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
$(\mathbb {P}^1)^{2g}//\text {PGL(2)}$
. Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer
$(g-1)$
-varieties over
$\mathbb {P}^g$
inside the ramification locus of the theta map.
Invariant vector bundles of rank 2 on hyperelliptic curves.