commutative diagram:

H 0 B m OO ¨ H 0 B m OO Y m OO ¸ H 0 B m OO Y m OOŽ . Ž . Ž .Ž . Ž .X X X Y X

x x xB ¨ B m OO Y ¸ B m OO Y .Ž . Ž .X Y

The hypothesis is that the vertical left-hand side arrow and the verticalright-hand side arrow are surjective.

LEMMA 2.5. Let X be a surface with geometric genus 0, let B and B be1 21Ž . 1Ž .two base-point-free line bundles, and let L s B m B . If H B s H B1 2 1 2

2Ž U . 1Ž mn.s 0 and H B m B s 0, then H M m B s 0 for all n G 1.2 1 L 1

Ž . mnProof. If we tensor exact sequence 1.1 with B and take global1sections, we obtain

a0 0 mn 0 mnH L m H B ª H L m BŽ . Ž . Ž .1 1

ª H 1 M m Bmn ª H 0 L m H 1 Bmn .Ž .Ž . Ž .L 1 1

1Ž mn.From Observation 2.2 it follows that the vanishing of H M m B isL 1equivalent to the surjectivity of the multiplication map a . In the casen s 1 the surjectivity of a follows trivially from our hypothesis andTheorem 1.3. The proof of the surjectivity of a goes by induction. Weshow here only the case n s 2. We consider the commutative diagram

H 0 B m H 0 B m H 0 L ª H 0 Bm2 m H 0 LŽ . Ž . Ž . Ž .Ž .1 1 1

6 6g a

d0 0 0 m26

H B m H B m L H B m L ,Ž . Ž . Ž .1 1 1

where the maps are the obvious ones coming from multiplication. To provethe surjectivity of a it suffices to prove that g and d are surjective. Thesurjectivity of g follows from the surjectivity of a when n s 1. To provethe surjectivity of d , again by Theorem 1.3, it is enough to check that

1Ž . 2Ž .H L s H B s 0. This follows from the hypothesis and from the2Observations 2.2 and 2.3.

LEMMA 2.6. Let X be a surface with geometric genus 0, let B and B be1 2two base-point-free line bundles, and let L s B m B be nonspecial. Let B1 2 1

1Ž m2 . 1Ž . 2Ž U .and B satisfy the conditions H B s H B s 0 and H B m B s2 1 2 2 12Ž m2 U .H B m B s 0. If P is any effecti e line bundle on X such that either1 2


1Ž . 1Ž .H P s 0 or P , OO, then H M m L m P s 0. In particular, if L isLample, L is normally generated.

Ž .Proof. If we tensor 1.1 with L m P and take global sections, weobtain

H 0 L m H 0 L m PŽ . Ž .a 0 m2 1 0 1ª H L m P ª H M m L m P ª H L m H L m P .Ž . Ž . Ž . Ž .L

1Ž .From Observation 2.2 it follows that H L m P vanishes. Therefore the1Ž .vanishing of H M m L m P is equivalent to the surjectivity of theL

multiplication map a . To prove the surjectivity of a we use the same trickas in the proof of the previous lemma. We write this commutative diagram:

H 0 B m H 0 B m H 0 L m P ª H 0 L m H 0 L m PŽ . Ž . Ž . Ž . Ž .2 1

6 6g a

d0 0 0 m26

H B m H L m B m P H L m P .Ž . Ž . Ž .2 1

It suffices then to prove that g and d are surjective and by Theorem 1.3 it1Ž . 2Ž U . 1Ž m2is enough to check that H B m P s H B m B m P s H B m2 2 1 1

. 2Ž m2 U .P s H B m B m P s 0. These vanishings follow trivially from the1 2hypothesis of the lemma and from Observations 2.2 and 2.3.

Ž . 1Ž .2.7 Proof of Proposition 2.1. Observation 2.2 implies that H L1Ž .vanishes. Thus from Lemma 2.6 it follows that H M m L s 0. ThisL

1Ž m2 .implies that the vanishing of H M m L is equivalent to the surjectivityLof the multiplication map

a0 0 0 m2H M m L m H L ª H M m L . 2.7.1Ž . Ž . Ž .Ž .L L

To prove the surjectivity of a we write this commutative diagram:

H 0 B m H 0 B m H 0 M m L ª H 0 L m H 0 M m LŽ . Ž . Ž . Ž . Ž .2 1 L L

6 6a

0 0 0 m2H B m H M m L m B ª H M m L .Ž . Ž . Ž .2 L 1 L

By Theorem 1.3 it is enough to check that

H 1 M m B s H 1 M m Bm2 s 0Ž . Ž .L 2 L 1

H 2 M m B m BU s H 2 M m Bm2 m BU s 0.Ž . Ž .L 2 1 L 1 2


The first two vanishings follow from Lemma 2.5. The other two followŽ .from sequence 1.1 and from Observations 2.2 and 2.3.

Ž .If L is ample, it follows from Lemma 1.2, 1.2.1 , and Lemma 1.4 that Lis normally presented.

The conditions on the vanishing of cohomology required in the state-ment of Proposition 2.1 are not so restrictive. For instance if we take B1and B equal and ample, the conditions on the vanishing of H 2 are2automatically satisfied for surfaces with geometric genus 0. If the surfacewe are considering is Enriques or elliptic ruled the vanishing of H 1 alsooccurs. The next corollary is an outcome of these observations.

Ž .COROLLARY 2.8. Let X be an Enriques surface, let char k s 0, and let Bbe an ample line bundle on X without base points. Then Bm2 is normallypresented.

Proof. Since K ' 0 and B is ample, v m B is also ample and byX X1Ž .Kodaira vanishing, H B s 0. Thus we can apply Proposition 2.1.

3. AMPLENESS, BASE-POINT-FREENESS, ANDCOHOMOLOGY OF LINE BUNDLES ON

ELLIPTIC RULED SURFACES

We have shown in Corollary 2.8 that Bm2 is normally presented if B isan ample, base-point-free line bundle over an Enriques surface. The sameresult is true in the case of elliptic ruled surfaces. However, in this case we

Žcan do much better. In fact we will be able to characterize cf. Theorem.4.2 those line bundles which are normally presented. From the statement

of Lemma 2.6 it is clear that the knowledge of the vanishing of highercohomology of line bundles on elliptic ruled surfaces will be crucial for thispurpose. On the other hand once we know that the tensor product of twobase-point-free line bundles is normally presented, knowing in additionwhich line bundles on an elliptic ruled surface are base-point-free willallow us to characterize those line bundles that are normally presented. Inthis light we will devote this section to recalling the vanishing of cohomol-ogy of line bundles and the characterization of base-point-free line bun-dles on elliptic ruled surfaces.

