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

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 $g\ge 11$ and $g\ne 12$. 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 $g\ne 2$, $mg\ge 11$ and $mg\ne 12$. Furthermore, we introduce the destabilising regions to improve Feyzbakhsh's analysis. We show that there are hyper-Kähler varieties as Brill-Noether loci of curves in every dimension.

27 pages, 7 figures