Toric Stacks II: Intrinsic Characterization of Toric Stacks

TL;DR

This paper provides an intrinsic characterization of toric stacks, showing that under certain conditions, any Artin stack with a dense torus action can be described by a stacky fan, extending classical toric variety theory.

Contribution

It establishes that all Artin stacks with a dense open torus action are equivalent to those arising from stacky fans, generalizing previous results for smooth DM stacks.

Findings

Any Artin stack with a dense torus action arises from a stacky fan.

Extends classical toric variety characterization to broader stack contexts.

Provides criteria for recognizing toric stacks intrinsically.

Abstract

The purpose of this paper and its prequel (Toric Stacks I) is to introduce and develop a theory of toric stacks which encompasses and extends the notions of toric stacks defined in [Laf02, BCS05, FMN10, Iwa09, Sat12, Tyo12], as well as classical toric varieties. While the focus of the prequel is on how to work with toric stacks, the focus of this paper is how to show a stack is toric. For toric varieties, a classical result says that any normal variety with an action of a dense open torus arises from a fan. In [FMN09, Theorem 7.24], it is shown that a smooth separated DM stack with an action of a dense open stacky torus arises from a stacky fan. In the same spirit, the main result of this paper is that any Artin stack with an action of a dense open torus arises from a stacky fan under reasonable hypotheses.