On hyperplane sections of 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(^P(g−1), I^2_C(2k))=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.