Nodal curves with general moduli on K3 surfaces

TL;DR

This paper explores the properties of nodal curves on low genus K3 surfaces, establishing conditions under which general curves can be realized as normalizations of nodal curves on such surfaces, using deformation theory.

Contribution

It demonstrates that for certain degrees and genera, general curves are normalizations of nodal curves on primitively polarized K3 surfaces, expanding understanding of their moduli.

Findings

General genus g curves are normalizations of d-nodal curves on K3 surfaces for specified p, g, and d.

The proof employs a local deformation-theoretic analysis of the moduli map from pairs (S,X) to the moduli space of curves.

The results connect the geometry of K3 surfaces with the moduli of algebraic curves in low genus.

Abstract

We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a d-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p-2, for p any integer between 3 and 11 and g = p - d between 2 and p. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs (S,X) to the moduli space of curves of genus g that associates to X the isomorphism class [C] of its normalization.