Nodal curves with general moduli on K3 surfaces

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 -nodal curve X sitting on a primitively polarized K3 surface S of degree 2p-2, for 2g=p-p11. 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 Mg that associates to X the isomorphism class [C] of its normalization.