On hyperplane sections on K3 surfaces

Abstract

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 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.

Global sections of twisted normal bundles of K3 surfaces and their hyperplane sections

“…It is well-known that smooth curves of Clifford index one are either trigonal or isomorphic to smooth plane quintics. The next two results give all possible values of h 0 (N C/P g−1 (−2)), or equivalently, cork Φ ω C ,ω ⊗2 C , by (2), for such curves.…”

“…Much attention has been devoted to canonical curves and K3 surfaces. We refer to the recent works [2,6], and recall, as another instance, that considerations as above led to the proof that a curve of genus g ≥ 11, g = 12 lying on a K3 surface, is generically contained in at most one such surface [9].…”

On Lagrangian fibrations by Jacobians, II

“…When g = 2 both the Debarre system X → P 2 and its dual X → P 2 have fibres of polarization type (1,3). If an abelian variety is not principally polarized then it will only be isogenous to its dual, not isomorphic, so X → P 2 is certainly not self-dual.…”

Wahl maps and extensions of canonical curves and K⁢3K3 surfaces

“…Let X ⊂ P n be a smooth, projective, irreducible, non-degenerate variety, not a quadric. SetIf X is r-extendable and α(X) < n, then r α(X).This introduced the idea, explicit in Voisin's article [38], that the elements of (coker(Φ C )) ∨ (or rather of ker( T Φ C )) should be interpreted as ribbons, or infinitesimal surfaces, embedded in P g and extending C: see Section 4.The following statement is a first converse to Theorem (0.1), and a central element of the proofs of our Theorems (2.1) and (2.18).(0.2) Theorem (Wahl [43], Arbarello-Bruno-Sernesi [3]). Let C be a smooth curve of genus g 11, and Clifford index Cliff(C) > 2.…”