Moduli of nodal curves on K3 surfaces

TL;DR

This paper investigates the modular properties of nodal curves on general K3 surfaces, establishing conditions for the existence of irreducible components with maximal rank moduli maps, advancing understanding of curve moduli on K3 surfaces.

Contribution

It provides new criteria for the existence of irreducible components of Severi varieties on K3 surfaces with maximal moduli map rank, nearly complete for all but finitely many cases.

Findings

Identifies conditions on p, m, δ for maximal rank moduli maps.

Proves optimality of results for m ≥ 5.

Completes the classification for all but finitely many cases.

Abstract

We consider modular properties of nodal curves on general $K3$ surfaces. Let $\mathcal{K}_p$ be the moduli space of primitively polarized $K3$ surfaces $(S,L)$ of genus $p\geqslant 3$ and $\mathcal{V}_{p,m,\delta}\to \mathcal{K}_p$ be the universal Severi variety of $\delta$--nodal irreducible curves in $|mL|$ on $(S,L)\in \mathcal{K}_p$. We find conditions on $p, m,\delta$ for the existence of an irreducible component $\mathcal{V}$ of $\mathcal{V}_{p,m,\delta}$ on which the moduli map $\psi: \mathcal{V}\to \mathcal{M}_g$ (with $g= m^2 (p -1) + 1-\delta$) has generically maximal rank differential. Our results, which for any $p$ leave only finitely many cases unsolved and are optimal for $m\geqslant 5$ (except for very low values of $p$), are summarized in Theorem 1.1 in the introduction.