We introduce now some notation and recall some elementary factsabout elliptic ruled surfaces. Proofs for the statements of this paragraph

w xcan be found in H, Sect. V.2 . In this and the next section X will denote aŽ .smooth elliptic ruled surface, i.e., X s P EE , where EE is a vector bundle

of rank 2 over a smooth elliptic curve C. We will assume EE to benormalized, i.e., EE has global sections but twists of it by line bundles of


negative degree do not. Let p denote the projection from X to C. We setŽ . 2OO e s H EE and e s ydeg e G y1. We fix a minimal section C such0

Ž . Ž . Ž .that OO C s OO 1 . The group Num X is generated by C and by the0 PŽEE . 0class of a fiber, which we will denote by f. If a is a divisor on C, a f willdenote the pullback of a to X by the projection from X to C. Sometimes,when deg a s 1, we will write, by an abuse of notation, f instead of a f.The canonical divisor K is linearly equivalent to y2C q e f , and henceX 0numerically equivalent to y2C y ef.0

PROPOSITION 3.1. Let L be a line bundle on X, numerically equi alent toaC q bf.0

If e s y1:0Ž . 1Ž . 2Ž .a b h L h L h L

b ) yar2 ) 0 0 0a G 0 b s yar2 ? ? 0

b - yar2 0 ) 0 0

a s y1 any b 0 0 0

b ) yar2 0 ) 0 0a F y2 b s yar2 0 ? ?

b - yar2 0 0 ) 0

If e G 0:0Ž . 2Ž .a b h L h L

b ) 0 ) 0 0a G 0 b s 0 ? 0

b - 0 0 0

a s y1 any b 0 0

b ) ye 0 0a F y2 b s ye 0 ?

b - ye 0 ) 01Ž .a b h L

b ) ae 0a G 0 b s ae ?

b - ae ) 0

a s y1 any b 0

Ž .b ) e a q 1 0Ž .a F y2 b s e a q 1 ?Ž .b - e a q 1 ) 0


0Ž .Proof. If a - 0 it is obvious that h L s 0. If a G 0, one obtains the0Ž . 1Ž .statements for h L and h L by pushing down L to C and computing

the cohomology there. In the case of e F 0, we use the fact that thewsymmetric powers of E are semistable bundles Mi, Corollary 3.7 and Sect.

x5 . Then we use the fact that, if F is a semistable bundle over an ellipticŽ . 0Ž . 1Ž .curve and deg F ) 0, then h F ) 0 and h F s 0 and the fact that if

Ž . 0Ž . 1Ž .deg F - 0, then h F s 0 and h F ) 0. In the case e ) 0 the compu-tation of cohomology on C is elementary, since E is decomposable. If

1 1Ž .a s y1, p#L s R p#L s 0; hence H L s 0.The other statements in the proposition follow by duality.

The last proposition means that the vanishing of cohomology of linebundles on X is an almost numerical condition, in the sense that in mostcases we can decide whether or not a particular cohomology groupvanishes by simply looking at the numerical class to which the line bundlebelongs. As a matter of fact, in those numerical classes in which we cannotdecide, there exist line bundles for which certain cohomology groupsvanish and line bundles for which it does not. We will study in more detailthis situation in the case e s y1, because we will need for the sequel toknow exactly for which line bundles the cohomology vanishes. Concretely,this knowledge will allow us to use Proposition 2.1 and Proposition 5.4 inthe proofs of Theorem 4.2 and Theorem 5.7, respectively. It will be used as

w xwell in GP . Also we will show the existence of a smooth elliptic curvenumerically equivalent to 2C y f.0

PROPOSITION 3.2. Let X be a ruled surface with in¨ariant e s y1. Then

Ž .3.2.1 There exist only three effecti e line bundles in the numerical classŽ Ž . .of 2C y f. They are OO 2C y e q h f , where the h ’s are the nontri ial0 0 i i

0Ž .degree 0 di isors corresponding to the three nonzero torsion points in Pic C .< Ž . . <The unique element in 2C y e q h f is a smooth elliptic cur e E .0 i i

Ž .3.2.2 For each n ) 1, there are only four effecti e line bundles numeri-Ž . Ž Ž . . Žcally equi alent to n 2C y f . They are OO 2nC y n e q h f and OO 2nC0 0 i 0

. Ž . Žy ne f . The only smooth elliptic cur es and indeed the only irreducible. < <cur es in these numerical classes are general members in 4C y 2e f .0

The number of linearly independent global sections of these line bundles issummarized in the following table:

n G 0 0 1 2 3 . . . nn

0Ž Ž ..h OO 2nC y ne f 1 0 2 1 . . . 3 y n q 10 2n

0Ž Ž Ž . ..h OO 2nC y n e q h f 0 1 1 2 . . . n y0 i 2


Proof. For any p g C we consider the following exact sequence:wp0 0 06

0 ª H OO 2C y pf ª H OO 2C H OO 2CŽ . Ž . Ž .Ž . Ž . Ž .0 0 p f 0

ª H 1 OO 2C y pf ª 0.Ž .Ž .0

Pushing forward the morphism

H 0 OO 2C m OO ª OO 2CŽ . Ž .Ž .0 0

to C we obtain

w0 2 2H S EE m OO ª S EE ª Q ª 0.Ž . Ž .Ž . C

0Ž 2Ž ..Note that the restriction of w to the fiber of H S EE m OO over p isC0Ž Ž ..precisely w . Thus the points p for which H OO 2C y pf / 0 arep 0

0Ž Ž ..exactly the ones where the rank of w drops. Note that h OO 2C s 30Žpush down the bundle to C, use the same semistability considerations asin the sketch of the proof of Proposition 3.1 to obtain the vanishing of H 1,

. 2Ž .and then, use Riemann]Roch . The rank of S EE is also 3, so if the rankof w never dropped, w would be an isomorphism, which is not true,

2Ž .because, since e s y1, the degree of S EE is 3. Therefore there exists0Ž Ž ..p g C such that H OO 2C y pf / 0. We fix such a point p and some0

< <effective divisor E inside 2C y pf . Since 2C y f cannot be written as0 0sum of two nonzero numerical classes both containing effective divisors, E

Ž .is irreducible and reduced. By adjunction, p E s 1 and since E domi-anates C, E is indeed a smooth elliptic curve.

We prove now by induction on n the following statement: for eachn G 0, there are finitely many effective line bundles numerically equivalentto 2nC y nf. The result is obviously true for n s 0. Take now n ) 0. We0fix a divisor d9 of degree n y 1. What we want to prove is that the number

0Ž Ž ..of points z g C such that H OO 2nC y d f / 0 is finite, where d s d900Ž ŽŽ . Ž . ..q z. We may assume that H OO 2n y 2 C y d9 y p q z f s 0, since,0

by the induction hypothesis, there are only finitely many points z forwhich this does not happen. We tensor the sequence

0 ª OO y2C q pf ª OO ª OO ª 0 3.2.3Ž . Ž .0 E

Ž . 0Ž ŽŽ . Žby OO 2nC y d f and take global sections. Since H OO 2n y 2 C y d0 0. .. ŽŽ . Žy p f is 0 and the degree of the push forward of OO 2n y 2 C y d y0

. . 1Ž ŽŽ . Ž . ..p f to C is 0, it follows that H OO 2n y 2 C y d9 y p q z f s 0.00Ž Ž .. 0Ž Ž ..Hence we obtain that H OO 2nC y d f s H OO 2nC y d f . The0 E 0

Ž . 0Ž Ždegree of OO 2nC y d f is zero; therefore z is such that H OO 2nC yE 0 0.. Ž . Ž . Ž .d f / 0 iff OO 2nC y d f , OO , i.e., iff OO 2nC y d9 f , OO zf .E 0 E E 0 E

There are only finitely many such points z, since otherwise, we will have


that all the fibers of the 2 : 1 morphism from E onto C induced by thedegree 2 divisor which is obtained as the restriction of 2nC y d9 f to E0are members of the same g1.2

The last statement implies that the length of Q is finite, and equal to2Ž .deg S EE s 3. We claim that Q is in fact supported in three distinct

2Ž .points p , p , p . If not, there would exist a global section of S EE1 2 3vanishing at some point q to order greater or equal than 2. In particular,Ž .OO 2C y 2 qf would be effective, which contradicts Proposition 3.1. Our0

aim now is to identify p , p , p . Let E be the unique element of1 2 3 i< <2C y p f . We saw before that E is a smooth elliptic curve. Pushing0 i i

