Vector bundles, dualities, and classical geometry on a curve of genus two

Abstract

Let $C$ be a curve of genus two. We denote by $SU_C(3)$ the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over $C$, and by $J^d$ the variety of line bundles of degree $d$ on $C$. In particular, $J^1$ has a canonical theta divisor $\Theta$. The space $SU_C(3)$ is a double cover of $P^8=|3\Theta|$ branched along a sextic hypersurface, the Coble sextic. In the dual $\check{P}^8=|3\Theta|^*$, where $J^1$ is embedded, there is a unique cubic hypersurface singular along $J^1$, the Coble cubic. We prove that these two hypersurfaces are dual, inducing a non-abelian Torelli result. Moreover, by looking at some special linear sections of these hypersurfaces, we can observe and reinterpret some classical results of algebraic geometry in a context of vector bundles: the duality of the Segre-Igusa quartic with the Segre cubic, the symmetric configuration of 15 lines and 15 points, the Weddle quartic surface and the Kummer surface.