# The hyperelliptic theta map and osculating projections

## TL;DR

This paper provides a new geometric perspective on the theta map for moduli spaces of rank 2 semistable vector bundles on hyperelliptic curves, linking it to osculating projections and Kummer varieties.

## Contribution

It introduces a geometric description of the theta map via a fibration with fibers as GIT quotients and relates it to explicit osculating projections.

## Key findings

- Identification of the theta map restriction with degree two osculating projections
- Construction of a birational inclusion of Kummer $(g-1)$-folds into the ramification locus
- Description of a fibration of the moduli space with GIT quotient fibers

## Abstract

Let $C$ be a hyperelliptic curve of genus $g\geq 3$. In this paper we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on $C$ with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb{P}^1)^{2g}//PGL(2)$. Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree two osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$-folds over $\mathbb{P}^g$ inside the ramification locus of the theta map.