Ž .down to C the exact sequence 3.2.3 we obtain

0 ª OO ª p#OO ª R1p#OO y2C q p f ª 0.Ž .C E 0 ii

<Since p is unramified and E is connected, it follows that p#OO s OOE i E Ci i

[ LL for some line bundle LL such that LL m2 s OO , but LL / OO . UsingC CŽ .relative duality and projection formula one obtains that 2 p y e ; 0 buti

p y e ¤ 0. This proves the first part of the proposition.iFor the second part, remember that we have already proven the exis-

tence of only finitely many effective line bundles. Therefore, for anyp g C, we have the exact sequence

0 ª H 0 S2 n EE m OO yn q 1 p m OOŽ .Ž .Ž .C C

ª S2 n EE m OO yn q 1 p ª Q9 ª 0.Ž .Ž .C

2 n ŽŽ . .The length of Q9 is equal to the degree of S EE m OO yn q 1 p , whichCis 4m q 1 if n s 2m and 4m q 3 if n s 2m q 1, but it is also equal tothe sum of the dimensions of the linear spaces of global sections of linebundles in the numerical class of 2nC y nf. Then, the rest of the0statement in 3.2.2 and the numbers in the table follow from comparing thelength of Q9 with the sum of the dimensions of the linear spaces generatedby sections corresponding to reducible divisors numerically equivalent to2nC y nf.0

Ž .3.2.4 We will fix once and for all a smooth elliptic curve E in thenumerical class of 2C y f.0

Ž .3.2.5 For a different proof of the existence of a smooth elliptic curvew xin the numerical class of 2C y f see Ho2, Corollary 2.2 .0

In the case e G 0 we are interested in finding sections of p whoseŽself-intersection is near to that of C they will play in the sequel a role0

.similar to that of E :

PROPOSITION 3.3. Let X be an elliptic ruled surface with in¨ariant e G 0.< <The general member of C y e f is a smooth elliptic cur e and those are the0

only smooth cur es in the numerical class of C q ef.0


Ž . < < 0Ž Ž ..Proof. If det EE / OO the dimension of C y e f is h OO [ OO ye y0< <1 s e. Since the dimension of ye9 f is e y 1 for any nontrivial divisor

< <ye9 of degree e on C it is clear that not all the elements in C y e f are0unions of C and e fibers. On the other hand this is the only way in which0

< < Žan element of C y e f can be reducible this is because for any divisor d0< <of degree d - e, the dimension of C q d f is d y 1, which implies that0

< < .any element of C q d f is the union of C and d fibers . Thus the0 0< <general member of C y e f is irreducible. Therefore, it maps surjectively0

onto C and hence it is a smooth elliptic curve. If e9 ' e but e9 / e , then

< < 0dim C y e9 f s h OO e y e9 [ OO ye9 y 1 s e y 1Ž . Ž .Ž .0

< <which means that all members of C y e9 f are reducible.0Ž .If det EE s OO, EE is an extension of OO by OO. Thus the member or

< < < < < <members of C y e f s C are smooth elliptic curves and C y e9 f s B0 0 0for any divisor e9 on C of degree 0 different from e.

Ž .3.3.1 We will fix once and for all a smooth elliptic curve E9 in thenumerical class of C q ef.0

w xPROPOSITION 3.4 H, V.2.20.b and V.2.21.b . Let L be a line bundle on Xin the numerical class of aC q bf.0

1If e s y1, L is ample iff a ) 0 and a ) y b.2

If e G 0, L is ample iff a ) 0 and b y ae ) 0.

We state now a proposition describing numerical conditions which implywbase-point-freeness. The proof follows basically the one given in Ho1,

xHo2 . In characteristic 0 the proposition can also be proven using Reider’sw xtheorem R .

PROPOSITION 3.5. Let L be a line bundle on X in the numerical class ofaC q bf.0

If e s y1, a G 0, a q b G 2, and a q 2b G 2, then L is base-point-free.If e G 0, a G 0, and b y ae G 2, then L is base-point-free.

Proof. First we consider the case e s y1. In the first place we provethe proposition when a s 0 and b G 2, when a s b s 1, and when a s 2and b s 0. The first case follows easily from the fact that line bundles onelliptic curves whose degrees are greater than or equal to 2 are base-point-free. For the other two cases we use the fact that there are only one

Ž w x.or two minimal sections through a given point of X cf. Ho2 . On theother hand, for a given point p g C, there are infinitely many effective

< <reducible divisors in C q pf , namely, those consisting of the union of a0divisor linearly equivalent to C q t f and a divisor linearly equivalent to0Ž .p y t f , where t is a degree 0 divisor on C. Hence, the intersection of all


those reducible divisors is empty. Analogously, for a given divisor n of< <degree 0 there are infinitely many effective reducible divisors in 2C q n f ,0

namely, those consisting of the union of a divisor linearly equivalent toŽ . Ž .C q n q t f and a divisor linearly equivalent to C q n y t f , and the0 0

same argument goes through.Now we use Lemma 2.4. The base-point-free line bundle B will be

Ž .numerically equivalent to bf b G 2 , C q f , or 2C and Y will be E0 0Ž Ž .. Ž Ž Ž ..defined in 3.2.4 or C note that since deg L m OO Y G 2, it follows0 Y

Ž . .that L m OO Y is base-point-free . Iterating this process we obtain theYresult. The only place where we have to be careful in the application ofLemma 2.4 iteratively is in making sure that the base-point-free linebundles we keep obtaining are nonspecial. This problem is taken care of byProposition 3.1.

The case e G 0 is easier. The line bundle L is base-point-free if it is inthe numerical class of bf, when b G 2. Then we get the result for anyother bundle satisfying the conditions in the proposition by using Lemma

Ž Ž ..2.4. The curve Y in Lemma 2.4 will be E9 defined in 3.3.1 . AgainProposition 3.1 assures us that the line bundles we obtain are nonspecial.

Remark 3.5.1. The numerical condition of Proposition 3.5 characterizesthose equivalence classes consisting entirely of base-point-free line bun-dles.

Proof. A line bundle that satisfies the above numerical conditions isbase-point-free by virtue of Proposition 3.5. To prove the other implica-tion, consider a base-point-free line bundle L in the numerical class of

Ž .aC q bf, which does not satisfy the above conditions. If e X s y1, the0restriction of L to the elliptic curve E is a base-point-free line bundle.Hence, since its degree is equal to a q 2b - 2, it must be the trivial linebundle, which implies that a q 2b s 0. Then it follows from Proposition3.2 that the general member of the numerical class is not base-point-freeŽ . Ž .in fact it is not even effective! . If e X G 0, for the same reason asabove, the restriction of L to C is trivial. This is only possible if0

Ž Ž ..L s OO n C y e f .0

Ž .3.5.2 The proof of the previous remark suggests that there existnontrivial base-point-free line bundles with self-intersection 0. That isindeed the case. For example, if e s y1, the divisors 2nC q ne f for any0even number n greater than 0 are base-point-free; if X s C = P1, the

Ž .divisors nC and if e G 1, the divisors n C y e f are base-point-free.0 0Hence base-point-freeness cannot be characterized numerically. However,if we assume L to be ample, then the numerical conditions in Proposition3.5 do give a characterization of base-point-freeness:


Remark 3.5.3. Let L be a line bundle on X in the numerical class ofaC q bf.0

If e s y1, the line bundle L is ample and base-point-free iff a G 1,a q b G 2, and a q 2b G 2.

