The Coble Quadric

Given a smooth genus three curve $C$, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in $\mathbb{P}^8$ as a hypersurface whose singular locus is the Kummer threefold of $C$; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric fourform in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover SU$_C(2, L)$, the moduli space of rank two stable vector bundles on C with fixed determinant of odd degree L, as a subvariety of $G(2, 8)$. In fact, each point $p \in C$ defines a natural embedding of SU$_C(2, \mathca…