Toric Stacks I: The Theory of Stacky Fans

The purpose of this paper and its sequel (Toric Stacks II) 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. In this paper, we define a \emph{toric stack} as a quotient of a toric variety by a subgroup of its torus (we also define a generically stacky version). Any toric stack arises from a combinatorial gadget called a \emph{stacky fan}. We develop a dictionary between the combinatorics of stacky fans and the geometry of toric stacks, stressing stacky phenomena such as canonical stacks and good moduli space morphisms. We also show that smooth toric stacks carry a moduli interpretation extending the usual moduli interpretations of $\mathbb{P}^n$ and $[\mathbb{A}^1/\mathbb{G}_m]$. Indeed, smooth toric stacks precisely solve moduli problems specified by (generalized) effective Cartier divisors with given linear relations and given intersection relations. Smooth toric stacks therefore form a natural closure to the class of moduli problems introduced for smooth toric varieties and smooth toric DM stacks in [Cox95] and [Per08], respectively. We include a plethora of examples to illustrate the general theory. We hope that this theory of toric stacks can serve as a companion to an introduction to stacks, in much the same way that toric varieties can serve as a companion to an introduction to schemes.Comment: 36 pages (update to match published version

Formal GAGA for good moduli spaces

We prove formal GAGA for good moduli space morphisms under an assumption of "enough vector bundles" (which holds for instance for quotient stacks). This supports the philosophy that though they are non-separated, good moduli space morphisms largely behave like proper morphisms.Comment: 16 pages (updated to match published numbering

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.Comment: 20 pages (update to match published version

There is no degree map for 0-cycles on Artin stacks

We show that there is no way to define degrees of 0-cycles on Artin stacks with proper good moduli spaces so that (i) the degree of an ordinary point is non-zero, and (ii) degrees are compatible with closed immersions.Comment: 3 page

When is a variety a quotient of a smooth variety by a finite group?

Non UBCUnreviewedAuthor affiliation: California Institute of TechnologyPostdoctora

Toric Stacks

The first purpose of this dissertation is to introduce and develop a theory of toric stacks which encompasses and extends the notions of toric stacks defined by Lafforgue, Borisov-Chen-Smith, Fantechi-Mann-Nironi, Iwanari, and Satriano, as well as classical toric varieties. In addition to introducing a broader class of smooth toric stacks, the definition we introduce allows singularities. The second purpose is to characterize toric stacks in a "bottom up" fashion, similar to the treatment of smooth toric Deligne-Mumford stacks by Fantechi-Mann-Nironi and the characterization of toric varieties as "abstract toric schemes" which are reduced, separated, and normal