If e G 0, the line bundle L is ample and base-point-free iff a G 1 andb y ae G 2.

Proof. If L satisfies the numerical conditions in the statement of theremark, then by Propositions 3.4 and 3.5 it is ample and base-point-free.Now assume that L is ample and base-point-free. If e s y1, fromProposition 3.4, it follows that a G 1. On the other hand, since L isbase-point-free, its restriction to any curve in X is also base-point-free.Consider the curves C and a smooth curve E in the numerical class of02C y f. The restriction of L to each of them has degree a q b and0a q 2b, respectively. The fact that the restriction of L to C is base-0point-free implies that either a q b G 2 or the restriction of L to C is0trivial. The latter is impossible since L is ample. Analogously the fact thatL is ample and that the restriction of L to E is base-point-free impliesthat a q 2b G 2. If e G 0, by Proposition 3.4, a G 1. Since L is as well

Žbase-point-free, by restricting L to C we obtain that b y ae s deg L m0.OO G 2.C0

4. NORMAL PRESENTATION ON ELLIPTIC RULEDSURFACES

We recall that in this section X denotes a ruled surface over an ellipticcurve and we continue to use the notation introduced at the beginning ofSection 3. We have just seen which line bundles on X are ample andwhich are base-point-free. The next question to ask would be: ‘‘which linebundles are very ample and which are normally generated?’’ This problem

w xwas solved by Y. Homma in Ho1, Ho2 , who proved that a line bundle Lon X is normally generated iff it is very ample. She also characterizes

Ž .those line bundles see Fig. 1 for the case e s y1 . Homma proves as wellthat in the case of a normally generated line bundle L, the ideal corre-sponding to the embedding induced by L is generated by quadratic andcubic forms. Thus the next question is to identify those line bundles whichare normally presented.Ž .4.1 Throughout the remaining part of this section we assume thatŽ .char k / 2.

We will use Proposition 2.1 and the results from Section 3 to character-ize the line bundles on X which are normally presented.


THEOREM 4.2. The condition of normal presentation depends only onnumerical equi alence. More precisely, let L be a line bundle on X numericallyequi alent to aC q bf. If e s y1, L is normally presented iff a G 1,0a q b G 4, and a q 2b G 4. If e G 0, L is normally presented iff a G 1 andb y ae G 4.

Proof. First we prove that if a line bundle L satisfies the numericalconditions in the statement, it is normally presented. To this end we willuse Proposition 2.1. The idea is to write L as tensor product of two linebundles B and B satisfying the numerical conditions in Proposition 3.5,1 2

2Ž U . 2Ž U .and such that H B m B s H B m B s 0. Let us exhibit the line1 2 2 1bundles B and B in the different cases:1 2

If e s y1, L can be written as tensor product of B and B , where the1 2couple B and B satisfies one of the following numerical conditions:1 2

Ž . Ž4.2.1 B ' C q nf and B ' 2 f or C q f , for some n G 1 in this1 0 2 0.case, 1 F a F 2 and a q b G 4 .

Ž .4.2.2 B ' 2C and B ' 2C q lf or C q nf , for some l G 0, and1 0 2 0 0Ž .some n G 1 in this case 3 F a F 4 and a q b G 4 .

Ž . Ž .4.2.3 B ' 2C q m 2C y f and B ' 2C q lf or C q nf , for1 0 0 2 0 0Žsome m G 1, some l G 1, and some n G 1 in this case, a G 5 and

.a q 2b ) 4 .Ž . Ž . Ž4.2.4 B ' 2C q m 2C y f and B ' 2C , for some m G 1 in1 0 0 2 0

.this case, a G 5 and a q 2b s 4 .

Ž . Ž . Ž .If B and B satisfy 4.2.1 , 4.2.2 , or 4.2.3 , Proposition 3.1 implies that1 22Ž U . 2Ž U . Ž . 2ŽH B m B s H B m B s 0. If B and B satisfy 4.2.4 , H B m1 2 2 1 1 2 1

U . 2Ž U .B vanishes but H B m B might not be zero. However, from Proposi-2 2 1Ž .tion 3.2 it follows that, given L ' 4C q m 2C y f , m G 1, we can0 0

Ž .choose B ' 2C q m 2C y f and B ' 2C such that L s B m B1 0 0 2 0 1 22Ž U .and H B m B s 0; hence we are done in this case.2 1

If e G 0, L can be written as tensor product of B and B , where,1 2

Ž . ? @ Ž . u vif a is even: B ' ar2 C q br2 f and B ' ar2 C q br2 f1 0 2 0and

? @ ?Ž . @ u v uŽif a is odd: B ' ar2 C q b y e r2 f and B ' ar2 C q b q1 0 2 0. ve r2 f.

Proposition 3.4 implies that L is ample, Proposition 3.5 implies that B1and B are base-point-free, and Propositions 3.1 and 3.2 imply that2

1Ž . 1Ž . 2Ž U . 2Ž U .H B s H B s H B m B s H B m B s 0, so from Proposi-1 2 1 2 2 1tion 2.1 it follows that L is normally presented.

Now we will suppose that L is normally presented but does not satisfythe numerical conditions in the statement and we will derive a contradic-tion. We can assume that a G 1. Otherwise L would not be ample. If


e s y1, we can also assume that a q b s 3 and a q 2b s 3. If not therestriction of L to either the minimal section C or the curve E defined in0Ž .3.2.4 would not be very ample. Analogously, if e G 0, we can assume thatb y ae s 3. Otherwise, the restriction of L to C would not be very ample.0

To obtain the contradiction we follow the same strategy: we will see thatthe assumption of L being normally presented forces its restriction to bothC and E to be also normally presented, and we will derive from that the0

Ž .contradiction. If e G 0 let P denote the line bundle OO C . If e s y1 and0Ž .L ' C q 2 f or 2C q f , let P denote OO C . If e s y1 and L ' 3C q0 0 0 0

Ž . Ž .m 2C y f , m G 0, let P denote OO E . Let p denote a smooth elliptic0< <curve in P . We claim that

22H M m L m P* s 0. 4.2.5Ž .H Lž /

Ž . 2Ž m2 .To prove 4.2.5 we will prove instead the fact that H M m L m P* sL0. Consider the following exact sequence, which arises from exact se-

Ž .quence 1.1 ,

H 1 M m Lm2 m P* ª H 2 Mm2 m L m P*Ž . Ž .L L

ª H 0 L m H 2 M m L m P* . 4.2.6Ž . Ž . Ž .L

1Ž . 1Ž m2 .Since H L s 0, the vanishing of H M m L m P* is equivalent toLthe surjectivity of the map

H 0 L m H 0 Lm2 m P* ª H 0 Lm3 m P* .Ž . Ž . Ž .

