The Stacks project

112.5.10 Toric stacks

Toric stacks provide a great class of examples and a natural testing ground for conjectures due to the dictionary between the geometry of a toric stack and the combinatorics of its stacky fan in a similar way that toric varieties provide examples and counterexamples in scheme theory.

• Borisov, Chen and Smith: The orbifold Chow ring of toric Deligne-Mumford stacks [bcs]

  Inspired by Cox's construction for toric varieties, this paper defines smooth toric DM stacks as explicit quotient stacks associated to a combinatorial object called a stacky fan.

• Iwanari: The category of toric stacks [iwanari_toric]

  This paper defines a toric triple as a smooth Deligne-Mumford stack $\mathcal{X}$ with an open immersion $\mathbf{G}_ m \hookrightarrow \mathcal{X}$ with dense image (and therefore $\mathcal{X}$ is an orbifold) and an action $\mathcal{X} \times \mathbf{G}_ m \rightarrow \mathcal{X}$. It is shown that there is an equivalence between the 2-category of toric triples and the 1-category of stacky fans. The relationship between toric triples and the definition of smooth toric DM stacks in [bcs] is discussed.

• Iwanari: Integral Chow rings for toric stacks [iwanari_chow]

  Generalizes Cox's $\Delta $-collections for toric varieties to toric orbifolds.

• Perroni: A note on toric Deligne-Mumford stacks [perroni]

  Generalizes Cox's $\Delta $-collections and Iwanari's paper [iwanari_chow] to general smooth toric DM stacks.

• Fantechi, Mann, and Nironi: Smooth toric DM stacks [fmn]

  This paper defines a smooth toric DM stack as a smooth DM stack $\mathcal{X}$ with the action of a DM torus $\mathcal{T}$ (ie. a Picard stack isomorphic to $T \times BG$ with $G$ finite) having an open dense orbit isomorphic to $\mathcal{T}$. They give a “bottom-up description” and prove an equivalence between smooth toric DM stacks and stacky fans.

• Geraschenko and Satriano: Toric Stacks I and II [gs_toric1] and [gs_toric2]

  These papers define a toric stack as the stack quotient of a toric variety by a subgroup of its torus. A generically stacky toric stack is defined as a torus invariant substack of a toric stack. This definition encompasses and extends previous definitions of toric stacks. The first paper develops a dictionary between the combinatorics of stacky fans and the geometry of the corresponding stacks. It also gives a moduli interpretation of smooth toric stacks, generalizing the one in [perroni]. The second paper proves an intrinsic characterization of toric stacks.