Mathematische Zeitschrift

Math. Z. (212 27:871 887 DOI 1.17/s29-1-83-2 Mathematische Zeitschrift Modular properties of nodal curves on K3 surfaces Mihai Halic Received: 25 April 21 / Accepted: 28 November 21 / Published online: 11 January 211 Springer-Verlag 211 Abstract In this article we are going to address the following issues: (1 the first is a rigidity property for pairs (S, C consisting of a general projective K 3 surface S,andacurve C obtained as the normalization of a nodal, hyperplane section Ĉ S. We prove that a non-trivial deformation of a pair (S, C induces a non-trivial deformation of C; (2the second question concerns the Wahl map of curves C obtained as above. We prove that the Wahl map of the normalization of a nodal curve contained in a general projective K 3 surface is non-surjective. In both cases, we impose upper bounds on the number of nodes of Ĉ. Keywords K 3 surfaces Nodal curves Deformations Wahl map Mathematics Subject Classification (2 14H1 14J28 14D15 Introduction Curves on K 3 surfaces have been investigated from various points of view, and there is an extensive literature concerning their properties. Most attention has been payed to the smooth curves. In a series of articles Mukai studied the properties of the morphism { (S, C S is a general projective K 3 surface, C is a smooth hyperplane section of genus g. } μ M g, which associates to a pair (S, C consisting of a general, projective K 3 surface the class of the curve C in the Deligne Mumford space. He proved in [11] that the morphism (μ is finite (μ The research has been funded by the King Fahd University of Petroleum and Minerals project FT16. M. Halic (B Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P. O. Box 546, Dhahran 31261, Saudi Arabia e-mail: mihai.halic@gmail.com

872 M. Halic for g 13, and then he went on proving that it is actually birational (see [12, Theorem 1.2]. These topics are surveyed in [13]and[1]. By contrast nodal curves on K 3 surfaces have received somewhat less attention. The existence of nodal curves on K 3 surfaces has been addressed in [1], and later on generalized in [7]. The deformation theory of nodal curves on K 3 surfaces has been treated in [14], and recently in [8]. The goal of this article is to answers the following questions: (i (ii The first one (see [8, Question 5.7(ii] concerns the finiteness of the forgetful morphism (μ, where one considers instead pairs (S, C, such that C is the normalization of a nodal curve Ĉ S. The second one is to find obstructions for embedding nodal curves into K 3 surfaces. More precisely, a result due to Wahl says that for a smooth curve C laying on a projective K 3 surface, the homomorphism 2 w C : H (C, K C H (C, KC 3 is non-surjective (see [2,15]. Thequestionraisedin[8, Question 5.5] is the following: suppose that C is the normalization of a nodal curve on a projective K 3 surface. Is it true that the homomorphism w C is still non-surjective? For these questions we have the following two answers: Theorem Let n and d be two positive integers (subject to the inequalities below. We define: Then hold: n2 25 if d = 1 and n 5; δ max (n, d = 2n 27 if d = 2 and n 14; 2(n 1(d 1 25 if d 3 and n = 11 or n 13. (i For δ δ max (n, d, the forgetful morphism (S, A K n, C M n δ, u : C S is a morphism μ (S, C, u s.t. u C S is a reduced, nodal curve with δ nodes M 1+(n 1d 2 δ which belongs to the linear system da. is generically finite onto its image. (ii Suppose (S, A is a polarized K 3 surface, with Pic(S = ZA, A 2 = 2(n 1. Consider a nodal, hyperplane section Ĉ of S of degree d, with δ δ max (n, d nodes. Let C be the normalization of Ĉ. Then the Wahl map of C is not surjective. A remark concerning the upper bound appearing in (ii above: there are few articles discussing the surjectivity properties of the normalization of nodal curves on surfaces. The reference [6] deals with the surjectivity of the Wahl map of plane nodal curves, where the authors impose a similar upper bound on the number of nodes. This article is structured as follows: In the first section we briefly recall basic facts concerning the deformation theory of curves on surfaces, and fix the notations used throughout the article.