The line bundle L is base-point-free by Proposition 3.5. Therefore, by1Ž . 2Ž .Theorem 1.3, it is enough to check that H L m P* s H P* s 0. The

vanishings follow from our choice of P and from Proposition 3.1, except2Ž . Ž .the vanishing of H P* when P , OO E , which follows from Proposition

Ž . 2Ž .3.2 and duality. Using 1.1 we obtain that H M m L m P* will vanish ifL1Ž m2 . 2Ž .H L m P* and H L m P* vanish. These two vanishings follow from

Ž . 2Ž m2 .Proposition 3.1. Therefore by 4.2.6 , H M m L m P* s 0 andL2Ž 2 .H H M m L m P* s 0.L

1Ž .Now since L is assumed to be normally presented and H L s 0 we1Ž 2 . Ž .have, by Lemma 1.2, that H H M m L s 0. Thus from 4.2.5 it followsL

that

2 21 1H M m OO m L s H M m L m OO s 0. 4.2.7Ž .Ž .H HL p L pž / ž /

ŽConsider the following commutative diagram which holds for any base-point-free line bundle L and for any nontrivial line bundle P such that


1Ž . .H L m P* s 0 :

0 0

x x0 0Ž . Ž .0 ª H L m P* m OO ª H L m P* m OO ª 0p p

x x x0 Ž .0 ª M m OO ª H L m OO ª L m OO ª 0L p p p

x x x00 ª M ª H L m OO m OO ª L m OO ª 0Ž .Lm OO p p pp

x x x0 0 0

From the left-hand side vertical sequence we obtain the surjection

2 2

M m OO m L ª M m L.Ž .H HL p Lm OOp

Since p is a curve, it follows that

21H M m L s 0. 4.2.8Ž .H Lm OOž /p

The line bundle L m OO on p is normally generated because p is anpŽ . Ž .elliptic curve and deg L m OO s 3. Thus Lemma 1.2 and 4.2.8 imply thatp

L m OO is normally presented, which is impossible since the completep

linear series of L m OO embeds p as a plane cubic!p

Ž .4.2.9 It follows in particular from Proposition 4.2 that the normallygenerated line bundles on X, or more precisely, their numerical classes,

Ž .form a convex set in Num X , as shown in Fig. 1, in which we describeŽ . Ž .Num X when e s y1 we do not draw the similar picture for e G 0 .

Now we reformulate Theorem 4.2 and state some corollaries, which willhelp us to put our result in perspective:

THEOREM 4.3. Let X be an elliptic ruled surface. A line bundle L on X isnormally presented iff it is ample and can be written as the tensor product oftwo line bundles B and B such that e¨ery line bundle numerically equi alent1 2to any of them is base-point-free.

Proof. If L is normally presented, it is obviously ample and satisfies thenumerical conditions of Proposition 4.2. In the proof of that propositionwe showed that a line bundle satisfying the mentioned numerical condi-tions can be written as the tensor product of two base-point-free linebundles B and B satisfying the conditions of Remark 3.5.1. Hence these1 2


FIG. 1. Cross means that all the members in the numerical class are base-point-free,Ž .dashed or lined square means that the corresponding coordinate ring is presented by

Ž .quadratic forms, dashed or lined disc means normally presented, annulus means normallygenerated, blank disc means ample, gray or hashed disc means ample and base-point-free.

B and B are such that all the line bundles in their numerical classes are1 2base-point-free.

On the other hand, assume that L is ample and isomorphic to B m B1 2and that any line bundle numerically equivalent to either B or B is1 2base-point-free. Let B be in the numerical class of a C q b f and L ini i 0 ithe numerical class of aC q bf. If e s y1, by Remark 3.5.1, a q b G 20 i iand a q 2b G 2. Thus we obtain that a q b G 4 and a q 2b G 4. Sincei i


L is ample, a G 1. Hence, from Proposition 4.2 it follows that L isnormally presented. If e G 0 one argues in a similar fashion.

COROLLARY 4.4. Let X be as abo¨e. Let B be an ample and base-point-ifree line bundle on X for all 1 F i F q. If q G 2, then B m ??? m B is1 qnormally presented and if q - 2, in general B m ??? m B is not normally1 qpresented.

Proof. It follows from Remarks 3.5.1 and 3.5.3 that a line bundlenumerically equivalent to any of the B is base-point-free. From Remarki3.5.3 and Proposition 3.4 it follows that L is ample. Thus, by Theorem 4.3,L is normally presented.

COROLLARY 4.5. Let X be as abo¨e. Let A be an ample line bundle on Xifor all 1 F i F q. If q G 4, then A m ??? m A is normally presented and if1 qq - 4, in general A m ??? m A is not normally presented.1 q

Proof. From Proposition 3.4 and Remark 3.5.3 it follows that the tensorproduct of two ample line bundles is ample and base-point-free. Hence thecorollary follows from Corollary 4.4.

COROLLARY 4.6. Let X be an elliptic ruled surface. Let A be an ampleiline bundle on X for all 1 F i F q.

If e s y1 and q G 5, then v m A m ??? m A is normally presented. IfX 1 qe s y1 and q - 5, in general v m A m ??? m A is not normally pre-X 1 qsented.

If e s 0 and q G 4, then v m A m ??? m A is normally presented. IfX 1 qe s 0 and q - 4, in general v m A m ??? m A is not normally presented.X 1 q

If e G 1 and q G 3, then v m A m ??? m A is normally presented. IfX 1 qe G 1 and q - 3, in general v m A m ??? m A is not normally presented.X 1 q

Proof. Let A be in the numerical class of a C q b f and v m Ai i 0 i X 1m ??? m A in the numerical class of aC q bf. If e s y1, A is ample iffq 0 i

Ž .a G 1 and a q 2b G 1 cf. Proposition 3.4 . In particular we also havei i ithat if A is ample, then a q b G 1. Since v is numerically equivalenti i i Xto y2C q f it follows that0

a G q y 2 G 3 ) 1, a q b G q y 1 G 4, a q 2b G q G 4.

Hence by Theorem 4.2, v m A m ??? m A is normally presented.X 1 qŽ .If e s 0, A is ample iff a G 1 and b G 1 cf. Proposition 3.4 . Sincei i i

v is numerically equivalent to y2C it follows that a G q y 2 ) 1 andX 0b y ae G q G 4. Hence by Theorem 4.2, v m A m ??? m A is normallyX 1 qpresented.

Ž .If e G 1, A is ample iff a G 1 and b y a e G 1 cf. Proposition 3.4 .i i i iSince v is numerically equivalent to y2C y ef it follows that a G q yX 0


2 G 1 and b y ae G q q e G 4. Then by Theorem 4.2, v m A m ??? m AX 1 qis normally presented.

Ž . Ž . Ž .The line bundles v m OO 4C , if e X s y1, v m OO 3C q 3 f , ifX 0 X 0Ž . Ž Ž . . Ž .e X s 0, and v m OO 2C q 2 e q 1 f , if e X G 1, are not normallyX 0

Ž .presented cf. Theorem 4.2 . Thus our bound is sharp.

We want to compare our results to the results known for curves. FujitaŽ w x.and St.-Donat cf. F, S-D proved that, on a curve, any line bundle of

degree bigger than or equal to 2 g q 2 is normally presented. The results inthis section as well as Theorem 2.1 are analogous in different ways to theresult by Fujita and St.-Donat. The approach taken up to now to general-ize this result has been to look at adjoint linear series. Along this line itwas conjectured by Mukai that on any surface X, v m A m ??? m AX 1 qshould be normally presented for all q G 5 and A ample line bundles.iCorollary 4.6 shows that this conjecture holds if X is an elliptic ruled

Ž .surface and that it is sharp if the invariant e X s y1. One disadvantageof this generalization is that it only gives information about a small class ofline bundles. The possible ways of generalization indicated by Theorem4.4:

Ž .4.7 Let X be a surface. If L is the product of two ample linebundles B and B , such that every line bundle B numerically equivalent1 2to either B or B is base-point-free, then L is normally presented;1 2

or by Proposition 2.1:

Ž .4.8 Let X be a surface. If L is ample and the product of twobase-point-free and nonspecial line bundles, then L is normally presented;

or maybe by some combination of the two, take in account a larger class ofŽ .line bundles in general. In subsequent articles we prove that both 4.7 and

Ž .4.8 hold for K3 surfaces.We remark that Theorem 4.3, which is stronger than Corollary 4.4, can

also be seen as an analogue of Fujita and St.-Donat’s theorem, since thelatter can be rephrased as follows:

Let L be a line bundle on a curve. Every line bundle numericallyequivalent to L is normally presented iff L is ample and the tensorproduct of two line bundles B and B such that every line bundle B1 2numerically equivalent to B or B is base-point-free.1 2

However, the veracity of Theorem 4.3 seems to depend on the particularproperties of elliptic ruled surfaces and the corresponding statement isfalse on K3 surfaces.

