Canonical Artin Stacks over Log Smooth Schemes

TL;DR

This paper extends the theory of toric Artin stacks over log smooth schemes, generalizes the Chevalley-Shephard-Todd theorem for diagonalizable groups, and shows toroidal embeddings as good moduli spaces of smooth log smooth Artin stacks.

Contribution

It develops a new framework for toric Artin stacks over log smooth schemes and generalizes classical theorems to this setting.

Findings

Toric Artin stacks extend toric Deligne-Mumford stacks.

Generalization of Chevalley-Shephard-Todd theorem to diagonalizable group schemes.

Toroidal embeddings are good moduli spaces of smooth log smooth Artin stacks.

Abstract

We develop a theory of toric Artin stacks extending the theories of toric Deligne-Mumford stacks developed by Borisov-Chen-Smith, Fantechi-Mann-Nironi, and Iwanari. We also generalize the Chevalley-Shephard-Todd theorem to the case of diagonalizable group schemes. These are both applications of our main theorem which shows that a toroidal embedding X is canonically the good moduli space (in the sense of Alper) of a smooth log smooth Artin stack whose stacky structure is supported on the singular locus of X.