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

Abstract

We consider, under suitable assumptions, the following situation: $\mathcal{B}$ is a component of the moduli space of polarized surfaces and ${\mathcal{V}}_{m,\delta }$ is the universal Severi variety over $\mathcal{B}$ parametrizing pairs $(S,C)$, with $(S,H)\in \mathcal{B}$ and $C\in |mH|$ irreducible with exactly $\delta$ nodes as singularities. The moduli map $\mathcal{V}\to {\mathcal{M}}_g$ of an irreducible component $\mathcal{V}$ 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.

submitted paper

References in corpus (2)

• Projective Degenerations of K3 Surfaces, Gaussian Maps, and Fano Threefolds
• On the Severi varieties of surfaces in P^3