We generalize Corollaries 4.4 and 4.5 to higher syzygies in a forthcomingŽ w x.article cf. GP , by proving that the product of p q 1 or more ample and

base-point-free line bundles satisfies the property N .p


5. KOSZUL ALGEBRAS

In the previous section we determined which line bundles on an ellipticruled surface are normally presented. A question to ask is whether thecoordinate ring of the embedding induced by those line bundles is aKoszul ring, since it is well known that a variety with a Koszul homoge-neous coordinate ring is projectively normal and defined by quadrics. Theanswer to this question is affirmative not only in the case of elliptic ruledsurfaces, but in all other cases with which we have dealt throughout thiswork, since we are able to prove that the coordinate ring corresponding toa line bundle satisfying the conditions of Proposition 2.1 is Koszul.

We introduce now some notation and some basic definitions: given aŽ . ` 0Ž mn.line bundle L on a variety X, we recall that R L s [ H X, L .ns0

DEFINITION 5.1. Let R s k [ R [ R [ . . . be a graded ring and k a1 2RŽ .field. R is a Koszul ring iff Tor k, k has pure degree i for all i.i

Now we will give a cohomological interpretation, due to Lazarsfeld, ofŽ .the Koszul property for a coordinate ring of type R L . Let L be a globally

generated line bundle on a variety X. We will denote M Ž0., L [ L andM Ž1., L [ M m L s M Ž0., L m L. If M Ž1., L is globally generated, we de-L Mnote M Ž2., L [ M Ž1., L m L. We repeat the process and define inductivelyMM Žh., L [ M Žhy1., L m L, if M Žhy1., L is globally generated. Now we areMready to state the following

w xLEMMA 5.2 P, Lemma 1 . Let X be a projecti e äriety oër an algebraicclosed field k. Assume that L is a base-point-free line bundle on X such that

Žh., L 1Ž Žh., Lthe ëctor bundles M are globally generated for eëry h G 0. If H Mms. Ž .m L s 0 for eëry h G 0 and eëry s G 0 then R L is a Koszul k-alge-

1Ž ms.bra. Moreoër, if H L s 0 for eëry s G 1 the conërse is also true.

Now we will prove a general result analogous to Proposition 2.1. Butbefore that, we state the following well-known

OBSERVATION 5.3. Let FF be a locally free sheaf oër a scheme X and A0Ž mn. 0Ž .an ample line bundle. If the multiplication map H FF m A m H A ª

0Ž mnq1.H FF m A surjects for all n G 0, then FF is globally generated.

THEOREM 5.4. Let X be a surface with p s 0, let B and B be twog 1 2base-point-free line bundles, and let L s B m B be ample. If1 2

H 1 B s H 1 B s H 2 B m BU s H 2 B m BU s 0,Ž . Ž . Ž . Ž .1 2 1 2 2 1

then the following properties are satisfied for all h G 0:

Ž . Žh., L5.4.1 M is globally generated;Ž . 1Ž Žh., L mb1 mb2 .5.4.2 H M m B m B s 0 for all b , b G 0;1 2 1 2Ž . 1Ž Žh., L U .5.4.3 H M m B s 0 where j s 1, 2;j


Ž . 1Ž Žh., L U .5.4.4 H M m B m B s 0 where i s 1, 2 and j s 2, 1;i jŽ . 1Ž Žh., L m2 U .5.4.5 H M m B m B s 0 where i s 1, 2 and j s 2, 1.i j

1Ž Žh., L ms. Ž .In particular H M m L s 0 for all h, s G 0, and R L is a Koszulk-algebra.

Proof. We prove the lemma by induction on h.If h s 0, property 5.4.1 means that L is globally generated, which is true

1Ž mŽb1q1 . mŽb2q1 ..by hypothesis. Properties 5.4.2 to 5.4.5 mean that H B m B1 21Ž mb i.s H B s 0 where b , b G 0, b s 1, 2, 3, and i s 1, 2. These vanish-i 1 2 i

ings occur by hypothesis and Observation 2.2.Now consider h ) 0 and assume that the result is true for all 0 F h9 F h

y 1. Let L9 denote Bmb1 m Bmb2 , BU , B m BU , or Bm2 m BU accordingly.1 2 j i j i jIf we tensor

0 ª M Žhy1. , L ª H 0 M Žhy1. , L m OO ª M Žhy1. , L ª 0 cf. 1.1Ž . Ž .Ž .M

by L m L9 and take global sections, we obtain

H 0 M Žhy1. , L m H 0 L m L9Ž . Ž .a 0 Žhy1. , Lª H M m L m L9Ž .

ª H 1 M Žh. , L m L9 ª H 0 M Žhy1. , L m H 1 L m L9 . 5.4.6Ž . Ž . Ž . Ž .

1Ž .Since, by Observation 2.2, H L m L9 s 0, properties 5.4.2 to 5.4.5 areequivalent to the surjectivity of the multiplication map a in the differentcases. First we prove property 5.4.5. Consider the following commutativediagram:

0 Žhy1., L 0 0 0 6 0 Žhy1., L 0 m3Ž . Ž . Ž . Ž . Ž . Ž .H M mH B mH B mH B H M mH Bi i i i

6w1

0 Žhy1., L 0 0

6

Ž . Ž . Ž .H M m B m H B m H Bi i ia

6w2

w30 Žhy1., L m2 0 0 Žhy1., L m36Ž . Ž . Ž .H M m B m H B H M m B .i i i

To show the surjectivity of a it suffices then to show the surjectivity ofw , w , and w . To prove that these three maps are surjective we use1 2 3Theorem 1.3. For example, to see that w is surjective it is enough by1

1Ž Žhy1., L U . 2Ž Žhy1., LTheorem 1.3 to show that H M m B s 0 and H M mi


y2 .B s 0. We argue analogously for the other two maps and thus concludeithat in order to show the surjectivity of a it is enough to check that

5.4.7 H 1 M Žhy1. , L m BU s H 1 M Žhy1. , L s 0Ž . Ž .Ž .i

H 1 M Žhy1. , L m B s 0Ž .iand

5.4.8Ž .U2 Žhy1. , L y2 2 Žhy1. , LH M m B s H M m B s 0Ž . Ž .i i

H 2 M Žhy1. , L s 0.Ž .

