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 Syz(C)(X-a, Y-a, Z(a)), a >= 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(P2 vertical bar C) of arbitrary length and a problem of Brenner regarding primes with strongly semistable reduction.