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

## TL;DR

This paper investigates the degeneration of differentials and the moduli space of nodal curves on K3 surfaces, providing new conditions for the maximal rank of the moduli map and applying these to give a novel proof of existing results.

## Contribution

It introduces sufficient conditions for the existence of components where certain cohomology vanishings hold, specifically applied to K3 surfaces, offering a new proof of recent results.

## Key findings

- Established conditions for the maximal rank of the moduli map.
- Provided a new proof of a recent result on nodal curves on K3 surfaces.
- Applied degeneration techniques to moduli spaces of curves.

## 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.