Mukai's program (reconstructing a K3 surface from a curve) via wall-crossing

TL;DR

This paper extends Mukai's program to reconstruct K3 surfaces from curves with specific genus conditions using wall-crossing and Bridgeland stability, revealing a new approach via Fourier-Mukai transforms of Brill-Noether loci.

Contribution

It generalizes Mukai's reconstruction method for K3 surfaces from curves by employing wall-crossing and Bridgeland stability conditions, connecting the surface to vector bundles on the curve.

Findings

Reconstruction of K3 surfaces via Fourier-Mukai transforms.

Application of wall-crossing with Bridgeland stability.

Extension of Mukai's program to new genus conditions.

Abstract

Let $C$ be a curve of genus $g \geq 11$ such that $g-1$ is a composite number. Suppose $C$ is on a K3 surface whose Picard group is generated by the curve class $[C]$. We use wall-crossing with respect to Bridgeland stability conditions to generalise Mukai's program to this situation: we show how to reconstruct the K3 surface containing the curve $C$ as a Fourier-Mukai transform of a Brill-Noether locus of vector bundles on $C$.