GAUSSIAN MAPS OF PLANE CURVES WITH NINE SINGULAR POINTS. Dedicated with gratitude to the memory of Sacha Lascu

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1 GAUSSIAN MAPS OF PLANE CURVES WITH NINE SINGULAR POINTS EDOARDO SERNESI Dedicated with gratitude to the memory of Sacha Lascu Abstract. We consider nonsingular curves which are the normalization of plane curves with nine ordinary singular points, viewing them as embedded in the blow-up X of the projective plane along their singular points. For a large class of such curves we show that the gaussian map relative to the canonical line bundle has corank one. The proof makes essential use of the geometry of X. 1. Introduction This paper contains some elementary computations motivated by the attempt to understand the behaviour of the gaussian-wahl maps of smooth curves contained in the blow-up X of P 2 at nine general points. We will exclusively consider smooth curves on X which are strict transforms of irreducible plane curves singular at all the nine points that are blown up. This accounts for the title of this note. Recent work by several authors has shown the interest of such curves in Brill-Noether theory. In outline the situation is the following. Let C be a nonsingular projective curve of genus g 12. Denote by Φ K : H 0 (C, ω C ) H 0 (C, ωc) 3 the so-called gaussian map, or Wahl map, relative to the canonical line bundle (see 2 for the definition). Denote by J X the anticanonical elliptic curve. We know a priori that for a smooth curve C X we have cork(φ K ) 1. This is because, after blowing up the points of C J and blowing down the proper transform of J, we obtain a singular surface S containing C which is a fake K3 surface, namely a regular Gorenstein surface with trivial dualizing sheaf. This implies that S can be embedded in P g with canonical curve sections, one of them being C, and therefore Φ K is not surjective, by [W2], Theorem 7.1. The surface X contains, among others, certain curves C of arbitrary genus, called Du Val curves, which are Brill-Noether-Petri general under suitable generality assumptions, as shown in [ABFS]; later Arbarello and Bruno 1

2 2 EDOARDO SERNESI [AB] have shown that on general Du Val curves of odd genus 11 the map Φ K has corank exactly one. In the attempt to recover their result by direct computation we were led to study Φ K for a general class of curves contained in X, not just for those considered in [ABFS]. We consider curves belonging to E-excellent divisor classes in the sense of [H1] (see 3 for the definitions). Our main result is Theorem 9, stated in 6, which asserts that Φ K has corank one for a large class of such curves. Its statement requires some notation that will be given in 3 and therefore we do not reproduce it here. Unfortunately the Du Val curves remained unaccessible to the methods used here and therefore they are not covered by Theorem 9, even though they were the original motivation of our computations. Nodal curves are not part of the statement either. We refer to Remark 10 for details and a discussion. A few words about the techniques. The method of proof consists in viewing Φ K as the composition of a suitable gaussian map on X with two other maps defined on X. We compute that the corank of the third map is one, and that the two others are surjective thus proving that Φ K has corank one. This strategy is exactly the one prescribed in [DM], Lemma 2.6 and applied in [CLM2] to prove the surjectivity of Φ K for the normalization of a plane nodal curve with sufficiently many nodes. Acknowledgements. We thank the referee for his careful review of the manuscript. The author is a member of GNSAGA-INDAM and of the PRIN Project Geometry of Projective Varieties funded by the Italian MIUR. 2. Generalities on Gaussian maps We work over C. Given line bundles L, M on a nonsingular projective variety X we consider: R(L, M) = ker[h 0 (L) H 0 (M) H 0 (LM)] Then we have a canonical map: Φ L,M : R(L, M) H 0 (Ω 1 XLM) called the Gaussian map, or Wahl map, of L, M, which is defined as follows. Let X X be the diagonal and p 1, p 2 : X X X the projections. Then R(L, M) = H 0 (X X, p 1L p 2M I ) Since I O = Ω 1 X, the restriction to : p 1L p 2M I p 1L p 2M I O

