The hyperelliptic theta map and osculating projections

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.