Toric Stacks II: Intrinsic Characterization of Toric Stacks

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.