Nodal Curves with General Moduli on K3 Surfaces

Abstract

We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a δ-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p − 2, for 2 ≤ g = p − δ < p ≤ 11. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs (S, X) to the moduli stack of curves ℳ g that associates to X the isomorphism class [C] of its normalization.