3 GAUSSIAN MAPS OF PLANE CURVES WITH NINE SINGULAR POINTS 3 induces Φ L,M on global sections. The exact sequence: 0 p 1L p 2M I 2 p 1L p 2M I p 1L p 2M I O 0 shows that the vanishing: H 1 (X X, p 1L p 2M I 2 ) = 0 is a sufficient condition for the surjectivity of Φ L,M. In case L = M we have R(L, L) = I 2 (X) 2 H 0 (L), where I 2 (X) = ker[s 2 H 0 (L) H 0 (L 2 )] and Φ L,L is zero on I 2 (X). Therefore Φ L,L is equivalent to its restriction to 2 H 0 (L), which is denoted by Φ L : H 0 (L) H 0 (Ω 1 XL 2 ) In particular, for a non-hyperelliptic curve C we are interested in Φ K,K or rather in Φ K : H 0 (C, ω C ) H 0 (C, ωc) 3 where O C (K) = ω C is the canonical invertible sheaf. Suppose that C X where X is a nonsingular regular surface such that H 0 (X, K X ) = 0. Then H 0 (C, ω C ) = H 0 (X, K X + C) and Φ K can be described as a composition: where Φ K = H 0 (ρ) Φ KX +C Φ KX +C : H 0 (C, ω C ) = H 0 (X, O X (K X +C)) H 0 (X, Ω 1 X(2K X +2C)) and ρ : Ω 1 X (2K X + 2C) ωc 3 is the map obtained by twisting the natural map Ω 1 X ω C by O X (2K X +2C). Since ρ fits into the following diagram: Ω 1 X (2K X + 2C) γ ρ Ω 1 X C (2K X + 2C) one can decompose H 0 (ρ) = H 0 (δ) H 0 (γ) and study the two maps H 0 (δ) and H 0 (γ) separately. Therefore Φ K can be studied by analyzing the three maps Φ KX +C, H 0 (γ), H 0 (δ) separately. This is what we will do in a specific situation that will be described next. We refer to [DM], δ ω 3 C

4 4 EDOARDO SERNESI Lemma 2.6, for an outline of an analogous strategy for curves lying on a rational ruled surface. 3. The set-up Let Z = {z 1,..., z 9 } P 2 be a set of nine distinct points lying on a unique cubic J, which we assume to be nonsingular, and let σ : X := Bl Z P 2 P 2 be the blow-up along Z. We denote by E j = σ 1 (z j ) the j-th ( 1)-curve, by E = j E j the exceptional divisor and by l = σ 1 (λ) the inverse image of a general line λ P 2. The set of divisor classes E = {O X (l), O X (E 1 ),..., O X (E 9 )} is a so-called exceptional configuration for X, in the terminology of [H1]. A canonical divisor on X is K X = 3l + E. Let n 1, 0 ɛ 2 and b = (b 1,..., b 9 ) where the b j s are integers such that 0 b 1 b 9 n. Consider the linear system on X: 9 nk 9 X + ɛl + b j E j = (3n + ɛ)l (n b j )E j Let j=1 L n,ɛ,b = O X ( nk X + ɛl + The following intersection numbers 9 (1) L n,ɛ,b L n,ɛ,b = ɛ(6n+ɛ)+ b j (2n b j ), are readily computed. j=1 j=1 9 b j E j ) j=1 Proposition 1. Let n, ɛ, b be as above. (i) If 1 ɛ 2 or b 9 1 then L n,ɛ,b, h 1 (X, L n,ɛ,b ) = 0 = h 2 (X, L n,ɛ,b ) K X L n,ɛ,b = 3ɛ and a general C L n,ɛ,b is irreducible and nonsingular. (ii) If 1 ɛ 2 or j b j 2 then L n,ɛ,b is globally generated. Proof. (i) From (1) we see that K X L n,ɛ,b < 0. Therefore, in the terminology of [H1], the divisor class of L is E-excellent. Then the vanishing and non-emptiness follow from [H1], Thm The irreducibility is proved in [N], Prop. 8, and the nonsingularity follows from Bertini s Theorem. (ii) follows from [H1], Lemma j=1 b j

