Finite torsors over strongly $F$-regular singularities

Carvajal-Rojas, Javier

Épijournal de Géométrie Algébrique, Tome 6 (2022)

Résumé

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.

Détail

DOI : 10.46298/epiga.2022.7532

Classification : 13A35, 13A50, 14F35, 14G17, 14L15, 16T05