Birational geometry of the Mukai system of rank two and genus two

TL;DR

This paper analyzes the birational geometry of the Mukai system of rank two on genus two K3 surfaces, identifying wall structures, describing birational maps, and interpreting exceptional loci via Brill--Noether theory.

Contribution

It determines the walls in the movable cone and decomposes the birational map to the Hilbert scheme into flops, providing new geometric insights.

Findings

Identified walls in the movable cone for the Mukai system.

Decomposed the birational map into a sequence of flops.

Interpreted exceptional loci using Brill--Noether loci.

Abstract

Using the techniques of Bayer--Macr\`i, we determine the walls in the movable cone of the Mukai system of rank two for a general K3 surface $S$ of genus two. We study the (essentially unique) birational map to $S^{[5]}$ and decompose it into a sequence of flops. We give an interpretation of the exceptional loci in terms of Brill--Noether loci.