The Coble Quadric

TL;DR

This paper explores the geometric construction of the Coble quartic and introduces the Coble quadric, linking moduli spaces of vector bundles on genus three curves to special quadrics in Grassmannians.

Contribution

It generalizes the construction of the Coble quartic to a new quadratic hypersurface, the Coble quadric, associated with pointed genus three curves and their vector bundle moduli spaces.

Findings

The Coble quadric is uniquely associated with the moduli space SU$_C(2, ext{O}(p))$.

Each point on the curve defines a natural embedding of the moduli space into a Grassmannian.

The Coble quadric is singular exactly along the embedded moduli space for generic points.

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 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$_C(2, \mathcal{O}(p))$ 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 SU$_C(2, \mathcal{O}(p))$, and thus deserves to be coined the Coble quadric of the pointed curve $(C, p)$.