An explicit example of Frobenius periodicity

TL;DR

This paper demonstrates a specific Frobenius periodicity for the restriction of the cotangent bundle on Fermat curves in certain characteristics, revealing explicit examples of Frobenius periodicity in algebraic geometry.

Contribution

It provides an explicit example of Frobenius periodicity for vector bundles on Fermat curves in characteristic p, expanding understanding of Frobenius behavior in algebraic geometry.

Findings

Frobenius pull-back of the bundle is isomorphic to the bundle itself up to tensoring.

The result applies to Fermat curves of degree 2d in characteristic p ≡ -1 mod 2d.

Establishes a concrete example of Frobenius periodicity for a class of vector bundles.

Abstract

In this note we show that the restriction of the cotangent bundle $\Omega_{\mathbb P}^2$ of the projective plane to a Fermat curve $C$ of degree $d$ in characteristic $p \equiv -1 \mod 2d$ is, up to tensoration with a certain line bundle, isomorphic to its Frobenius pull-back. This leads to a Frobenius periodicity $F^*({\mathcal E}) \cong {\mathcal E} $ on the Fermat curve of degree 2d, where ${\mathcal E}= {\rm Syz}(U^2,V^2,W^2)(3)$.