Frobenius inverse image of the semi-stable boundary in the moduli space of vector bundles

TL;DR

This paper investigates the behavior of rank 2 vector bundles with trivial determinant under Frobenius pullback on general curves, providing a scheme-theoretic description of the inverse image of the semi-stable boundary in the moduli space.

Contribution

It offers a novel scheme-theoretic description of the Frobenius inverse image of the semi-stable boundary for certain vector bundles on genus 2 curves.

Findings

Frobenius pullback of certain bundles is non-stable on general curves.

Application of theta divisor results to describe inverse images.

Explicit description of the semi-stable boundary in genus 2 case.

Abstract

We study stable rank 2 vector bundles with trivial determinant whose Frobenius pull back is non stable over a general curve of genus g>1. In genus 2, we apply recent results about the theta divisor associated to the bundle B of locally exact differential forms and we derive a scheme-theoretic description of the Frobenius inverse image of the semi-stable boundary of the moduli space.