Degeneration of differentials and moduli of nodal curves on $K3$ surfaces

We consider, under suitable assumptions, the following situation: $\mathcal B$isacomponentofthemodulispaceofpolarizedsurfacesand$\mathcal V_{m,\delta}$istheuniversalSeverivarietyover$\mathcal B$ parametrizing pairs $(S,C)$, with $(S,H)\in \mathcal{B}$ and $C\in \mathrm{\mid }mH\mathrm{\mid }$ irreducible with exactly $\delta$ nodes as singularities. The moduli map $\mathcal V\to \mathcal M_g$ofanirreduciblecomponent$\mathcal V$of$\mathcal V_{m,\delta}$ is generically of maximal rank if and only if certain cohomology vanishings hold. Assuming there are suitable semistable degenerations of the surfaces in $\mathcal{B}$, we provide sufficient conditions for the existence of an irreducible component $\mathcal{V}$ where these vanishings are verified. 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.