Modular properties of nodal curves on K 3 surfaces 873 The second section and the appendix contain the technical results used for addressing the two questions mentioned above. In Proposition 2.1 we give an effective upper bound for the number of nodes of a nodal curve Ĉ S, such that the pull-back of the tangent bundle of S to the normalization of Ĉ is still stable. The appendix contains a diagram chasing which is needed for proving the non-surjectivity of the Wahl map. The third and the fourth sections contain respectively the proof of the first and the second main result. 1 Description of the problem Throughout the article we will work over the field C of complex numbers. Most of the material appearing in this section is contained in the references [1]and[8]. Here we will introduce only those objects, and recall those properties, which are essential for our presentation. Definition 1.1 (i We say that a polarized K 3 surface (S, A is Picard general if Pic(S = ZA, with A X ample. (ii In this case the self-intersection number A 2 = 2(n 1, with n 3, and for generic (S, A the linear system A induces an embedding S P n. We say that a morphism S between π irreducible algebraic varieties is a family of Picard general, polarized K 3 surfaces, if there is a relatively ample line bundle A S such that fibres (S t, A t, t, are Picard general K 3 surfaces. In this case the function t At 2 is constant. Theorem 1.1 (i Let K n be the set of Picard general, polarized K 3 surfaces. Then K n can be endowed with the structure of a smooth stack, whose local charts are given by the local Kuranishi models of its points. (ii For any g 1,letM g be the Deligne-Mumford stack of smooth and irreducible curves of genus g. For d 1 and δ (n 1d 2,wedefine Vn,δ d := (S, A K n, C M n δ, and u : C S is a morphism (S, C, u s.t. u C S is a reduced, nodal curve with δ nodes, which belongs to the linear system da. Then Vn,δ d can be endowed with the structure of an analytic stack, which admits two forgetful morphisms M g(d δ μ Vn,δ d κ K n with g(d := 1 + (n 1d 2. (1.1 (iii There is a non-empty open subset K n K n such that (V d n,δ := κ 1 ( K n is smooth, the projection (V d n,δ K n is submersive, and all the irreducible components of (V d n,δ are 19 + g(d δ dimensional. Note that for d = 1 we recover the situation studied in [8, Sect. 4]: the stack V n,δ introduced in loc.cit., Definition 4.3. ( V 1 n,δ coincides with

874 M. Halic Proof (i The detailed construction can be found in [3, chap. VIII, Sect. 12]. (ii The analytic stack structure is obtained as follows: for a family (S, A of π Picard general K 3 surfaces, Vn,δ d (S is naturally an open subscheme of the Kontsevich Manin space of stable maps M g(d δ (S; β, with suitable β H 2 (S; Z such that π β =. (iii The proof is ad litteram thesameas[8, Proposition 4.8]. According to [7], there is a non-empty open subset K n K n such that for all (S, A K n the linear system da contains irreducible, nodal curves with δ nodes. Consider a point (S, C, u κ 1 ( K n, and denote Ĉ := u C. Then the short exact sequence T S Ĉ :=Ker(ˆλ induces the long exact sequence in cohomology: ι ˆλ T S u (u T S /T } {{ C } = K C H (C, K C H 1 (S, T S Ĉ H 1 (ι H 1 (S, T S } {{ } = k 2 H 1 (ˆλ H 1 (C, K C } {{ } = k H 2 (S, T S Ĉ The cohomology group H 1 S, T S Ĉ is naturally isomorphic to the Zariski tangent space T V d n,δ,(s,c,u, and the homomorphism H 1 (ι can be identified with the differential of κ at (S, C, u (see diagram (1.2 below. Hence: Image ( H 1 (ι T Kn,[S] dim ( Image ( H 1 (ι 19, Image ( H 1 (ι = Ker H 1 (ˆλ dim ( Image ( H 1 (ι 19. We deduce that Image(dκ (S,C,u = Image ( H 1 (ι = T Kn,[S], and H 2 (S, T S Ĉ =. The Zariski tangent space of Vn,δ d is described in [8, Sect. 4]. For a triple (S, C, u V d n,δ, we consider the blow-up S σ S at the δ double points of Ĉ = u(c. Then the morphism u can be lifted to a morphism ũ into S, which is a closed embedding: S ũ u C S. The infinitesimal deformations of (S, C, u are controlled by the (locally free sheaf (σ T S C defined in the diagram below: σ

