Finite torsors over strongly $F$-regular singularities

TL;DR

This paper studies finite torsors over strongly F-regular singularities, establishing conditions for their extension, and explores implications for class groups and Picard groups in algebraic geometry.

Contribution

It proves the existence of finite covers where torsors extend and derives new transformation rules for the F-signature, impacting the understanding of class and Picard groups.

Findings

Existence of finite local covers with torsor extension properties

A generalized transformation rule for F-signature under finite extensions

Bound on the torsion of the class group related to F-signature

Abstract

We investigate finite torsors over big opens of spectra of strongly $F$-regular germs that do not extend to torsors over the whole spectrum. Let $(R,\mathfrak{m},k)$ be a strongly $F$-regular $k$-germ where $k$ is an algebraically closed field of characteristic $p>0$. We prove the existence of a finite local cover $R \subset R^{\star}$ so that $R^{\star}$ is a strongly $F$-regular $k$-germ and: for all finite algebraic groups $G/k$ with solvable neutral component, every $G$-torsor over a big open of $\mathrm{Spec} R^{\star}$ extends to a $G$-torsor everywhere. To achieve this, we obtain a generalized transformation rule for the $F$-signature under finite local extensions. Such formula is used to show that that the torsion of $\mathrm{Cl} R$ is bounded by $1/s(R)$. By taking cones, we conclude that the Picard group of globally $F$-regular varieties is torsion-free. Likewise, it shows that canonical covers of $\mathbb{Q}$-Gorenstein strongly $F$-regular singularities are strongly $F$-regular.