Seshadri constants and the geometry of surfaces

Transcription

1 J. reine angew. Math. 564 (2003), Journal für die reine und angewandte Mathematik ( Walter de Gruyter Berlin New York 2003 Seshadri constants and the geometry of surfaces By Michael Nakamaye*) at Albuquerque 0. Introduction In this paper we study how locally defined numerical invariants can carry global geometric information about algebraic surfaces. The local invariants which we will study are Seshadri constants which measure the positivity of a line bundle near a point. Definition 1. bundle on X. Then Suppose X is a smooth projective variety, x A X, and A an ample line c 1 ðaþ X C eðx; AÞ ¼ inf C C x mult x ðcþ ; here the infimum runs over all integral curves C H X passing through x. This numerical definition is equivalent to a more intuitive geometric definition. In particular, eðx; AÞ is the supremum of all non-negative rational numbers a such that the linear series jnaj separates na-jets at x for n su ciently large and divisible. Note that if L is a nef line bundle on X then Definition 1 still makes sense and eðx; LÞ is defined accordingly. When L is nef but not ample, however, eðx; LÞ can take the value zero at some or all points of X. Given the local nature of this definition, it is surprising that Seshadri constants can carry global information about a variety X. Indeed, at special points Seshadri constants rarely carry interesting global information: for example, suppose p : X! Y is the blow-up of a smooth surface at a point with exceptional divisor E. Then for any ample line bundle A on Y and any su ciently large positive integer n, e x; p ðnaþð EÞ ¼ 1 at all points x A E. At a very general point, however, this type of behavior cannot occur as the numerical equivalence class of any cycle through a very general point moves to cover the entire variety: recall that a point x A X is called very general if x belongs to the complement of a countable union of closed, proper subvarieties of X. Thus at a very general point one can hope for Seshadri constants to carry some global information. *) Partially supported by NSF Grant DMS

2 206 Nakamaye, Seshadri constants In particular, one global property which might be captured via Seshadri constants is whether or not X admits a dominant morphism to a variety of smaller dimension. Indeed suppose p : X! Y is a surjective map of projective varieties with dimðy Þ f 1 and let A be an ample line bundle on Y. IfB is an ample line bundle on X and h A X a general point then e h; p ðnaþþeb f 1 for e f 1 as one can see by intersecting with a curve C contained in the fibre p 1 pðhþ. Note that the line bundles p ðnaþþeb form an unbounded family as n grows. We will show that the converse is also true in case X is a surface: namely, if there exists an unbounded family of ample line bundles whose Seshadri constant at a very general point is bounded, then X admits a dominant morphism to a curve. In order to state our result formally, we require a few definitions. We let SðXÞ ¼fA A NEðXÞ n Q ample j eðh; AÞ e 1g; where h A X is a very general point and NEðXÞ denotes divisor classes modulo numerical equivalence. For an ample Q-divisor A we let mðaþ ¼ sup fmult h ðdþjd A DivðXÞ n Q e ectiveg: D1A To see that the invariant mðaþ is well-defined, choose a very ample divisor B and let d ¼ dimðxþ. For any Q-divisor D 1 A, we can choose general divisors B 1 ;...; B d 1 A jbj, containing h, so that the intersection supportðdþ X B 1 X XB d 1 is proper. It then follows that mult h ðdþ e c 1 ðaþ X c 1 ðbþ d 1 establishing that the supremum exists in the definition of mðaþ. Theorem 2. Let X be a smooth projective surface and define mðx Þ¼ sup mðaþ: A A SðXÞ Then mðxþ > 2 if and only if X admits a surjective morphism f : X! C to a curve C. Note in particular that Theorem 2 implies that mðxþ > 2 if and only if mðxþ ¼y. Indeed, whenever X fibres over a variety Y it is easy, as above, to produce a family of line bundles fa i g with bounded Seshadri constant at a very general point but where mða i Þ!y as i! y. One interesting corollary of Theorem 2 is the following: Corollary 3. Suppose A is an ample line bundle on a surface X satisfying