Ž .The vanishings in 5.4.7 follow from the assumption that propertiesŽ .5.4.2 and 5.4.3 hold for h y 1. Proving 5.4.8 is not hard. For instance, to

obtain

5.4.8.1 H 2 M Žhy1. , L m By2 s 0Ž . Ž .i

Ž .we consider the following sequence that we obtain from 1.1 :

H 1 M Žhy2. , L m B m BUŽ .j i

ª H 2 M Žhy1. , L m By2 ª H 2 B m BU .Ž .Ž .i j i

Ž .Hence it is clear that in order to show 5.4.8.1 it is enough to check that1Ž Žhy2., L U . 2Ž U .H M m B m B s 0 and that H B m B s 0. Arguing in aj i j i

Ž .similar way for the remaining vanishings in 5.4.8 , we conclude that inŽ .order to prove 5.4.8 it is enough to check

5.4.9Ž .U1 Žhy2. , L 1 Žhy2. , LH M m B m B s H M m B s 0Ž . Ž .j i j

H 1 M Žhy2. , L m B m B s 0Ž .1 2

5.4.10 H 2 B m BU s H 2 B s H 2 B m B s 0.Ž . Ž .Ž . Ž .j i j 1 2

Ž .The vanishings in 5.4.9 follow from the assumption that propertiesŽ .5.4.2 and 5.4.4 hold for h y 2. Statement 5.4.10 follows by hypothesis and

Observation 2.3.The proof of properties 5.4.3 and 5.4.4 is analogous. In fact, notice that

we have implicitly proven both when we showed the surjectivity of w and1w .2

Now we prove property 5.4.2. The argument is similar to the one wehave used to prove 5.4.5 and we will only sketch it here in little detail. To


show the surjectivity of the map a

H 0 M Žhy1. , L m H 0 Bmb1q1 m Bmb2q1Ž . Ž .1 2

ª H 0 M Žhy1. , L m Bmb1q1 m Bmb2q2Ž .1 2

Ž Ž ..cf. 5.4.6 , one can write a diagram similar to the one in the proof of5.4.5. Then it is enough to prove the surjectivity of the following map,

Ž .which is a composition of multiplication maps we assume b G b :2 1

mb q1 mb yb1 2 10 Žhy1. , L 0 0 0H M m H B m H B m H BŽ . Ž . Ž . Ž .1 2 2

6w

0 Žhy1. , L mb q1 mb q21 2H M m B m B .Ž .1 2

We show the surjectivity of the composite map w by showing thesurjectivity of each of its components. The first component is

mb mb yb1 2 10 Žhy1. , L 0 0 0 0 0w x w xŽ . Ž . Ž . Ž . Ž . Ž .H M mH B mH B m H B mH B m H B1 2 1 2 2

6

w1

mb mb yb1 2 10 Žhy1. , L 0 0 0 0w x w xŽ . Ž . Ž . Ž .H M m B m H B m H B m H B m H B .Ž .1 2 1 2 2

Hence by Theorem 1.3 it is enough to check the vanishings of the1Ž Žhy1., L U . 2Ž Žhy1., L y2 .cohomology groups H M m B and H M m B . For1 1

the surjectivity of the second component

mb mb yb1 2 10 Žhy1. , L 0 0 0 0w x w xŽ . Ž . Ž . Ž .H M m B m H B m H B m H B m H BŽ .1 2 1 2 2

6

w2

mb mb yb1 2 10 Žhy1. , L 0 0 0w x w xŽ . Ž . Ž .H M m B m B m H B m H B m H B ,Ž .1 2 1 2 2

again by Theorem 1.3 it is enough to check the vanishings of the groups1Ž Žhy1., L U . 2Ž Žhy1., L y2 .H M m B m B and H M m B m B . We use the1 2 1 2

same argument for the remaining components of w and conclude that inorder to prove the surjectivity of w, it suffices to check

Ž . 1Ž Žhy1., L mb1 mb 2 .5.4.11 H M m B m B s 0, for all b and b satisfy-1 2 1 2ing one of the following conditions:

Ž .5.4.11.1 y1 F b F b y 1 and b s b q 1,1 1 2 1Ž .5.4.11.2 1 F b F b q 1 and b s b y 2,1 1 2 1Ž .5.4.11.3 b s b q 1 and b F b F b y 1;1 1 1 2 2


and

Ž . 2Ž Žhy1., L mg 1 mg 2 .5.4.12 H M m B m B s 0, for all g and g satisfy-1 2 1 2ing one of the following conditions:

Ž .5.4.12.1 y2 F g F b y 2 and g s g q 2,1 1 2 1Ž .5.4.12.2 1 F g F b q 1 and g s g y 3,1 1 2 1Ž .5.4.12.3 g s b q 1 and b y 1 F g F b y 2.1 1 1 2 2

Ž . 1Ž Žhy1., L U .The vanishings in 5.4.11 , except the vanishings of H M m B11Ž Žhy1., L U .and H M m B m B , follow from the assumption that property1 2

1Ž Žhy1., L U .5.4.2 holds for h y 1. The vanishing of H M m B follows from1the assumption that property 5.4.3 holds for h y 1. The vanishing of

1Ž Žhy1., L U .H M m B m B follows from the assumption that property 5.4.41 2Ž .holds for h y 1. To prove the vanishings in 5.4.12 we consider the

Ž .following sequence that we obtain from 1.1 :

H 1 M Žhy2. , L m BmŽg 1q1 . m BmŽg 2q1 .Ž .1 2

ª H 2 M Žhy1. , L m Bmg 1 m Bmg 2 ª H 2 BmŽg 1q1 . m BmŽg 2q1 . .Ž . Ž .1 2 1 2

Hence it is enough to show that these cohomology groups vanish:

Ž . 1Ž Žhy2., L mŽg 1q1 . mŽg 2q1 ..5.4.13 H M m B m B s 0 and1 2

Ž . 2Ž mŽg 1q1 . mŽg 2q1 ..5.4.14 H B m B s 0,1 2

Ž .for all g and g satisfying one of the conditions from 5.4.12.1 to1 2Ž .5.4.12.3 .

Ž . 1Ž Žhy2., LStatement 5.4.13 , except for the vanishings of H M m B m2y1 . 1Ž Žhy2., L m2 y1.B and H M m B m B , follows from the assumption that1 1 2

1Ž Žhy2., L y1.property 5.4.2 holds for h y 2. The vanishing of H M m B m B2 1follows from the assumption that property 5.4.4 holds for h y 2. The

1Ž Žhy2., L m2 y1.vanishing of H M m B m B follows from the assumption that1 2Ž .property 5.4.5 holds for h y 2. All the vanishings in 5.4.14 follow by

hypothesis and Observation 2.3.Finally we prove property 5.4.1. By Observation 5.3, it is enough to show

that the map

H 0 M Žh. , L m Lmn m H 0 L ª H 0 M Žh. , L m Lmnq1 5.4.15Ž . Ž . Ž . Ž .surjects for all n G 0. For that it suffices to prove the surjectivity of themap

H 0 M Žh. , L m Lmn m H 0 B m H 0 B ª H 0 M Žh. , L m Lmnq1Ž . Ž . Ž . Ž .1 2

for all n G 0. Using Theorem 1.3, it is enough to check

5.4.16 H 1 M Žh. , L m Bmny1 m Bmn s 0Ž . Ž .1 2

H 1 M Žh. , L m Bmnq1 m Bmny1 s 0Ž .1 2


and

5.4.17 H 2 M Žh. , L m Bmny2 m Bmn s 0Ž . Ž .1 2

H 2 M Žh. , L m Bmnq1 m Bmny2 s 0.Ž .1 2

Ž .The vanishings in 5.4.16 follow from the fact, which we have justŽ .proved, that properties 5.4.2 to 5.4.4 hold for h. To prove 5.4.17 , again by

Ž .1.1 , it is enough to show that

5.4.18 H 1 M Žhy1. , L m Bmny1 m Bmnq1 s 0Ž . Ž .1 2

H 1 M Žhy1. , L m Bmnq2 m Bmny1 s 0Ž .1 2

and

5.4.19Ž .2 mny1 mnq1 2 mnq2 mny1H B m B s H B m B s 0.Ž . Ž .1 2 1 2

Ž .The vanishings in 5.4.18 follow from the assumption that propertiesŽ .5.4.2, 5.4.4, and 5.4.5 hold for h y 1 and 5.4.19 follows by hypothesis and

Observation 2.3.1Ž Žh., L ms.In particular, it follows from property 5.4.2 that H M m L s 0

for all h, s G 0. Thus, as a consequence of Lemma 5.2, the coordinate ringŽ .R L is a Koszul k-algebra.