Modular properties of nodal curves on K 3 surfaces 875 (σ T S ( C (σ T S ( C (1.2 (σ T S C :=Ker(λ σ λ T S N u r T C u T S N u The sheaf N u, defined by the last row of the diagram, is called the normal sheaf to the map u. The Zariski tangent space to Vn,δ d at (S, C, u is isomorphic to H 1 ( S,(σ T S C = H 1 (S, T S Ĉ. After these preparations, we can finally precise the topic of this article. The first issue, related to [8, Question 5.7 (ii], is: Question 1.1 Is the morphism V d enough? n,δ μ M g(d δ (generically finite for g(d δ large The ( second issue, related to [8, Question 5.5], is the following: one can see that dim Vn,δ d < dim M g(d δ for n sufficiently large. Question 1.2 Is there any obstruction for a point C M g(d δ to belong to the image of μ? We will address these questions in Theorems 3.1 and 4.2, respectively. However, we do not know if the bounds in there are optimal or how far are they from being optimal. 2 A vanishing result In this section we prove the main technical ingredient needed for addressing the Question 1.1. Lemma 2.1 Let Ĉ be a reduced, irreducible, nodal curve with δ nodes, and denote by ν : C Ĉ its normalization. Consider a vector bundle E Ĉ, and a quotient line bundle q : ν E Q. Then there is a torsion-free quotient ˆq : E ˆQ of rank one such that Q = ν ˆQ/(ν ˆQ tor, and whose degree satisfies the inequality deg C Q degĉ ˆQ deg C Q + δ. Here we define degĉ ˆQ := χ( ˆQ + p a (Ĉ 1, with p a (Ĉ the arithmetic genus of Ĉ. Proof We denote by ˆx 1,..., ˆx δ the nodes of Ĉ, and by {x 11, x 21 },...,{x 1δ, x 2δ } C their pre-images by ν. For j {1,...,δ} we let q 1 j : E ˆx j = (ν E x1 j Q x1 j = k, q2 j : E ˆx j = (ν E x2 j Q x2 j = k. and consider the set δ := { j {1,...,δ} Ker(q 1 j = Ker(q 2 j }. The construction of the sheaf ˆQ Ĉ is done in two steps, described below:

876 M. Halic Step 1 j δ,thatisker(q 1 j = Ker(q 2 j. In this case there is an isomorphism τ j such that the diagram below commutes: E ˆx j q 1 j Q x1 j τ j E ˆx j q 2 j Q x2 j After identifying Q x1 j and Q x2 j using τ j, we get an invertible sheaf on the nodal curve C/{x 1 j = x 2 j } whose degree equals deg C Q. We repeat this procedure until we eliminate all the pairs of points {x 1 j, x 2 j } with j δ. The result will be a nodal curve C, a naturally induced morphism ν : C Ĉ, and an invertible sheaf Q C of degree deg C Q,which is still quotient of (ν E. Step 2 j δ,thatisker(q 1 j = Ker(q 2 j. The homomorphism (ν E Q is surjective, and therefore ν q : E ν O C ˆQ := (ν Q is the same. Then ˆQ Ĉ is a torsion free sheaf of rank one (since Q is so, and degĉ ˆQ = deg C Q + δ. We claim that the composed homomorphism E OĈ E ν O C = ν (u E ˆq ˆQ. is still surjective. Indeed, let ˆx ( {ˆx j j δ } be a double point of Ĉ. The (analytic germ of (Ĉ, ˆx is isomorphic to Spec k[[u,v]] uv.sincee,(, Ĉ is locally free, we have E ˆx = k[[u,v]] r, uv and this isomorphism depends on the choice of a frame at ˆx. By hypothesis holds Ker(q 1 j = Ker(q 2 j, and therefore Ker q 1 : E ˆx Q x1 + Ker q2 : E ˆx Q x2 = E ˆx. Hence we may choose the frame in such a way that (q 1, q 2 : E ˆx Q x1 Q x2 corresponds to the homomorphism k[[u,v]] r 2 ( k[[u,v]] k[[u,v]] pr 2 k[[u,v]] k[[u,v]] h k[[u,v]] k[[u,v]]. uv uv uv uv uv v u } {{ } =E ˆx With respect to these choices the homomorphism E ˆx k[[u,v]] r and k[[u,v]] 2 k[[u,v]] k[[u,v]] k[[u,v]] uv k[[u,v]] uv pr 2 k[[u,v]] 2 ı k[[u,v]] 2 k[[u,v]] k[[u,v]] uv ˆq ˆx ˆQ ˆx equals ν q ı pr 2 : forgets the first r 2 components, ( k[[u,v]] k[[u,v]] ν q k[[u]] k[[v]], where k[[u,v]] v u ı( f, g = ( f (u, + f (,v,g(u, + g(,v and (ν q (h 1 (u + k 1 (v, h 2 (u + k 2 (v = h 1 (u + k 2 (v. Our claim becomes now clear.