3 Nakamaye, Seshadri constants 207 qffiffiffiffiffiffiffiffiffiffiffiffiffi c 1 ðaþ 2 p > ffiffiffi 3 eðh; AÞ: Then there exists a non-trivial fibration p : X! C such that the general fibre F h is Seshadriexceptional for A. Note that an inequality like that of Corollary 3 holds for any ample line bundle on a smooth surface, namely qffiffiffiffiffiffiffiffiffiffiffiffiffi eðh; AÞ e c 1 ðaþ 2 : With somewhat more delicate analysis one can also establish the following generalization of a result proven for abelian varieties in [N]: Theorem 4. Suppose A is an ample line bundle on X satisfying A 2 > 1 and eðh; AÞ¼1. Then there exists a non-trivial fibration p : X! C such that the general fibre F h is smooth and Seshadri-exceptional for A. The methods we use to prove Theorems 2 and 4 are very close to those of [EL], namely we exploit the fact that any curve C H X passing through a very general point moves in a non-trivial family. Theorem 4 also used the Kodaira-Spencer construction of [EL]. Note that by work of Oguiso [O], Seshadri constants can not increase under specialization and thus if h is a very general point then eðh; AÞ is the maximal value achieved by the Seshadri constants eðx; AÞ as x varies over all points of X. Hwang and Keum [HK] study this maximal value, which they denote by mðaþ, and, following the methodology of [EKL], prove that when mðaþ is too small relative to the volume of A then X admits a fibration like that produced in Corollary 3, Theorem 4, and Corollary 6. The work of Hwang and Keum is very closely related to the present paper, analysing the higher dimensional case and providing several interesting examples. Acknowledgments. It is a real pleasure to thank the referee for numerous helpful comments and suggestions which greatly improved the quality of exposition in this paper. The referee also kindly shared with me the very interesting paper of Hwang and Keum [HK] which has not, to my knowledge, been published. 1. Main results Proof of Theorem 2. One direction of Theorem 2 is trivial. Namely, suppose that there exists a surjective map f : X! C to some curve C. Let h A X be a very general point and let P ¼ fðhþ. We denote by F P the fibre of X over P. Then F P is nef and eðh; F P Þ¼0. Thus, if A is any ample line bundle on X we have, for a su ciently small eðh; aa þ nf P Þ e 1; En g 0 and this is clearly a family S of divisors for which mðsþ is unbounded. For the other direction of Theorem 2, we require the notion of a Seshadri exceptional subvariety.

4 208 Nakamaye, Seshadri constants Definition 5. Suppose A is an ample line bundle on a variety X and x A X. An irreducible subvariety V containing x is called Seshadri exceptional at x relative to A if eðx; AÞ ¼ deg AðVÞ mult x ðvþ : A result of Campana and Peternell [CP] asserts that therepalways exists a Seshadri exceptional subvariety. In particular, on a surface X, ifeðh; AÞ < ffiffiffiffiffiffi A 2 then X itself can not be Seshadri exceptional and thus there must exist a Seshadri exceptional curve. Suppose mðxþ > 2. Then by definition we can find an ample divisor A A SðXÞ and an a > 0 with ð2:1þ mðaþ > 2 þ a: Since eðh; AÞ e 1 it follows that X is not Seshadri exceptional for A and thus there is a Seshadri exceptional curve C satisfying ð2:2þ A C mult h ðcþ e 1: Let m ¼ mult h ðcþ. Applying [EL], Corollary 1.2, gives ð2:3þ C 2 f mðm 1Þ: We begin by showing ð2:4þ C is smooth at h and C 2 ¼ 0: Suppose first that m f 2. We claim that ð2:5þ eðh; CÞ f m 1: To prove (2.5), note first that C 2 =mult h ðcþ f m 1 by (2.3) and for any curve C 0 3 C through h one has C C 0 =mult h ðc 0 Þ f m. Thus we have established (2.5). Note that since C 2 > 0 and C is irreducible, it follows that C is big and nef. Consequently, there exists an e ective divisor D so that C dd is ample for any d > 0. Since h is general, we can assume that h is not contained in D. By (2.5), given e > 0 for all d su ciently small eðh; C ddþ > m 1 e: So if p : Y! X denotes the blow-up of X at h with exceptional divisor E then p ðc ddþ ðm 1 eþe is ample. Thus for k su ciently large and divisible the linear series jkðc ddþnih kðm 1 eþ j has an isolated base point at h and its general member is irreducible: here I h H O X is the ideal sheaf of the point h. Hence for any e > 0 there exists a non-trivial family fd t g of Q-divisors, irreducible in a neighborhood of h, numerically equivalent to C with

