Partial desingularizations of good moduli spaces of Artin toric stacks

TL;DR

This paper introduces Reichstein transforms as birational modifications of Artin stacks, demonstrating that they preserve the toric structure and can produce Deligne-Mumford stacks through a canonical sequence, linking to Kirwan's desingularization.

Contribution

It establishes that Reichstein transforms of Artin toric stacks remain toric and provides a canonical process to obtain Deligne-Mumford stacks, connecting to existing desingularization methods.

Findings

Reichstein transforms preserve the Artin toric stack structure.

A canonical sequence of transforms yields a Deligne-Mumford toric stack.

The procedure relates to Kirwan's partial desingularization in the projective case.

Abstract

We define Reichstein transforms to be certain birational transformations of Artin stacks with good moduli spaces. Our main technical result is that the Reichstein transform of an Artin toric stack is again an Artin toric stack. This leads to our main theorem which states that for Artin toric stacks there is a canonical sequence of Reichstein transforms which produces a Deligne-Mumford toric stack. When the good moduli space of an Artin toric stack is projective our procedure can be interpreted in terms of Kirwan's partial desingularization of geometric invariant theory quotients.