The Coble quadric

Abstract

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 four-form in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover $\operatorname {\mathrm {SU}}_C(2,L)$ , 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 $\operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}(p))$ in $G(2,8)$ . We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of $\operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}(p))$ and thus deserves to be coined the Coble quadric of the pointed curve $(C,p)$ .