5 Nakamaye, Seshadri constants 209 ð2:6þ mult h ðd t Þ f m 1 e: By (2.1), there is a Q-divisor F, numerically equivalent to A, with mult h ðfþ > 2 þ a. Choose a divisor D t in the family above which meets F properly, except possibly along D. By (2.2) On the other hand, using (2.6) we have A C e m: A C ¼ F D t f mult h ðfþ mult h ðd t ÞþdD F f ð2 þ aþðm 1 eþ OðdÞ: For d and e su ciently small, this is a contradiction unless m ¼ 1. We conclude that the first part of (2.4) holds, namely that C is smooth at h. To prove the second part of (2.4), note first that C 2 f 0 since C passes through a very general point of X. Moreover, since C is irreducible C must be nef so that eðh; CÞ is well-defined. We assume that C 2 f 1 and derive a contradiction. Since C is irreducible we have Since C is smooth at h we also have C C 0 mult h ðc 0 Þ f 1; EC 0 3 C: C C mult h ðcþ f 1: Thus eðh; CÞ f 1. In particular, for e > 0 we see as in (2.6) above that there is a non-trivial family of divisors fd t g, numerically equivalent to C and locally irreducible at h, with mult h ðd t Þ > 1 e. Suppose as above that F is a Q-divisor numerically equivalent to A with mult h ðfþ > 2 þ a. Arguing exactly as above we see that F cannot intersect D t properly, a contradiction since the family of divisors D t has no base points other than h. Thus we have established the second half of (2.4), namely that C 2 ¼ 0, and we now have a curve which is a candidate for giving a fibration of X. Since the curve C passes through a very general point h A X we can apply [Ko], Proposition 2.5 to obtain a non-trivial family of curves in X, parametrized by a variety S, which are numerically equivalent to C. More precisely, there is a scheme U with a flat map p : U! S to a reduced scheme S of finite type and a map f : U! X such that f : p 1 ðsþ!x is birational for all s A S and C ¼ f p 1 ðtþ for some t A S. We replace S with a suitable smooth a ne curve T and U with p 1 ðtþ so that the new family is also non-trivial. Consider the graph S H U T of p and let S 0 H X T be the image of S via the morphism f id. Let p 2 : S 0! T denote the projection to the second factor. Shrinking T if necessary, we can assume that p 1 2 ðtþ is a curve for all t A T. Then for x A T the divisors C x ¼ p2 ðxþ are algebraically equivalent and satisfy C x 2 ¼ 0 since each curve C x is numerically equivalent to C.

6 210 Nakamaye, Seshadri constants Choose a general point x A T and consider the map f : T! Pic 0 ðxþ given by fðyþ ¼O X ðc y C x Þ where C x and C y are the curves corresponding to x; y A T. The map f naturally induces a map f n : T n! Pic 0 ðxþ where T n is the n th Cartesian product of T and for n su ciently large f n is not injective. This means that for some m e n there are points fp i g m and fq ig m, mutually distinct, satisfying P m ðc Pi C Qi Þ linearly equivalent to 0: Consider the linear system P m C Pi : This system contains the two e ective divisors, Pm C Pi and Pm C Qi. By hypothesis, these two divisors are disjoint and thus the linear series P m C Pi is base point free. In particular, using the pencil of divisors spanned by Pm C Pi and Pm C Qi gives a map c : X! P 1 which contracts C x for all x A T since C Pi C x ¼ 0. On the other hand, the map is P m surjective because the corresponding linear series has projective dimension one. This completes the proof of Theorem 2. Note that we have the following immediate corollary to Theorem 2: Corollary 6. Suppose X is a smooth projective surface admitting no non-trivial fibration over a curve. Then pffiffiffiffiffiffi e 2 eðh; AÞ A 2 for all ample line bundles A. p ffiffiffiffiffiffi Proof of Corollary 6. Suppose that A 2 > 2 eðh; AÞ. Then it follows that there is a Q-divisor D numerically equivalent to A such that mult h ðdþ > 2eðh; AÞ. Arguing as in the proof of Theorem 2 shows that if C is Seshadri exceptional for A at h then C 2 ¼ 0 and one obtains a fibration p ffiffiffiffiffiffi of X over a curve. Corollary 6 can also be restated as follows: if A is ample on X and A 2 > 2eðh; AÞ then X fibres over a curve with general fibre Seshadri exceptional for A. This is slightly weaker than Corollary 3 which we prove now. Proof of Corollary 3. According to the proof of Theorem 2, it su ces to produce a Q-divisor D numerically equivalent to A with mult h ðdþ > 2eðh; AÞ. However, using [EKL], Proposition 2.3, we see that for any n > 0 a divisor D A jnaj with mult h ðdþ > n eðh; AÞþa has multiplicity at least na along C h. Note that it is critical for

