On hyperplane sections of K3 surfaces
Abstract
Let C be a Brill-Noether-Petri curve of genus g\geq 12. We prove that C lies on a polarized 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\geq 8, and if the Clifford index of C is greater than 2, then H^1(P^{g-1}, I_C^2(k))=0, for k\geq 3. We prove this conjecture for g\geq 11. The second conjecture states that a Brill-Noether-Petri curve of genus g\geq 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.
Cite
@article{arxiv.1507.05002,
title = {On hyperplane sections of K3 surfaces},
author = {Enrico Arbarello and Andrea Bruno and Edoardo Sernesi},
journal= {arXiv preprint arXiv:1507.05002},
year = {2016}
}
Comments
Title, abstract, and introduction changed (previous title: "On two conjectures by J. Wahl"). Several typos corrected. Exposition improved in various instances, according to referee's suggestions. The paper will appear in "Algebraic Geometry"