### On hyperplane sections on K3 surfaces

Enrico Arbarello
2017 Algebraic Geometry
Let C be a Brill-Noether-Petri curve of genus g 12. We prove that C lies on a polarised K3 surface, or on a limit thereof, if and only if the Gauss-Wahl map for C is not surjective. The proof is obtained by studying the validity of two conjectures by J. Wahl. Let I C be the ideal sheaf of a non-hyperelliptic, genus g, canonical curve. The first conjecture states that if g 8 and if the Clifford index of C is greater than 2, then H 1 (P g−1 , I 2 C (k)) = 0 for k 3. We prove this conjecture for g
more » ... is conjecture for g 11. The second conjecture states that a Brill-Noether-Petri curve of genus g 12 is extendable if and only if C lies on a K3 surface. As observed in the introduction, the correct version of this conjecture should admit limits of polarised K3 surfaces in its statement. This is what we prove in the present work. On hyperplane sections of K3 surfaces Lazarsfeld's theorem also shows that it is not possible to characterise the locus of curves lying on K3 surfaces by looking at line bundles on them. One may look at vector bundles, and then the situation is completely different. In fact, following an idea of Mukai, the authors of this paper where able to describe a general K3 surface in terms of an exceptional Brill-Noether locus for vector bundles on one of its hyperplane sections [ABS14]. Here we take a different point of view. The starting point is a theorem by Wahl [Wah87] proving that for a smooth curve C of genus g 2 lying on a K3 surface S, the Gaussian map (also called the Gauss-Wahl map or Wahl map) is not surjective. A beautiful geometric proof of this fact was given by Beauville and Merindol [BM87]. This theorem is all the more significant in light of the fact that for a general curve, the Wahl map is surjective. This was proved, with different methods, by Ciliberto, Harris, Miranda [CHM88], and by Voisin [Voi92]. In this last paper Voisin, for the first time, suggests to study the non-surjectivity of the Wahl map under the Brill-Noether-Petri condition, in order to characterise hyperplane sections of K3 surfaces (see [Voi92, Remark 4.13(b)], where the author also refers to Mukai). In [CU93], Cukierman and Ulmer proved that when g = 10, the closure of the locus in M 10 of curves that are canonical sections of K3 surfaces coincides with the closure of the locus of curves with non-surjective Wahl map. This locus has been studied in more detail by Farkas-Popa in [FP05], where they gave a counterexample to the slope conjecture. Our aim is to prove the following theorem. Theorem 1.1. Let C be a Brill-Noether-Petri curve of genus g 12. Then C lies on a polarised K3 surface, or on a limit thereof, if and only if its Wahl map is not surjective. Our investigation was sparked by a remarkable paper by Wahl [Wah97], where the author analyses the significance of the non-surjectivity of his map from the point of view of deformation theory. Let C ⊂ P g−1 be a canonically embedded curve of genus g 3. Wahl finds a precise connection between the cokernel of ν and the deformation theory of the affine cone over C. Denoting by the coordinate ring of this cone, Wahl considers the graded cotangent module T 1 A and shows that T 1 He also gives a precise interpretation of the graded pieces of the obstruction module T 2 A : where I C/P is the ideal sheaf of C ⊂ P g−1 . The modules T 1 A and T 2 A govern the deformation theory of the cone Spec A. The canonical curve C ⊂ P g−1 is said to be extendable if it is a hyperplane section of a surface S ⊂ P g which is not a cone. The main theorem of [Wah97] is the following. Theorem 1.2 (Wahl). Let C ⊂ P g−1 be a canonical curve of genus g 8, with Cliff(C) 3. Suppose that H 1 P g−1 , I 2 C/P (k) = 0 , k 3 . (1.1) Then C is extendable if and only if ν is not surjective. 563 E. Arbarello, A. Bruno and E. Sernesi Wahl then conjectures that every canonical curve of Clifford index greater than or equal to 3 satisfies the vanishing condition (1.1). We recall that the Clifford index of a curve C is the minimum value of deg D − 2 dim |D| taken over all linear systems |D| on C such that dim |D| 1 and dim |K C − D| 1. With a slight restriction on the genus, this is what we prove in the first part of the present paper. Theorem 1.3. Let C ⊂ P g−1 be a canonically embedded curve of g 11. Suppose Cliff(C) 3. Let I C/P ⊂ O P g−1 be the ideal sheaf of C. Then H 1 P g−1 , I 2 C/P (k) = 0 for all k 3. From Theorem 1.2, one gets the following obvious corollary. Corollary 1.4. Let C ⊂ P g−1 be a canonical curve of genus g 11 with Cliff(C) 3. Then C is extendable if and only if ν is not surjective. On hyperplane sections of K3 surfaces