Birational geometry of Beauville-Mukai systems I: the rank three and genus two case

TL;DR

This paper investigates the birational geometry of rank three Beauville-Mukai systems on genus two K3 surfaces, revealing their relation to Hilbert schemes via flops and describing associated Brill-Noether loci.

Contribution

It establishes a connection between Beauville-Mukai systems and Hilbert schemes through explicit flop sequences and characterizes the exceptional loci as Brill-Noether loci.

Findings

Relation between Beauville-Mukai systems and Hilbert schemes via flops

Description of exceptional loci as Brill-Noether loci

Brill-Noether type results for sheaves in the system

Abstract

We study wall-crossing for the Beauville-Mukai system of rank three on a general genus two K3 surface. We show that such a system is related to the Hilbert scheme of ten points on the surface by a sequence of flops, whose exceptional loci can be described as Brill-Noether loci. We also obtain Brill-Noether type results for sheaves in the Beauville-Mukai system.