Mukai's program for non-primitive curves on K3 surfaces

TL;DR

This paper extends Mukai's program to non-primitive curves on K3 surfaces, demonstrating that the surface can be recovered from such curves under broader conditions and introducing destabilising regions to enhance the analysis.

Contribution

It proves Mukai's program holds for irreducible curves in non-primitive linear systems with specific genus and degree conditions, and introduces destabilising regions for improved analysis.

Findings

Mukai's program is valid for irreducible curves with certain genus and degree constraints.

Hyper-K"ahler varieties appear as Brill-Noether loci in every dimension.

The analysis is enhanced by the concept of destabilising regions.

Abstract

Mukai's program seeks to recover a K3 surface $X$ from any curve $C$ on it by exhibiting it as a Fourier-Mukai partner to a Brill-Noether locus of vector bundles on the curve. In the case $X$ has Picard number one and the curve $C\in |H|$ is primitive, this was confirmed by Feyzbakhsh for $g\geq 11$ and $g\neq 12$. More recently, Feyzbakhsh has shown that certain moduli spaces of stable bundles on $X$ are isomorphic to the Brill-Noether locus of curves in $|H|$ if $g$ is sufficiently large. In this paper, we work with irreducible curves in a non-primitive ample linear system $|mH|$ and prove that Mukai's program is valid for any irreducible curve when $g\neq 2$, $mg\geq 11$ and $mg\neq 12$. Furthermore, we introduce the destabilising regions to improve Feyzbakhsh's analysis. We show that there are hyper-K\"ahler varieties as Brill-Noether loci of curves in every dimension.