\'Etale fundamental groups of strongly $F$-regular schemes

TL;DR

This paper proves that strongly F-regular schemes have a special finite cover that aligns their étale fundamental groups with those of their regular locus, extending the understanding of their algebraic and geometric structure.

Contribution

It establishes the existence of a finite, generically Galois, étale-in-codimension-one cover for strongly F-regular schemes, aligning their étale fundamental groups with those of their regular parts.

Findings

Existence of a finite, étale-in-codimension-one cover for strongly F-regular schemes.

The étale fundamental groups of the cover and its regular locus agree.

Extension of finite étale covers from the regular locus to the entire scheme.

Abstract

We prove that a strongly $F$-regular scheme $X$ admits a finite, generically Galois, and \'etale-in-codimension-one cover $\widetilde X \to X$ such that the \'etale fundamental groups of $\widetilde X$ and $\widetilde X_{reg}$ agree. Equivalently, every finite \'etale cover of $\widetilde X_{reg}$ extends to a finite \'etale cover of $\widetilde X$. This is analogous to a result for complex klt varieties by Greb, Kebekus and Peternell.