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arXiv · 1410.0872

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

Daniel Brinkmann·Almar Kaid
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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 $Ω_{P^2}|_C$ of arbitrary length and a problem of Brenner regarding primes with strongly semistable reduction.

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Daniel Brinkmann, Almar Kaid. 2014-10-03. Rank-2 syzygy bundles on Fermat curves and an application to Hilbert-Kunz functions. https://arxiv.org/abs/1410.0872

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