On vector bundles destabilized by Frobenius pull-back

TL;DR

This paper investigates the stratification of the moduli space of vector bundles on algebraic curves in positive characteristic, focusing on bundles destabilized by Frobenius pull-back, providing classifications, dimension counts, and connections to opers.

Contribution

It offers a complete classification of Frobenius destabilized rank two bundles in characteristic two and explores the structure and dimensions of Frobenius strata, including new constructions and connections to opers.

Findings

Frobenius destabilized bundles form a locus of dimension 3g-4 in rank two cases.

Complete classification of rank two semi-stable bundles destabilized by Frobenius in characteristic two.

Construction of Frobenius destabilized bundles in specific ranks and characteristics.

Abstract

Let X be an irreducible smooth projective curve of genus at least two over an algebraically closed field k of characteristic p>0. In this paper we study the natural stratification, defined using the absolute Frobenius of X, on the moduli space of vector bundles on X of suitable rank. In characteristic two we provide a complete classification of rank two semi-stable vector bundles whose Frobenius pull-back is not semi-stable. We also obtain fairly good information about the strata of the Frobenius stratification, including the irreducibility and the dimension of each non-empty Frobenius stratum. In particular we show that the locus of Frobenius destabilized bundles has dimension 3g-4 in the moduli space of semi-stable bundles of rank two. We also construct stable bundles that are destabilized by Frobenius in the following situations: characteristic p=2 and rank four, (2) characteristic p=rank=3, (3) characteristic p=rank=5 and g at least three. We also explore (in any characteristic) the connection between Frobenius destabilized bundles and (pre)-opers, this approach allows us to reinterpret some of our results in terms of pre-opers and also allows us to construct Frobenius destablised bundles from certain pre-opers (or opers). The other result we obtain is (for characteristic two): we show that the Gunning bundle descends under Frobenius when genus g is even. If g is odd, then the Gunning bundle twisted by any odd degree line bundle also descends.