Rank-2 syzygy bundles on Fermat curves and an application to Hilbert-Kunz functions

TL;DR

This paper investigates the Frobenius pull-backs of certain syzygy bundles on Fermat curves, classifies their stability properties, and applies these results to compute Hilbert-Kunz functions and Frobenius periodicities.

Contribution

It provides a detailed description of Frobenius pull-backs of syzygy bundles on Fermat curves and applies this to problems in Hilbert-Kunz theory and stability of vector bundles.

Findings

Frobenius pull-backs classified by Harder-Narasimhan filtration or periodicity

Determination of Frobenius periodicities of the restricted cotangent bundle

Insights into primes with strongly semistable reduction

Abstract

In this paper we describe the Frobenius pull-backs of the syzygy bundles $Syz_C(X^a, Y^a, Z^a)$, $a \geq 1$, on the projective Fermat curve C of degree n in characteristics coprime to n, either by giving their strong Harder-Narasimhan Filtration if $Syz_C(X^a, Y^a, Z^a)$ is not strongly semistable or in the strongly semistable case by their periodicity behavior. Moreover, we apply these results to Hilbert-Kunz functions, to find Frobenius periodicities of the restricted cotangent bundle $\Omega_{P^2}|_C$ of arbitrary length and a problem of Brenner regarding primes with strongly semistable reduction.