Opers of higher types, Quot-schemes and Frobenius instability loci

TL;DR

This paper explores the relationship between opers of higher types, Quot-schemes, and Frobenius instability loci in the moduli space of semi-stable vector bundles over algebraic curves in characteristic p, proposing conjectures for their structure and dimensions.

Contribution

It introduces a conjectural generalization linking higher type opers to Quot-schemes of Frobenius direct images and proposes a formula for the dimension of the Frobenius instability locus.

Findings

Identified maximal Frobenius instability strata with opers of type 1.

Proposed a conjectural correspondence between higher type opers and Quot-schemes.

Suggested a formula for the dimension of the Frobenius instability locus.

Abstract

In this paper we continue our study of the Frobenius instability locus in the coarse moduli space of semi-stable vector bundles of rank $r$ and degree $0$ over a smooth projective curve defined over an algebraically closed field of characteristic $p>0$. In a previous paper we identified the "maximal" Frobenius instability strata with opers (more precisely as opers of type $1$ in the terminology of the present paper) and related them to certain Quot-schemes of Frobenius direct images of line bundles. The main aim of this paper is to describe for any integer $q \geq 1$ a conjectural generalization of this correspondence between opers of type $q$ (which we introduce here) and Quot-schemes of Frobenius direct images of vector bundles of rank $q$. We also give a conjectural formula for the dimension of the Frobenius instability locus.