On the Moduli of Orthogonal Bundles on a Nodal Hyperelliptic Curve

Let X be a complete irreducible hyperelliptic curve of arithmetic genus g with an ordinary node as its only singularity. We find explicit descriptions of the moduli spaces of rank 4 orthogonal bundles with a Z/2Z-action (and of a certain fixed topological type) on X in terms of spaces associated to the singular pencil of quadrics determined by X.