Maximally Frobenius-destabilized vector bundles over smooth algebraic curves

TL;DR

This paper investigates vector bundles over algebraic curves in positive characteristic that are maximally destabilized by the Frobenius morphism, establishing their existence, uniqueness, and applications to ample vector bundles.

Contribution

It introduces the concept of maximally Frobenius-destabilized vector bundles, proves their existence and uniqueness in characteristic 3, and applies these findings to the study of ample vector bundles.

Findings

Existence of rank 2 maximally Frobenius-destabilized bundles in characteristic 3

Uniqueness of such bundles up to line bundle twisting

Application to the theory of ample vector bundles in all characteristics

Abstract

Vector bundles in positive characteristics have a tendency to be destabilized after pulling back by the Frobenius morphism. In this paper, we closely examine vector bundles over curves that are, in an appropriate sense, maximally destabilized by the Frobenius morphism. Then we prove that such bundles of rank 2 exist over any curve in characteristic 3, and are unique up to twisting by a line bundle. We also give an application of such bundles to the study of ample vector bundles, which is valid in all characteristics.