An explicit computation of a family of trivialising \'etale covers

TL;DR

This paper explicitly computes certain étale covers of Fermat curves that trivialize specific vector bundles, providing concrete examples in algebraic geometry over fields of characteristic p.

Contribution

It presents explicit calculations of étale covers for Fermat curves that trivialize particular vector bundles, advancing understanding of vector bundle trivializations in positive characteristic.

Findings

Explicit étale covers for Fermat curves are constructed.

These covers trivialize the specified vector bundles.

The work provides concrete examples in algebraic geometry.

Abstract

We explicitly compute \'etale covers of the smooth Fermat curves $Y_{p+1} = \Proj k[u,v,w]/(u^{p+1} + v^{p+1} - w^{p+1})$ which trivialise the vector bundles $\Syz(u^2, v^2, w^2)(3)$, where $k$ is a field of characteristic $p \geq 3$.