Stable bundles as Frobenius morphism direct image

TL;DR

This paper characterizes when the direct image of a stable bundle under the Frobenius morphism remains stable on a smooth projective curve in positive characteristic, linking stability to the bundle's origin.

Contribution

It proves a precise criterion for the stability of Frobenius direct images, establishing a direct correspondence with stable bundles on the original curve.

Findings

Stability of Frobenius direct images is characterized by the bundle being a direct image of a stable bundle.

The invariant I(E) equals (p-1)(2g-2) if and only if E is a Frobenius direct image.

The result applies to smooth projective curves of genus g ≥ 2 in characteristic p > 0.

Abstract

Let X be a smooth projective curve of genus $g\geq 2$ defined over an algebraically closed field k of characteristic $p>0$ and let $F:X\rightarrow X_{1}$ be the relative k-linear Frobenius map. We prove (Theorem 1.1) E is a stable bundle on $X_{1}$ with $I(E)= (p-1)(2g-2)$ if and only if E is the direct image of some stable bundle W on $X$.