Frobenius morphisms of noncommutative blowups

TL;DR

This paper introduces a class of noncommutative blowups in positive characteristic with a flat Frobenius morphism, leading to Kunz regularity, and connects these concepts to algebraic varieties with specific singularities.

Contribution

It defines Frobenius morphisms for noncommutative blowups, proves their flatness, and demonstrates their existence for all varieties over algebraically closed fields of positive characteristic.

Findings

Noncommutative blowups in the studied class are Kunz regular due to flat Frobenius morphisms.

Existence of Kunz regular noncommutative blowups for all varieties over algebraically closed fields of positive characteristic.

Varieties with F-pure and FFRT singularities have associated Kunz regular noncommutative blowups.

Abstract

We define the Frobenius morphism of certain class of noncommutative blowups in positive characteristic. Thanks to a nice property of the class, the defined morphism is flat. Therefore we say that the noncommutative blowups in this class are Kunz regular. One of such blowups is the one associated to a regular Galois alteration. As a consequence of de Jong's theorem, we see that for every variety over an algebraically closed field of positive characteristic, there exists a noncommutative blowup which is Kunz regular. We also see that a variety with F-pure and FFRT (finite F-representation type) singularities has a Kunz regular noncommutative blowup which is associated to an iteration of the Frobenius morphism of the variety.