Pencils of quadrics and hyperelliptic curves

Abstract: Connections between the complex geometrry of a hyperelliptic curve $C$ and the internal geometry of the base locus of the associated pencil of quadrics are classical and trace back to Andre Weil. There is a rational description of of the moduli space of rank 2 stable bundles with odd determinant on a  smooth hyperelliptic curve $C$ of genus $g$ in terms of the Grassmannian of $g-1$ dimensional linear subspaces contained in the base locus of the associated pencil of quadrics due to Ramanan. We explain a twist of this construction which leads to connections between period index bounds for the unramified Brauer classes on $K(C)$, $K$ being a totally imaginary number field and the existence  of rational points on the Grasmannians in the associated pencil of quadrics.