Mukai's program for curves on a K3 surface

TL;DR

This paper explores how to reconstruct K3 surfaces from curves lying on them using Mukai's ideas, focusing on specific genera and employing Fourier-Mukai transforms of vector bundles.

Contribution

It demonstrates a method to reconstruct K3 surfaces from curves via Fourier-Mukai transforms of Brill-Noether loci, extending Mukai's approach.

Findings

Reconstruction of K3 surfaces from certain curves.

Application of Fourier-Mukai transforms to Brill-Noether loci.

Extension of Mukai's ideas to new genus cases.

Abstract

Let C be a general element in the locus of curves in M_g lying on some K3 surface, where g is congruent to 3 mod 4 and greater than or equal to 15. Following Mukai's ideas, we show how to reconstruct the K3 surface as a Fourier-Mukai transform of a Brill-Noether locus of rank two vector bundles on C.