Modular properties of nodal curves on K 3 surfaces 877 The proposition below is the main result of this section. It is an application of Bogomolov s effective restriction theorem for stable vector bundles over surfaces (see [9, Sect. 7.3]. Proposition 2.1 (a Let (S, A be a Picard general K 3 surface, with A 2 = 2(n 1, and let E S be a stable vector bundle of rank r 2, and c 1 (E =. LetĈ j S be a reduced and irreducible, nodal curve having δ double points, with Ĉ da. We denote by C ν u:= j ν Ĉ its normalization, and by C S the composed morphism. Suppose that one of the conditions below are satisfied: (i d = 1 or r = d = 2, and δ c 2 (E, or (r 1(n 1d 2 1 r (ii r 2 and d 2, with (r, d = (2, 2, and δ 2(n 1d c 2 (E Then H (C, u E =. r(n 1+1 r 1. (b Assume that the rank of E is r = 2.Thenu E C is a stable vector bundle too. s Proof (a Suppose that there is a non-zero section O C u E. Then there is an effective divisor on C, such that s extends to a monomorphism of vector bundles O C ( u E ; equivalently, we obtain an epimorphism of vector bundles u E q Q := O C (, with deg C Q. According to Lemma 2.1, the line bundle Q C determines the torsion free sheaf of rank one ˆQ Ĉ satisfying deg C Q degĉ ˆQ deg C Q + δ, together with the epimorphism E OĈ ˆQ. We denote by G := Ker(ε where: E E OĈ ε j ˆQ. It is a locally free sheaf (a vector bundle of rank r over S. We compute its numerical invariants using [9, Proposition 5.2.2]: c 1 (G = [Ĉ] = da, c 2 (G = c 2 (E + (n 1d 2 + χ( ˆQ = c 2 (E + δ + deg C Q c 2 (E + δ, [ (G := 2rc 2 (G (r 1c1 2 (G = 2r c 2 (G r 1 ] (n 1d 2 r [ 2r c 2 (E + δ r 1 ] (n 1d 2. r The hypothesis implies that (G <. Therefore [9, Theorem 7.3.4] implies the existence of a subsheaf G G of rank r, with torsion free quotient, such that: ξ G,G := c 1(G r c 1(G >, and ξg 2 r,g (G r 2 (r 1. (2.1 The sheaf G is contained in E, which is stable by hypothesis, hence c 1 (G A <. Since S is a Picard general K 3 surface, it follows that c 1 (G = ma with m 1. Moreover, the inequality ξ G,G > implies < r d rm 1 (r 1d rm m (r 1d 1. r

878 M. Halic In particular r+1 r 1 d. Ford = 1andforr = d = 2 this is a contradiction, coming from the assumption that u E C has non-zero sections. In higher degrees, we must go further, and use the second inequality in (2.1: (b ( d r 1 2 ( d 2(n 1 r 1 r m 2 r 2(n 1 2(r 1(n 1d2 r(c 2 (E + δ r 2 (r 1 r(n 1 δ 2(n 1d c 2 (E r 1. This inequality contradicts our hypothesis again. Since c 1 (E =, u E C is unstable if and only if it admits a (maximal destabilizing quotient line bundle u E Q with deg C Q. (Here we use r = 2.Byrepeating ad litteram the previous proof, we obtain a contradiction. Hence u E C is indeed a stable rank two vector bundle. ρ Remark 2.1 For r 3, by applying the result to E,forρ = 1,...,r 1 (which are all stable, it follows that u E C is a stable vector bundle too, as soon as the number δ of nodes is small enough. However, the formula for the upper bound of the number of nodes is lengthy, and we did not work it out explicitly. The tangent bundle T S of S plays a privileged role for proving the rigidity of nodal curves on K 3 surfaces. In this case, the previous theorem becomes: Corollary 2.1 Suppose that S and u : C S are as in Proposition 2.1. If either (i d = 1 and δ δ max (n, 1 := n 2 25 (hence n 5, or (ii d = 2 and δ δ max (n, 2 := 2n 27 (hence n 14, or (iii d 3 and δ δ max (n, d := 2(n 1(d 1 25 (hence (n 1(d 1 13, then u T S C is a stable, rank two bundle, and therefore H (C, u T S =. Remark 2.2 The upper bound obtained above obeys the rule δ max (n, d d A 2, d 1. However it is not clear how to interpret this fact. We speculate that this asymptotic behavior reflects some positivity property of the tangent bundle T S S.Webringsomeevidencein this direction in Corollary 3.1. Question 2.1 Suppose that S,andu : C S are as above, and that the genus of C is at least two. Is it true that u T S C has no section? Although it looks elementary, we are unable to answer this question. A positive answer would allow to extend the rigidity results obtained in Sects. 3 and 4. 3 The rigidity result In this section we are going to give a (partial positive answer to the Question 1.1. Theorem 3.1 The morphism μ : V d n,δ M g(d δ is generically finite onto its image for all triples (d, n,δsatisfying n 13 and δ δ max (n, d, with δ max (n, d as in Corollary 2.1.

