On hyperplane sections on K3 surfaces

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 (Pg-1; I-C(2) (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.