Universal flattening of Frobenius

TL;DR

This paper introduces a sequence of blowups called F-blowups, derived from iterated Frobenius morphisms, which under certain conditions stabilize to produce resolutions like the G-Hilbert scheme for tame quotient singularities.

Contribution

It defines the concept of F-blowups for varieties in positive characteristic and explores their stabilization and resolution properties, connecting to known schemes such as the G-Hilbert scheme.

Findings

Sequence of F-blowups stabilizes under certain conditions.

F-blowups can produce minimal or crepant resolutions.

For tame quotient singularities, the sequence yields the G-Hilbert scheme.

Abstract

For a variety $X$ of positive characteristic and a non-negative integer $e$, we define its $e$-th F-blowup to be the universal flattening of the $e$-iterated Frobenius of $X$. Thus we have the sequence (a set labeled by non-negative integers) of blowups of $X$. Under some condition, the sequence stabilizes and leads to a nice (for instance, minimal or crepant) resolution. For tame quotient singularities, the sequence leads to the $G$-Hilbert scheme.