Abstract
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
$(C,p)$
.
Publisher
Cambridge University Press (CUP)