Modular properties of nodal curves on K 3 surfaces 879 Proof We must prove that for any smooth, quasi-projective curve, and for any morphism U Vn,δ d such that μ U : M g(d δ is constant, the morphism U is constant itself. Such a morphism U is equivalent to the following data: a smooth and irreducible curve C of genus g := g(d δ; a smooth family (S, A of π Picard general K 3 surfaces; a family of morphisms over the 1-dimensional base U=(u t t C := C S = (S t t pr π (3.1 such that Ĉ t := u t (C S t are nodal curves with δ ordinary double points, and Ĉ t da t for all t. We must prove that, up to isomorphism, S t and u t are independent of t. Step 1 First notice that we may assume that T is trivializable. Otherwise we cover with trivializable open subsets. We denote by / t a trivializing section of T. The differentials of the various morphisms in (3.1 fit into the diagram: prc T C T C = prc T C pr T pr T = O C U U s (3.2 U T S/ U π T S O C Notice that s := U ( / t is a section of U T S C.Wedenoteby s t := s {t} C H (C, u t T S. The diagram (3.1 commutes, and therefore the tangential map π : T S St T, t sends s into the trivializing section / t H (, T. It follows that the second row in (3.2 is split, that is U T S = O C U T S/. With respect to this splitting s = (s, s,where s H ( C, U T S/. By hypothesis Ĉ t S t are nodal curves for all t. Therefore Corollary 2.1 implies that u t T S t C has no non-trivial sections, that is s {t} C = forallt. We deduce that s =, or intrinsically U ( / t = (s,. (3.3 Step 2 We interpret the result locally on S: there are local coordinates (t, z,w on S such that the morphism U is given by U(t, x = (t, z(t, x, w(t, x, (t, x C. } {{ } = u t (x The equality (3.3 becomes: dz (t,x ( / t = and dw (t,x ( / t =, (t, x C. It follows that z(t, x = z(x and w(t, x = w(x, it means that the vertical component u t (x of U is independent of the parameter t. Consider an arbitrary t. Suppose that ˆx Ĉ t is a double point, and let x 1, x 2 C t u be the corresponding pair of points identified by the normalization map C t Ĉ t. Then for all t holds u t (x 1 = (z(t, x 1, w(t, x 2 = (z(t, x 1, w(t, x 2 = u (x 1 = u (x 2 = =u t (x 2,

88 M. Halic that is the morphism u t will identify the same pairs of points of C.Sincet was arbitrary, we conclude that the curves Ĉ t := u t (C S t are all isomorphic to Ĉ := Ĉ t. Step 3 The previous step reduces the initial problem to the study of deformations of pairs (S, Ĉ consisting of a K 3 surface S, and a nodal curve Ĉ S (that is we ignore the normalization ν : C Ĉ. More precisely, we must prove that for any commutative diagram Ĉ j =( j t t S = (S t t π such that Ĉ = j t (Ĉ S t, t are nodal curves, (3.4 the family (S, Ĉ, j is trivial. The deformations of the pair (S, Ĉ are controlled by the locally free sheaf T S Ĉ,defined similarly as in (1.2 (see[8, Sect. 2]. It fits into the exact sequence T S ( Ĉ T S Ĉ TĈ := ν T C. The deformation j appearing in (3.4 keeps the nodal curve Ĉ fixed, as an abstract curve. Therefore the infinitesimal deformation induced by j corresponds to an element ( ( ê Ker H 1 (S, T S Ĉ H 1 (Ĉ, TĈ = Image H 1 (S, T S ( Ĉ H 1 (S, T S Ĉ. Accordingto[5, Lemma 2.3] (see also [11], H 1 (S, T S ( Ĉ = for a general (S, A with A 2 = 2(n 1 24. It follows that ê =, which means that the deformation of the pair (S, Ĉ is trivial. Remark 3.1 The first step of the previous proof can be interpreted and proved at the level of the Zariski tangent space of Vn,δ d.lete H 1 ( S,(σ T S C be the element corresponding to the deformation (3.1. By diagram chasing in (1.2 at the level of the long exact sequences in cohomology, we obtain the following (self-explanatory diagram: H (C,T C =! f e H 1 S,(σ T S C H 1 (C, T C f H 1 S,(σ T S ( C It is injective by Corollary 2.1. e H 1 ( S,σ T S More precisely, we have the inclusion Ker ( H 1 (r ( H (C, T C = == H 1 S,(σ H (C, u TS = T S ( C H 1 ( S,σ T S = H 1 (S, T S which shows that we can identify the deformation (3.1 with the induced infinitesimal deformation of S, corresponding to e = π (e. This is the differential theoretic counterpart of the claim that the section s in the proof of 3.1 has the form (s,. The third step can be proved by differential methods too. However the second step is not proved at the tangential level: it uses effectively the fact that we are considering a 1-dimensional deformation.