Ž .Note that if h s 1 and n s 1, the multiplication map 5.4.15 is actuallyŽ . 1Ž .the same as 2.7.1 . Moreover, the fact that H M m L s 0 is a specialL

case of 5.4.2. Hence, in the course of proving Theorem 5.4, we havereproved Proposition 2.1 and therefore Theorem 5.4 may be seen as ageneralization of the cited proposition.

Even though the above theorem is stated for surfaces with p s 0, thegŽ .same proof or indeed a simpler one works for curves. Thus we obtain the

following

THEOREM 5.5. Let C be a cur e, let B and let B be two nontri ial1 21Ž . 1Ž . Ž .base-point-free line bundles on C. If H B s H B s 0, then R L is1 2

Koszul.

Proof. The only properties of surfaces with p s 0 that we use in theg2Ž .proof of Theorem 5.4 are the fact that H OO s 0, and Observations 2.2X

and 2.3. Observation 2.2 is obviously still true if X is a curve. Observation2Ž . 2Ž U . 2Ž U .2.3 and the fact that H OO s H B m B s H B m B s 0 areX 1 2 2 1

trivially true for curves; hence the theorem follows from the proof ofTheorem 5.4.


Theorem 5.5 yields as a corollary the following result by David ButlerŽ w x.see also Po :

w xCOROLLARY 5.6 B, Theorem 3 . Let C be a cur e and let L be a lineŽ . Ž .bundle on C. If deg L G 2 g q 2, then R L is Koszul.

Ž .Proof. If deg L G 2 g q 2, then L can be written as a tensor productof two general line bundles of degree g q 1. Such line bundles arebase-point-free and nonspecial.

Theorem 5.4 yields these three results:

COROLLARY 5.7. Let X be an Enriques surface oër an algebraic closedfield of characteristic 0 and let B be an ample line bundle on X without base

Ž m2 .points. Then R B is Koszul.

Proof. The proof is analogous to the proof of Corollary 2.8.

THEOREM 5.8. Let X be an elliptic ruled surface. Let L be a line bundleon X numerically equi alent to aC q bf. If e s y1 and a G 1, a q b G 4,0

Ž .and a q 2b G 4, then R L is Koszul. If e G 0 and a G 1 and b y ae G 4,Ž .then R L is Koszul.

Proof. The proof is analogous to the proof of the first part of Theorem4.2.

Ž . Ž .5.9 It is well known that for an ample line bundle L, the fact of R LŽbeing Koszul implies formally the property of being normally presented cf.

w x.BF, 1.16 . Therefore Theorem 5.5 gives a different proof of Fujita andSt.-Donat’s theorem, Corollary 5.7 provides another proof of Corollary 2.8,and Theorem 5.8 provides another proof of the first part of Theorem 4.2.These proofs are less elementary, but in the case of Theorem 4.2, we havethe advantage of working also in characteristic 2.

Ž .If we assume that char k / 2, it follows from Proposition 4.2 that theŽ .property of R L being a Koszul algebra is characterized by the numerical

conditions in the statement of Theorem 5.8. We can restate this as we didin the case of Theorem 4.2:

THEOREM 5.9. Let X be as aboë and let L be a line bundle on X.Ž . Ž .Assume that char k / 2. Then R L is a Koszul algebra iff it is ample and

can be written as the tensor product of two line bundles B and B such that1 2eëry line bundle numerically equi alent to any of them is base-point-free.

Ž .5.9.1 Having in account that, on elliptic ruled surfaces, normalŽ .presentation only depends on numerical equivalence cf. Theorem 4.2 ,

Theorem 5.9 can be considered as analogous to Theorem 5.5. IndeedTheorem 5.5 can be rephrased as follows:

If a line bundle L on C is normally presented and so is every lineŽ .bundle numerically equivalent to L, then R L is Koszul.


ACKNOWLEDGMENTS

We would like to thank our advisor David Eisenbud for his encouragement and helpfuladvice. We are also glad to thank Aaron Bertram, Raquel Mallavibarrena, and GiuseppePareschi for helpful discussions.

REFERENCES

w xBF J. Backelin and R. Froberg, Veronese subrings, and rings with linear resolutions,¨Ž .Re¨. Roumaine Math. Pures Appl. 30 1985 , 85]97.

w xB D. Butler, Normal generation of vector bundles over a curve, J. Differential Geom.Ž .39 1994 , 1]34.

w xC G. Castelnuovo, Sui multipli di uni serie di gruppi di punti appatente ad una curvaŽ .algebraica, Rend. Circ. Mat. Palermo 7 1892 , 99]119.

w xF T. Fujita, Defining equations for certain types of polarized varieties, in ‘‘ComplexAnalysis and Algebraic Geometry,’’ pp. 165]173, Cambridge Univ. Press, Cam-bridge, UK, 1977.

w xGP F. J. Gallego and B. P. Purnaprajna, Higher syzygies of elliptic ruled surfaces, J.Ž .Algebra 186 1996 .

w xG M. Green, Koszul cohomology and the geometry of projective varieties, J. Differen-Ž .tial Geom. 19 1984 , 125]171.

w xGL M. Green and R. Lazarsfeld, Some results on the syzygies of finite sets andŽ .algebraic curves, Compositio Math. 67 1989 , 301]314.

w xH R. Hartshorne, ‘‘Algebraic Geometry,’’ Springer-Verlag, Berlin, 1977.w xHo1 Y. Homma, Projective normality and the defining equations of ample invertible

sheaves on elliptic ruled surfaces with e G 0, Natur. Sci. Rep. Ochanomizu Uni . 31Ž .1980 , 61]73.

w xHo2 Y. Homma, Projective normality and the defining equations of an elliptic ruledŽ .surface with negative invariant, Natur. Sci. Rep. Ochanomizu Uni . 33 1982 , 17]26.

w xMi Y. Miyaoka, The Chern class and Kodaira dimension of a minimal variety, in‘‘Algebraic Geometry}Sendai 1985,’’ Advanced Studies in Pure Math., Vol. 10, pp.449]476, North-Holland, Amsterdam.

w xMu D. Mumford, Varieties defined by quadratic equations, in ‘‘Corso CIME in Ques-tions on Algebraic Varieties, Rome, 1970,’’ pp. 30]100.

w xP G. Pareschi, Koszul algebras associated to adjunction bundles, J. Algebra 157Ž .1993 , 161]169.

w xPo A. Polishchuk, On the Koszul property of the homogeneous coordinate ring of aŽ .curve, J. Algebra 178 1995 , 122]135.

w xR I. Reider, Vector bundles of rank 2 and linear systems on an algebraic surface,( ) Ž .Ann. of Math. 2 127 1988 , 309]316.

w xS-D B. St.-Donat, Sur les equations definissant une courbe algebrique, C.R. Acad. Sci.´ ´Ž .Paris Ser. I Math. 274 1972 , 324]327.´

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