5 GAUSSIAN MAPS OF PLANE CURVES WITH NINE SINGULAR POINTS 5 We will compute the corank of the gaussian map of a general member of nk X +ɛl+ 9 j=1 b je j in certain cases when it is possible to apply the strategy outlined in Surjectivity of Φ KX +L n,ɛ,b We keep the notations introduced in 3. We will make use of the following result. Theorem 2. Let S be a nonsingular projective surface, S P 2. Let F, G, H be very ample invertible sheaves on S, and L = F G H. Then Φ KS +L is surjective. Proof. We outline the proof that can be found in [CLM1] or deduced from [BEL]. Let π : Y S S be the blow-up of S S along the diagonal and let Λ Y be the exceptional divisor. For any line bundle L on S denote by L i = (p i π) L, where p i : S S S is the i-th projection, i = 1, 2. The sufficient condition H 1 (S S, p 1(K S + L) p 2(K S + L)( 2 )) = 0 for the surjectivity of Φ KS +L is equivalent to (2) H 1 (Y, (K S + L) 1 (K S + L) 2 ( 2Λ)) = 0 We have: (K S + L) 1 (K S + L) 2 ( 2Λ) = K Y L 1 L 2 ( 3Λ) and therefore the required vanishing (2) will follow from the Kawamata- Vieweg vanishing theorem if we prove that L 1 L 2 ( 3Λ) is big and nef. As in [BEL], Claim 3.3, one proves that F 1 F 2 ( Λ), G 1 G 2 ( Λ) and H 1 H 2 ( Λ) are big and nef. Therefore L 1 L 2 ( 3Λ) = [F 1 F 2 ( Λ)] [G 1 G 2 ( Λ)] [H 1 H 2 ( Λ)] is big and nef. We can now prove the following: Theorem 3. Assume that n 4, 0 ɛ 2 and 2 b 1 b 9 n 2. Then Φ KX +L n,ɛ,b is surjective. Proof. Set F = G = O X ( K X + l), and H = O X ( (n 3)K X + (ɛ + 1)l + j (b j 1)E j ) Then L n,ɛ,b = F G H. Moreover F = G is very ample by [AH], Theorem 2.3. From the assumptions it follows that H is E-excellent and K X H 3. Therefore Thms. 1.1 and 2.1 of [H2] imply that H

6 6 EDOARDO SERNESI is very ample as well. The conclusion is now a consequence of Theorem The corank of H 0 (ρ) Lemma 4. Assume n 4, 0 ɛ 2 and 1 b 1 b 9 n 2. Then H i (X, σ Ω 1 P 2(2K X + C)) = 0, i = 1, 2 and H i (X, σ Ω 1 P 2(2K X + 2C)) = 0, i = 1, 2 where C L n,ɛ,b is general. Proof. By Prop. 1(i) the curve C is nonsingular. Consider 2K X + C 3l = (n 3)K X + ɛl + j (b j 1)E j From the assumptions and from [H1](0.3) it follows that (3) h 2 (X, O X (2K X + C 3l)) = 0 Moreover we have: 2K X + C 2l = (n 3)K X + (ɛ + 1)l + j (b j 1)E j and by Prop. 1(i) we have h i (X, O X (2K X + C 2l)) = 0, i = 1, 2, and therefore also (4) h i (X, O X (2K X + C kl)) = 0, 0 k 2, i = 1, 2 A fortiori we also have (5) h i (X, O X (2K X + 2C kl)) = 0, 0 k 2, i = 1, 2 and (6) h 2 (X, O X (2K X + 2C 3l)) = 0 as it follows from the exact sequences: 0 O X (2K X + C kl) O X (2K X + 2C kl) ω 2 C ( kl) 0 Now consider the pullback by σ of the Euler sequence, twisted by 2K X + C: 0 σ Ω 1 P 2 (2K X + C) V O X (2K X + C l) O X (2K X + C) 0 where V = H 0 (X, l). By Castelnuovo-Mumford regularity from (3) and (4) we deduce that V H 0 (X, O X (2K X + C l)) H 0 (X, O X ((2K X + C))