Modular properties of nodal curves on K 3 surfaces 881 As a by-product we obtain: Corollary 3.1 Suppose that S is a general K 3 surface with Pic(S = ZA, A 2 = 2(n 1. Then the global sections of 1 S A d separate its fibres over δ max (n, d general points on S. Proof According to [7], for any d 1and δ g(d there is a (g(d δ-dimensional family of irreducible nodal curves with exactly δ nodes. We apply this result for δ := δ max (n, d, and find a nodal curve Ĉ A d whose set of nodes is N ={ˆx 1,..., ˆx δ }; we denote by I N O S be the ideal of functions vanishing at N. At infinitesimal level, Theorem 3.1 implies that the homomorphism H 1 (r in diagram (1.2 is injective. Using the long exact sequence in cohomology corresponding to the first column of the diagram we deduce that H 1( S,(σ T S ( C = hence, by Poincaré duality, H 1( S, ( σ ( 1 S (Ĉ ( E =. (Here E stands for the exceptional divisor of σ. The projection formula implies that: H 1 S, 1 S (Ĉ I N =, or equivalently H S, 1 S (Ĉ evaln ˆx N 1 S, ˆx is surjective. The surjectivity property of the evaluation homomorphism at δ points is open. Therefore we find an open neighborhood U Sym δ (S of N, such that the homomorphisms H S, 1 evaln S (Ĉ 1 S, ˆx ˆx N are surjective for all N U. Concerning the Remark 2.2, one can interpret the corollary by saying that the number δ max (n, d is related to some kind of Seshadri constant of T S = 1 S (which is an intrinsic numerical invariant of S. 4 Application to the Wahl map The Wahl map for curves has been considered for the first time in [15]. The surjectivity of the Wahl map is an obstruction to embed a smooth curve into a K 3 surface. For an overview and further generalizations of these results, we invite the reader to consult [16]. Here we recall only those notions which are necessary for this article. Suppose that L V is a line bundle over some variety V. The Wahl map is by definition w L : 2 H (V, L H (V, 1 V L 2, s t sdt tds. (4.1 Equivalently, it is the restriction homomorphism H (res defined at the level of sections, induced by the exact sequence I 2 V (L L I V (L L res (I V /I 2 V L 2. } {{ } = 1 V

882 M. Halic Much attention has been payed to the case when V = C is a smooth projective curve, and L = K C,whereK C is the canonical line bundle of C. The importance of the map 2 w C : H (C, K C H (C, KC 3 relies in the fact that it is the only known obstruction to obtain a curve as a hyperplane section of a K 3 surface. More precisely: Theorem 4.1 (i (See [4]. The Wahl map w C is surjective for a general curve C of genus at least 12. (ii (See [2,15]. Suppose that C S is a smooth hyperplane section of some K 3 surface. Then the Wahl map is not surjective. In other words, a generic smooth curve C M g can not be realized as a hyperplane section of any K 3 surface, as soon as g 12. Remarkably, nodal curves escaped to the attention. Recently, in [8, Question 5.6], the authors asked whether there is an analogous obstruction for embedding nodal curves. We will use the results obtained in Sect. 2 to prove: Theorem 4.2 Let S be a Picard general K 3 surface, and let Ĉ S be a nodal curve of degree d with δ nodes, with δ δ max (n, d. Then the Wahl map 2 w C : H (C, K C H (C, KC 3 is not surjective. ( See Corollary 2.1 for the definition of δ max (n, d. The proof of the theorem is inspired from [2], but contains several modifications needed to include the double points. Let us recall from loc.cit. that the proof of 4.1(ii is based on the study of the diagram 2 H (S, O S (C ρ 2 H (C, K C w S H ( S, 1 S O S(2C H ( C, 1 S C K 2 C ρ 1 b w C H (C, KC 3. (4.2 The surjectivity of w C implies that of b, and one proves that this is impossible. The main difficulty for extending this argument to the nodal case is to find appropriate substitutes for the cohomology groups appearing in (4.2. Since this step is rather computational, and is based on diagram chasing, we have deferred it to Appendix A. Now we introduce the notations which will be used in this section. We denote by A S the ample generator of Pic(S. LetĈ S be a nodal curve of degree d (that is Ĉ A d with δ nodes, and let N ={ˆx 1,..., ˆx δ } S be its nodes. We denote by C ν Ĉ the normalization, and by x 1,1, x 1,2,...,x δ,1, x δ,2 C the pre-images by ν of ˆx 1,..., ˆx δ, respectively. Consider the blow-up S := Bl N (S σ S of S at the nodes of Ĉ,andletE = E 1 + +E δ be the exceptional divisor in S. Then the diagram ũ C S ν u Ĉ j S σ

