Nash blowups of toric varieties in prime characteristic

TL;DR

This paper explores Nash blowups of toric varieties over fields of prime characteristic, demonstrating their desingularization properties and providing combinatorial descriptions.

Contribution

It introduces a prime characteristic version of the logarithmic Jacobian ideal and links its blowup to Nash blowups, advancing understanding of singularity resolution in this setting.

Findings

Normalized Nash blowups desingularize normal toric surfaces

Prime characteristic logarithmic Jacobian ideal's blowup equals Nash blowup

Nash blowup of any toric variety can be described combinatorially

Abstract

We initiate the study of the resolution of singularities properties of Nash blowups over fields of prime characteristic. We prove that the iteration of normalized Nash blowups desingularizes normal toric surfaces. We also introduce a prime characteristic version of the logarithmic Jacobian ideal of a toric variety and prove that its blowup coincides with the Nash blowup of the variety. As a consequence, the Nash blowup of a, not necessarily normal, toric variety of arbitrary dimension in prime characteristic can be described combinatorially.