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 P⁸ 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 SUC(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∈ C defines a natural embedding of SUC(2,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 SUC(2,O(p)) and thus deserves to be coined the Coble quadric of the pointed curve $(C,p)$ .