7 GAUSSIAN MAPS OF PLANE CURVES WITH NINE SINGULAR POINTS 7 is surjective and therefore H 1 (X, σ Ω 1 P 2(2K X + C)) = 0 = H 2 (X, σ Ω 1 P 2(2K X + C)) because of (4) for k = 0, 1. The case of 2K X + 2C is similar, using (5) and (6) instead of (3) and (4). Proposition 5. Assume that n 4, and 0 ɛ 2. Consider the following possibilities: (a) 1 b 1 b 9 n 3. (b) 1 b 1 b 9 n 2. In either case let C L n,ɛ,b be general. Then: if (a) holds and if (b) holds. H i (X, Ω 1 X(2K X + C)) = 0, i = 1, 2 H i (X, Ω 1 X(2K X + 2C)) = 0, i = 1, 2 Proof. Consider the relative cotangent exact sequence twisted by 2K X + C: (7) 0 σ Ω 1 P 2 (2K X + C) Ω 1 X (2K X + C) ω E (2K X + C) 0 Then ω E (2K X + C) = ω E ( j (b j + 2 n)e j ) = j ω Ej ((b j + 2 n)e j ) Assume that (a) holds. Then n+b j +2 < 0 and we have H 1 (ω E (2K X + C)) = 0. Thanks to (7) and Lemma 4 this implies that H i (X, Ω 1 X(2K X + C)) = 0, i = 1, 2 and concludes the proof for the case 2K X + C. The case (b) is similar. Proposition 6. Assume that n 4, and 1 b 1 b 9 n 3. Then H 0 (γ) : H 0 (X, Ω 1 X(2K X + 2C)) H 0 (Ω 1 X C(2K X + 2C)) is surjective.

8 8 EDOARDO SERNESI Proof. The proposition is an immediate consequence of Proposition 5 and of the exact sequence (8) 0 Ω 1 X (2K X + C) Ω 1 X (2K X + 2C) The following result takes care of H 0 (δ). γ Ω 1 X C (2K X + 2C) 0 Proposition 7. Assume that ɛ 1 or b 9 1, and let n 3. Let C L n,ɛ,b be general. Then has corank one. H 0 (δ) : H 0 (C, Ω 1 X C(2K X + 2C)) H 0 (C, ω 3 C) Proof. The homomorphism δ is part of the conormal sequence of C X twisted by 2K X + 2C: 0 O C (2K X + C) Ω 1 X C (2K X + 2C) δ ω 3 C 0 Proposition 5 and the exact sequence (8) imply that Then: H 1 (C, Ω 1 X C(2K X + 2C)) = 0 coker(h 0 (δ)) = H 1 (C, O C (2K X + C)) = H 0 (C, O C ( K X )) Now consider the exact sequence on X: 0 O X ( K X C) O X ( K X ) O C ( K X ) 0 Then clearly H 0 (X, O X ( K X C)) = 0 and: h 1 (X, O X ( K X C)) = h 1 (X, O X (2K X + C)) = h 1 (L n 2,ɛ,b ) = 0 by Prop. 1 since n 3. Then h 0 ((C, O C ( K X )) = h 0 (X, O X ( K X )) = 1 Corollary 8. Let C L n,ɛ,b be general and assume that n 4 and 1 b 1 b 9 n 3. Then the map has corank one. H 0 (ρ) : H 0 (X, Ω 1 X(2K X + 2C)) H 0 (C, ω 3 C) Proof. The corollary follows directly from the fact that H 0 (ρ) = H 0 (δ) H 0 (γ) and from Propositions 6 and 7.

