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

Abstract

In this paper we describe the Frobenius pull-backs of the syzygy bundles $Sy{z}_C(X^a,Y^a,Z^a)$, $a\ge 1$, on the projective Fermat curve C of degree n in characteristics coprime to n, either by giving their strong Harder-Narasimhan Filtration if $Sy{z}_C(X^a,Y^a,Z^a)$ 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.

References in corpus (1)

• Semistability and Hilbert-Kunz multiplicities for curves