The hyperelliptic theta map and osculating projections

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