# The Wahl map of one-nodal curves on K3 surfaces

## TL;DR

This paper investigates the Wahl map of one-nodal curves on K3 surfaces, establishing surjectivity for certain genera, which advances understanding of the geometric properties of these curves.

## Contribution

It proves the surjectivity of the Wahl map for the normalization of one-nodal curves on K3 surfaces at specific genera, extending previous results.

## Key findings

- Surjectivity of the Wahl map for genus 40, 42, and ≥44.
- Normalization of one-nodal curves on K3 surfaces has specific geometric properties.
- Advances understanding of the Wahl map in the context of K3 surface curves.

## Abstract

We consider a general primitively polarized K3 surface $(S,H)$ of genus $g+1$ and a 1-nodal curve $\widetilde C\in |H|$. We prove that the normalization $C$ of $\widetilde C$ has surjective Wahl map provided $g=40,42$ or $\ge 44$.