7 Nakamaye, Seshadri constants 211 this result that the point h be very general. In particular, the cost of imposing order of vanishing n eðh; AÞþa at h is asymptotically at most! eðh; AÞ 2 þ aeðh; AÞ n 2 ; 2 eðh; AÞ2 the maximum cost coming of course when C h is smooth. Here n 2 represents the 2 asymptotic cost of imposing multiplicity neðh; AÞ at h and aeðh; AÞn 2 is the maximal cost of raising the multiplicity by an additional na. Thus one can always obtain a Q-divisor numerically equivalent to A with multiplicity at h arbitrarily close to eðh; AÞþ A2 eðh; AÞ 2 : 2eðh; AÞ In particular, the hypothesis of Corollary 3 is satisfied as soon as p ffiffiffiffiffiffi pffiffiffi A 2 > 3 eðh; AÞ: Proof of Theorem 4. We begin with some concrete examples of Corollary 3. Suppose A 2 f 4 and eðh; AÞ ¼1. Then by Corollary 3 the Seshadri exceptional curve of A at h must give a fibration of X over a curve. If A 2 ¼ 3 and eðh; AÞ ¼1 the argument for the case where A 2 f 4 fails if C h is smooth. In order to eliminate the possibility that C h is smooth we consider Ch 2.IfC h 2 ¼ 0 then we obtain the desired fibration. If C h 2 > 0 then eðh; C hþ f 1 and we find a contradiction arguing as in the second half of (2.4). We now consider the special case where A 2 ¼ 2. If C h is a Seshadri exceptional curve at h, then one readily establishes that there are only three possibilities. First C h is smooth with Ch 2 ¼ 0 in which case C h gives a fibration of X over a curve. Second, one could have Ch 2 f 1 and C h smooth at h. This would imply that A C h ¼ 1 which contradicts the Hodge index theorem: p A C f ffiffiffiffiffiffi p ffiffiffiffiffiffi A 2 C 2 : Finally, one could have C 2 h ¼ 2 and mult hðcþ ¼2: all other possibilities are eliminated as above using the Hodge index formula and (2.3). This last case is of course possible provided one allows C h to be reducible, namely one can take X ¼ C 1 C 2, the product of two curves and A ¼ F 1 þ F 2, the sum of the two fibres through h. In this case, both F 1 and F 2 are Seshadri exceptional and the divisor A satisfies A 2 ¼ 2 and mult h ðaþ ¼2. We would now like to establish that this is essentially the only such possibility. In particular, we will show that C h is reducible and thus both components of C h are smooth and give fibrations. We will assume that C h is irreducible for very general h A X and derive a contradiction. We first claim that for a very general point h we have the numerical equivalence ð4:1þ A 1 C h : Note that A C h ¼ 2 since mult h ðc h Þ¼2 and C h is Seshadri exceptional for A at h. Thus

8 212 Nakamaye, Seshadri constants A ða C h Þ¼0: One checks that ða C h Þ 2 ¼ 0 and thus the Hodge index theorem implies that A and C are numerically equivalent, establishing (4.1). Next we claim that the curve C h with mult h ðc h Þ¼2 and A C h ¼ 2 is unique. Suppose that there were two such curves C 1 and C 2. From the previous paragraph, we have that C 1 1 C 2 1 A. Thus C 1 C 2 ¼ 2 but since both C 1 and C 2 are singular at h this is only possible if C 1 ¼ C 2. Choose a positive number m so that ma is very ample and consider the corresponding Chow variety V parametrizing curves C H X of degree 2m relative to A. Note that any curve C with ma C ¼ 2m which is singular must either be one of the Seshadri exceptional curves C h or must have its singular point in a closed proper subvariety of X. Thus we can find a closed subvariety W H V parametrizing curves whose general member is one of the curves C h. Let S H W be an open subset parametrizing a family of the curves fc h g. We obtain an embedding S,! X since the curve C h singular at h is unique. Consider the universal curve D H X S defined by the property that D X X h ¼ C h h for all h A S. We let F denote a local equation defining D. Note that the divisor D is singular along the diagonal D by Bertini s theorem applied to the map p 2 : D! S. Suppose x 1 ; x 2 are local coordinates on X and t 1 ; t 2 the corresponding coordinates on S. We claim that for all general h A S there is a di erential operator q=qt, on the S factor, satisfying ð4:2þ q qt F j C h ¼ 0; where ðq=qtþf j C h locally represents the Kodaira-Spencer class along a one parameter family with tangent direction t at h. Let x A C h be a general point and choose q=qt so that q FðxÞ ¼0: qt this is possible since we have a two parameter family of partial derivatives in the coordinates on S. The Kodaira-Spencer construction in [EL], Corollary 1.2, would then produce a section of H 0 ðc; NÞ, where N denotes the normal bundle of C in X, vanishing at h and at x. Since mult h ðc h Þ¼2 this would imply that C 2 f 3 unless q qt F j C h ¼ 0, establishing (4.2). If A denotes an arc in S through h with tangent direction t then the family of curves p 1 2 ðaþ has a moving singular point and so as in [EL], 2, this implies that the corresponding Kodaira-Spencer class is non-trivial, a contradiction. We conclude that the curves C h must be reducible. The irreducible components of C h give the fibration in question, establishing Theorem 4. The question naturally arises whether or not the fibration in Theorem 2 can be detected in a more intrinsic fashion by studying the e ective cone more closely. In partic-

