Involutions on moduli spaces of vector bundles and GIT quotients

Abstract

Let $C$ be a hyperelliptic curve of genus $g\ge 3$. 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 $({\mathbb{P}}^1{)}^{2g}//PGL(2)$. 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 ${\mathbb{P}}^g$.

17 pages, 1 figure