Degeneration of differentials and moduli of nodal curves on K3 surfaces

We consider, under suitable assumptions, the following situation:$mathcal B$ is a component of the moduli space of polarized surfaces and $V_m$ is theuniversal Severi variety over B parametrizing pairs (S;C), with $(S;H) in B$ andC in |mH|$ irreducible with exactly $delta $ nodes as singularities. The moduli map$V o M_g$ of an irreducible component $V$ of $V_m$ is generically of maximal rankif and only if certain cohomology vanishings hold. Assuming there are suitablesemistable degenerations of the surfaces in $B$, we provide sufficient conditionsfor the existence of an irreducible component $V$ where these vanishings areverified. As a test, we apply this to $K3$ surfaces and give a new proof of aresult recently independently proved by Kemeny and by the present authors.