DEGENERATION OF DIFFERENTIALS AND MODULI OF NODAL CURVES ON K3 SURFACES

We consider, under suitable assumptions, the following situation: B is a component of the moduli space of polarized surfaces and V(m,d) is the universal Severi variety over B parametrising pairs (S, C), with S smooth, projective, irreducible surface, C a member of |O_S(mH)| irreducible with exactly d nodes as singularities. We assume there are suitable semistable degenerations of the surfaces in B. Then we give sufficient conditions for the existence of an irreducible component V of V(m,d) for which the moduli map fro V to the moduli space of genus g curve M(g) is generically of maximal rank, where g is the geometric genus of the curves in V(m,d). As a test, we apply this to K3 surfaces and give a new proof of a result recently independently proved by Kemeny and by the present authors.