Torification of diagonalizable group actions on toroidal schemes

TL;DR

This paper develops a method to convert diagonalizable group actions on toroidal schemes into toroidal actions through canonical blow-ups, ensuring the quotient remains toroidal, thus extending previous foundational results.

Contribution

It introduces a canonical torification process for arbitrary actions, generalizing earlier work to broader classes of group actions on toroidal schemes.

Findings

Canonical torification constructed for any group action

Quotients of toroidal schemes under toroidal actions are toroidal

Extends previous results of Abramovich-de Jong and others

Abstract

We study actions of diagonalizable groups on toroidal schemes (i.e. logarithmically regular logarithmic schemes). In particular, we show that for so-called toroidal actions the quotient is again a toroidal scheme. Our main result constructs for an arbitrary action a canonical torification - making the action toridal after an equivariant blowings up. This extends earlier results of Abramovich-de Jong, Abramovich-Karu-Matsuki-W{\l}odarczyk, and Gabber in various aspects.