Vector bundles on genus 2 curves and trivectors

TL;DR

This paper explores the relationship between vector bundles on genus 2 curves and invariant theory of trivectors, extending known complex-analytic results to arbitrary fields and highlighting new algebraic connections.

Contribution

It generalizes the connection between moduli spaces of bundles on genus 2 curves and invariant theory of trivectors to arbitrary fields.

Findings

Extended the known relationship to arbitrary fields.

Identified new links between vector bundles and invariant theory.

Laid groundwork for further in-depth study in follow-up work.

Abstract

Given a complex curve C of genus 2, there is a well-known relationship between the moduli space of rank 3 semistable bundles on C and a cubic hypersurface known as the Coble cubic. Some of the aspects of this is known to be related to the geometric invariant theory of the third exterior power of a 9-dimensional complex vector space. We extend this relationship to arbitrary fields and study some of the connections to invariant theory, which will be studied more in-depth in a followup paper.