THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

Abstract

AbstractLetCbe 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 onCwith 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.