Hecke cycles on moduli of vector bundles and orbital degeneracy loci

Abstract

Given a smooth genus two curve $C$, the moduli space SU${}_C(3)$ of rank three semi-stable vector bundles on $C$ with trivial determinant is a double cover in ${\mathbb{P}}^8$ branched over a sextic hypersurface, whose projective dual is the famous Coble cubic, the unique cubic hypersurface that is singular along the Jacobian of $C$. In this paper we continue our exploration of the connections of such moduli spaces with the representation theory of $G{L}_9$, initiated in \cite{GSW} and pursued in \cite{GS, sam-rains1, sam-rains2, bmt}. Starting from a general trivector $v$ in ${\wedge }^3{\mathbb{C}}^9$, we construct a Fano manifold $D_{Z_{10}}(v)$ in $G(3,9)$ as a so-called orbital degeneracy locus, and we prove that it defines a family of Hecke lines in SU${}_C(3)$. We deduce that $D_{Z_{10}}(v)$ is isomorphic to the odd moduli space SU${}_C(3,{\mathcal{O}}_C(c))$ of rank three stable vector bundles on $C$ with fixed effective determinant of degree one. We deduce that the intersection of $D_{Z_{10}}(v)$ with a general translate of $G(3,7)$ in $G(3,9)$ is a K3 surface of genus $19$.

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