Rank Four Symplectic Bundles Without Theta Divisors Over a Curve of Genus Two

Abstract

The moduli space M 2 of rank four semistable symplectic vector bundles over a curve X of genus two is an irreducible projective variety of dimension ten. Its Picard group is generated by the determinantal line bundle Ξ. The base locus of the linear system |Ξ| consists of precisely those bundles without theta divisors, that is, admitting nonzero maps from every line bundle of degree −1 over X. We show that this base locus consists of six distinct points, which are in canonical bijection with the Weierstrass points of the curve. We relate our construction of these bundles to another of Raynaud and Beauville using Fourier-Mukai transforms. As an application, we prove that the map sending a symplectic vector bundle to its theta divisor is a surjective map from M 2 to the space of even 4Θ divisors on the Jacobian variety of the curve.