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We investigate the modular properties of nodal curves on a low genus K3surface. We prove that a general genus g curve C is the normalization of ad-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p-2,for p any integer between 3 and 11 and g = p - d between 2 and p.  The proof is based on a local deformation-theoretic analysis of the map fromthe stack of pairs (S,X) to the moduli space of curves of genus g thatassociates to X the isomorphism class [C] of its normalization.

Nodal curves with general moduli on K3 surfaces