Modular properties of nodal curves on K 3 surfaces 883 is commutative, and ũ is an embedding. Note that the divisor := x 1,1 +x 1,2 + +x δ,1 +x δ,2 equals C E, hence O C ( = O S (E C = K S C. We deduce the existence of the short exact sequence T C ( 1 S O C K C, and the isomorphism K C = σ A d ( E C. (4.3 Lemma 4.1 Let S, C, and u : C S be as above, such that δ δ max (n, d. Then the exact sequence (4.3 is not split. Proof Assume that ũ T S = T C K C (. Then the diagram T C ũ T S K C ( T C u T S K C shows that K C ( u T S. Moreover, for δ δ max (n, d holds: deg C K C ( = 2 (g(c 1 δ = 2 (g(d 1 2δ. This contradicts Corollary 2.1,sayingthatu T S C is stable. We denote by L := O S (Ĉ = A d. The Wahl maps of C and S fit into the following commutative diagram, which is important in the subsequent constructions: 2 H S,σ w S L ( E ρ 2 H (C, K C surjective by Lemma A.1 H S, 1 S σ L ( 2E ρ 1 w C H (C, K 3 C. (4.4 Proof (of Theorem 4.2 We assume that there is a curve Ĉ S such that the Wahl map of its normalization is surjective. We define R(C,:= wc 1 ( H ( C, KC 3 ( 2 H (C, K C, and denote by w C, : R(C, H ( C, KC 3( the restriction of the Wahl map to it. Since w C is surjective, w C, is surjective too. Furthermore, we define R := ρ 1 (R(C, 2 H S,σ L ( E The rear face of the cube (A.3 constructed in the appendix gives the diagram: R ρ R(C, H 1 S, 1 S σ L 2 ( 3E w C, H ( C, K 3 C ( ( H C, 1 S C KC 2( b (4.5

884 M. Halic It will be the substitute in the case of nodal curves for the diagram (4.2. Indeed, the surjectivity of ρ, and of w C, implies the surjectivity of the homomorphism b. Butb is the restriction homomorphism at the level of sections in the exact sequence K C 1 S C KC 2 ( K C 3 (, (4.6 obtained by tensoring (4.3 with KC 2 (. Therefore the boundary map : H ( C, KC 3 ( H 1 (C, K C vanishes. By applying [2, Lemme 1], we deduce that the sequence (4.6 is split, hence (4.3 is split too. This contradicts the Lemma 4.1. Acknowledgments reference [5]. The author is grateful to the referee for the careful reading, and for pointing out the Appendix A: Diagram chasing We will use the same notations as in Sect. 4. Lemma ( A.1 The restriction homomorphisms H S,σ L ( E H (C, K C and H S,σ L ( 2E H (C, K C ( are both surjective. Proof We have the exact sequence over S: σ L 1 (2E O S O C. (i The first statement is obtained by tensoring it by σ ( L ( E, and using that H 1 S, O S (E = H 1 S, K S =. (ii The second statement ( is obtained by tensoring the exact sequence by σ L ( 2E, and using that H 1 S, O S =. There is a natural surjective restriction homomorphism 1 S res C K C, and its kernel F := Ker(res C is a locally free sheaf (a vector bundle over S of rank two. The following commutative diagram is essential for the proof of Theorem 4.2: F σ L 2 ( 3E 1 S σ L 2 ( 3E F σ L 2 ( 2E 1 S σ L 2 ( 2E F O E 1 S O E res C ρ 1, ρ 1 K 3 C ( K 3 C δ K C,x j,1 K C,x j,1 j=1 Actually the whole proof is based on the careful analysis of this diagram. (A.1

