Modular properties of nodal curves on K3 surfaces

TL;DR

This paper investigates the deformation properties and Wahl map behavior of nodal curves on general K3 surfaces, revealing rigidity and non-surjectivity results under certain nodal bounds.

Contribution

It establishes a rigidity property for pairs (S,C) and proves the non-surjectivity of the Wahl map for curves on general K3 surfaces, with bounds on nodes.

Findings

Deformation of (S,C) pairs induces deformation of C

Wahl map of such curves is non-surjective

Results hold under specific nodal bounds

Abstract

In this paper we partially address two issues: - The first is a rigidity property for pairs (S,C) consisting of a general projective K3 surface S, and a curve C obtained as the normalization of a nodal, hyperplane section of S. We prove that a non-trivial deformation of such a pair (S,C) induces a non-trivial deformation of C; - The second question concerns the Wahl map of curves C as above. We prove that the Wahl map of the normalization of a nodal curve contained in a general projective K3 surface is non-surjective. In both cases, we impose upper bounds on the number of nodes of the hyperplane section.