  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
d-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 from
the stack of pairs (S,X) to the moduli space of curves of genus g that
associates to X the isomorphism class [C] of its normalization.