Modular properties of nodal curves on K 3 surfaces 885 Every vector bundle on the projective line splits into the direct sum of line bundles. Hence the restriction of 1 S to each component E j, j = 1,...,δ, of the exceptional divisor E is the direct sum of line bundles. In fact 1 S O E j = 1 E j N E j S = O E j ( 2 O E j (1. Therefore 1 S O E notation O E ( 2 := = 1 E N E S = O E( 2 O E (1, where we use the shorthand δ j=1 O E j ( 2, and O E (1 := δ O E j (1. Since the homomorphism res C is the restriction of 1-forms on S to 1-forms on C, we deduce from the last line in (A.1 that j=1 F O E = O E ( 2 O E ( 1. (A.2 Lemma A.2 (i H (F O E = and H 1 (F O E = H 1 (O E ( 2 = C... C; } {{ } δ times (ii H S, ( = F σ L 2 ( 3E H S, F σ L 2 ( ( 2E is an isomorphism; (iii H 1 S, F σ L 2 ( 3E H 1 S, F σ L 2 ( 2E is injective. Proof (i It follows from (A.2. (ii and (iii Consider the long exact sequence in cohomology corresponding to the first column in (A.1. The claims follows from (i above. Standing hypothesis. From now on we will assume that the nodal curve Ĉ S has the property that the Wahl map of its normalization is surjective. w C : 2 H (C, K C H (C, K 3 C Lemma A.3 (i The homomorphisms H 1 S, F σ L 2 ( 2E H 1 S, 1 S σ L 2 ( 2E, and H 1 S, F σ L 2 ( 3E H 1 S, 1 S σ L 2 ( 3E (ii (iii are injective. The restriction homomorphisms ρ 1 : H S, 1 S σ L 2 ( 2E ρ 1, : H S, 1 S σ L 2 ( 3E are surjective. H S, 1 S σ L 2 ( 3E = ρ 1 1, { = s H S, 1 S σ L 2 ( 2E H ( C, KC 3, and H ( C, K 3 C ( ( H ( C, KC 3 ( ρ 1 (s H ( C, K 3 C ( }.

886 M. Halic Proof (i In the commutative diagram (4.4 the homomorphisms w C and ρ are surjective, hence ρ 1 is also surjective. The injectivity of the first homomorphism follows from the second line in (A.1. On the other hand, it follows from (A.1 thatwehavethe commutative square H 1 S, F σ L 2 ( 3E injective by A.2 (ii H 1 ( S, F σ L 2 ( 2E injective as ρ 1 surjective H 1 S, 1 S σ L 2 ( 3E H 1 S, 1 S σ L 2 ( 2E. (ii (iii Hence the upper homomorphism is injective too, as claimed. The surjectivity of ρ 1 has been proved already. For the second homomorphism, consider the long exact sequence in cohomology corresponding to the first line in (A.1, and use (i above. The first two rows of (A.1, together with (ii above imply that we have the commutative diagram H S, F σ L 2 ( 3E = H S, F σ L 2 ( 2E H S, 1 S σ L 2 ( 3E H S, 1 S σ L 2 ( 2E H ( C, K 3 C ( H ( C, KC 3 The claim is a consequence of the fact that the first vertical arrow is an isomorphism. We define R(C, := wc 1 ( H ( C, KC 3( 2 H (C, K C, and denote by w C, the restriction of the Wahl map to it. Then w C, : R(C, H ( C, KC 3( is surjective because w C is surjective. The cohomology groups introduced so far fit into the following commutative cube: R := ρ 1 w ( S,E (R(C, H S, 1 S σ L ( 3E (A.3 2 H S,σ w S L ( E H S, 1 S σ L ( 2E ρ surjective since ρ is so. R(C, 2 H (C, K C ρ ρ 1, w C, H ( C, K 3 The signs on various arrows denote inclusions. surjective by Lemma A.2 C ( w C H (C, K 3 ρ 1 C References 1. Beauville, A.: Fano threefolds and K 3 surfaces. In: The Fano Conference, University of Torino, pp. 175 184 2. Beauville, A., Mérindol, J.-Y.: Sections hyperplanes des surfaces K 3. Duke Math. J. 55, 873 878 (1987 3. Barth, W., Peters, C., Van de Ven, A.: Compact Complex Surfaces. Springer, Berlin (1984

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