9 ular, one can hope to detect such a fibration numerically as the fibre F is a nef class satisfying c 1 ðfþ 2 ¼ 0 and, moreover, curves in the equivalence class of F should be Seshadri exceptional for the appropriate choice of an ample line bundle. We see, however, in the following example, that numerical criteria alone are not su cient to identify a fibration. Example 7. Suppose E is an elliptic curve and consider X ¼ E E. Let F 1 ; F 2 be fibres of the first and second projections respectively and let D be the diagonal. Consider the divisor p D ¼ ffiffiffi 2 F1 þ ffiffiffi pffiffiffi p 6 3 F2 pffiffiffi pffiffi D: 2 þ 3 Then one checks that D 2 ¼ 0 and D A > 0 for A ¼ F 1 þ F 2. Thus D is a nef divisor on the boundary of the e ective cone. On the other hand the ray corresponding to D in the e ective cone of X can contain no points corresponding to integral divisor classes on X: to see this, note that D F 1 D D ¼ 3ð pffiffiffi pffiffiffi 3 2 Þ pffiffiffi pffiffiffi 3 þ 2 is irrational. Thus the divisor D is not associated to a surjective map f : X! C. A more natural example than Example 7 was given by Mumford (see [H]). In particular, Mumford constructs a surface X and a divisor D with D C > 0 for all irreducible curves C H X but where no multiple of D is e ective. In particular D is nef and satisfies c 1 ðdþ 2 ¼ 0 but D is not associated to a fibration of X. We also have the following phenomenon where a sequence of fibrations, suitably normalized, can actually converge to a distinct fibration: Example 8. Suppose again X ¼ E E for an elliptic curve E and for a positive integer m consider the morphism f m : E E! E; f m ðx; yþ ¼mx y: Let P A E be a point and let F m ¼ f 1 m ðpþ. Then the divisors F m are all on the boundary of the ample cone and F m F converges in NEðXÞ to the class, F a fibre of the projection jf m j jfj to the first factor. Thus a real nef class x with x 2 ¼ 0 does not necessarily carry specific geometric information about morphisms from X to a curve. One still has, however, the following interesting question. Question 9. Suppose X is a smooth projective surface and suppose that there exists a non-zero nef real class x with x 2 ¼ 0. Does X necessarily admit a surjective morphism f : X! C to a curve C? Question 9 naturally leads to the following Nakamaye, Seshadri constants 213

10 214 Nakamaye, Seshadri constants Question 10. Suppose X is a smooth projective surface and A an ample line bundle on X with A 2 ¼ 1. If h A X and p : Y! X the blow-up of X at h then does Y admit a non-trivial fibration over a curve C whose general fibre is numerically equivalent to a Seshadri exceptional curve of A? In particular, with the hypotheses of Question 10, by [EL] we know that eðh; AÞ f 1 and hence eðh; AÞ ¼1. Thus the line bundle L ¼ p Að EÞ is nef with L 2 ¼ 0. On the other hand, L is clearly not numerically equivalent to zero so Question 9 would provide the desired fibration. References [B] T. Bauer, Seshadri constants on algebraic surfaces, Math. Ann. 313 (1999), [CP] F. Campana and T. Peternell, Algebraicity of the ample cone of projective varieties, J. reine angew. Math. 404 (1990), [EL] L. Ein, R. Lazarsfeld, Seshadri constants on smooth surfaces, Astérisque 218 (1993), [EKL] L. Ein, O. Kuchle, R. Lazarsfeld, Local positivity of ample line bundles, J. Di. Geom. 42 (1995), [H] R. Hartshorne, Ample subvarieties of algebraic varieties, Lect. Notes Math. 156, Springer-Verlag, [HK] J.-M. Hwang and J.-H. Keum, Seshadri exceptional foliations, unpublished manuscript, [Ko] J. Kollár, Shafarevich Maps and Automorphic Forms, Princeton University Press, [N] M. Nakamaye, Seshadri constants on abelian varieties, Amer. J. Math. 118 (1996), [O] K. Oguiso, Seshadri constants in a family of surfaces, Math. Ann. 323 (2002), Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico nakamaye@math.unm.edu Eingegangen 17. November 2002, in revidierter Fassung 17. Januar 2003