Hecke cycles on moduli of vector bundles and orbital degeneracy loci

TL;DR

This paper explores the geometric and representation-theoretic properties of moduli spaces of vector bundles on genus two curves, constructing Fano manifolds as orbital degeneracy loci and relating them to Hecke lines and K3 surfaces.

Contribution

It introduces a novel construction of Fano manifolds in Grassmannians from trivectors, linking them to moduli spaces of vector bundles and their geometric features.

Findings

Constructed Fano manifolds as orbital degeneracy loci from trivectors.

Established isomorphism between these manifolds and moduli spaces of stable vector bundles.

Identified K3 surfaces as intersections with Grassmannian translates.

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 $GL_9$, 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 $19$.