Involutions on moduli spaces of vector bundles and GIT quotients

TL;DR

This paper provides a new geometric description of the theta map for rank 2 vector bundles on hyperelliptic curves, linking it to GIT quotients and explicit projections, revealing a birational equivalence involving Kummer varieties.

Contribution

It introduces a novel fibration of the moduli space with fibers as GIT quotients and identifies the theta map's restriction with explicit osculating projections, advancing understanding of the moduli space structure.

Findings

Describes a fibration of the moduli space with GIT quotient fibers.

Identifies the theta map restriction with explicit osculating projections.

Establishes a birational equivalence involving the ramification locus and Kummer varieties.

Abstract

Let $C$ be a hyperelliptic curve of genus $g \geq 3$. We give a new description of the theta map for moduli spaces of rank 2 semistable vector bundles with trivial determinant. In orther to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb{P}^1)^{2g}//\operatorname{PGL(2)}$. Then, we use recent results of Kumar to identify the restriction of the theta map to these GIT quotients with some explicit osculating projection. As a corollary of this construction, we obtain a birational equivalence between the ramification locus of the theta map and a fibration in Kummer $(g-1)$-varieties over $\mathbb{P}^g$.