Rank- $$2$$ 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 $${{\mathrm{Syz}}}_C(X^a,Y^a,Z^a)$$ , $$a\ge 1$$ , on the projective Fermat curve $$C$$ of degree $$n$$ in characteristics coprime to $$n$$ , either by giving their strong Harder–Narasimhan filtration if $${{\mathrm{Syz}}}_C(X^a,Y^a,Z^a)$$ 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 $$\Omega _{\mathbb {P}^2}|_C$$ of arbitrary length and a problem of Brenner regarding primes with strongly semistable reduction.