9 GAUSSIAN MAPS OF PLANE CURVES WITH NINE SINGULAR POINTS 9 6. Conclusions We can now prove our main result. Theorem 9. Assume that the following conditions hold: n 4, 2 b 1 b 9 n 3. Let C be a general element of nk X + ɛl + j b je j. Then C is irreducible and nonsingular and the gaussian map has corank one. Φ K : H 0 (C, ω C ) H 0 (C, ω 3 C) Proof. The irreducibility and nonsingularity follow from Proposition 1. The assumptions are stronger than those of both Corollary 8 and Theorem 3: it follows that Φ K is the composition of the surjective map Φ KX +C with the map H 0 (ρ) which has corank one. Then Φ K has corank one. Remark 10. Recall that a Du Val curve on X is a nonsingular curve C g gk X + E 9 for some g 2. The curve C g has genus g and, as proved in [ABFS], is Brill-Noether-Petri general under suitable assumptions of generality on the set Z of nine points that are blownup. In [AB] it is proved that for odd genera g 11 the gaussian map Φ K of C g has corank one. In our notation O X ( gk X + E 9 ) = L g,0,b with b = {0,..., 0, 1}. Therefore we see that the Du Val curves are not covered by Theorem 9 because its hypothesis requires that b j 0 for all j = 1,..., 9. Observe also that a curve C as in Theorem 9 is the proper transform of a plane curve C of degree 3n + 3 having a singularity of multiplicity 3 n b j n 2 at the point z j. In particular no nodal curves nor curves smooth at some z j are of this type. This last circumstance is expected a priori because if C is smooth at z 9 say, then C is embedded in Y := Bl {z1,...,z 8 }(P 2 ) and therefore Φ K is expected to have corank h 0 (Y, K Y ) = 2 (see [DM] and [W1]). References [AH] d Almeida J., Hirschowitz A.: Quelques plongements projectifs non spéciaux de surfaces rationelles, Math. Z. 211 (1992), [AB] Arbarello E., Bruno A.: Rank two vector bundles on polarized Halphen surfaces and the Gauss-Wahl map for du Val curves. arxiv: [ABFS] Arbarello E., Bruno A., Farkas G., Saccà G.: Explicit Brill-Noether- Petri general curves. Comment. Math. Helvetici 91 (2016),

10 10 EDOARDO SERNESI [BEL] [CLM1] Bertram A., Ein L., Lazarsfeld R.: Vanishing theorems, a theorem of Severi, and the equations defining projective varieties. J. AMS 4 (1991), Ciliberto C, Lopez A.F., Miranda R.: On the corank of gaussian maps for general embedded K3 surfaces. In Israel Mathematical Conference Proceedings. Papers in honor of Hirzebruch s 65th birthday, vol. 9, AMS Publications (1996), [CLM2] Ciliberto C, Lopez A.F., Miranda R.: On the Wahl map of plane nodal curves. In Complex Analysis and Algebraic Geometry, a volume in Memory of Michael Schneider. Thomas Peternell and Frank-Olaf Schreyer, editors. Walter de Gruyter, Berlin/New York p [DM] [H1] [H2] Duflot J., Miranda R.: The gaussian map for rational ruled surfaces. Trans. AMS 330 (1992), Harbourne B.: Complete linear systems on rational surfaces. Trans. AMS 289 (1985), Harbourne B.: Very ample divisors on rational surfaces. Math. Annalen 272 (1985), [N] Nagata M.: On rational surfaces. II. Mem. Coll. Sci. Kyoto (A) 33 (1960), [W1] Wahl J.: Gaussian maps on algebraic curves. J. Diff. Geometry 32 (1990), [W2] Wahl J.: The cohomology of the square of an ideal sheaf. J. Algebraic Geom. 6 (1997), address: sernesi@mat.uniroma3.it Dipartimento di Matematica e Fisica, Università Roma Tre, L.go S.L. Murialdo 1, Roma.