The purpose of this paper and its sequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature, as well as classical toric varieties.In this paper, we define a toric stack as the stack 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 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 P n and [A 1 /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 papers by Cox and Perroni, 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.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use TORIC STACKS I: THE THEORY OF STACKY FANS 1035This definition encompasses and extends the three kinds of toric stacks listed above:• Taking G to be trivial, we see that any toric variety X is a toric stack. • Smooth toric Deligne-Mumford stacks in the sense of [BCS05, FMN10, Iwa09] are smooth non-strict toric stacks which happen to be separated and Deligne-Mumford. See Remarks 2.17 and 2.18. • Toric stacks in the sense of [Laf02] are toric stacks that have a dense open point (i.e. toric stacks for which G = T 0 ). • A toric Artin stack in the sense of [Sat12] is a smooth non-strict toric stack which has finite generic stabilizer and which has a toric variety of the same dimension as a good moduli space. See Sections 4 and 6. • Toric stacks in the sense of [Tyo12] are toric stacks as well. This follows from the main theorem of [GS11b], stated below. See [GS11b, Remark 6.2]for more details.In this notation, the stack in Example 2.7 would be denoted [A 2 / ( 1 1 ) μ 2 ].Example 2.9. Again we have that X Σ = A 2 . This time β * = ( 1 0 ) : Z → Z 2 , which induces the homomorphism G 2 m → G m given by (s, t) → s. Therefore,We then have that X Σ,β = [(A 2 {(0, 0)})/ ( 1 1 ) G m ] = P 1 . Warning 2.11. Examples 2.6 and 2.10 show that non-isomorphic stacky fans (see Definition 3.2) can give rise to isomorphic toric stacks. The two presentations [(A 2 {(0, 0)})/ ( 1 1 ) G m ] and [P 1 /{e}] of the same toric stack are produced by different stacky fans. In Theorem B.3, we determine when different stacky fans give rise to the same toric stack.Example 2....
Abstract. The purpose of this paper and its prequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature, 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 a finite type scheme with an action of a dense open torus arises from a fan if and only if it is normal and separated. 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.
In casual discussion, a stack X is often described as a variety X (the coarse space of X ) together with stabilizer groups attached to some of its subvarieties. However, this description does not uniquely specify X , as is illustrated by Example 13. Our main result shows that for a large class of stacks one typically encounters, this description does indeed characterize them. Moreover, we prove that each such stack can be described in terms of two simple procedures applied iteratively to its coarse space: canonical stack constructions and root stack constructions.More precisely, if X is a smooth separated tame Deligne-Mumford stack of finite type over a field k with trivial generic stabilizer, it is completely determined by its coarse space X and the ramification divisor (on X) of the coarse space morphism X → X. Therefore, to specify such a stack, it is enough to specify a variety and the orders of the stabilizers of codimension 1 points. The group structures, as well as the stabilizer groups of higher codimension points, are then determined.Recall that a k-scheme U is said to have (tame) quotient singularities if it isétale locally a quotient of a smooth variety by a finite group (of order relatively prime to the characteristic of k). The coarse space of a smooth tame Deligne-Mumford stack has tame quotient singularities. We say that an algebraic stack Y has (tame) quotient singularities if there is anétale cover U → Y, where U is a scheme with (tame) quotient singularities. As explained in the next section, we can associate to every such stack a canonical smooth stack Y can over Y. Also discussed in the next section, for an effective Cartier divisor C on a stack Z and a positive integer n, there is a universal morphism to Z which ramifies to order n over C, called the n-th root of C, denoted n C/Z (or C/Z if there is no confusion about what n is). Theorem 1. Let X be a smooth separated tame Deligne-Mumford stack of finite type over a field k with trivial generic stabilizer. Let X denote the coarse space of X , D ⊆ X the ramification divisor of the coarse space map π : X → X, and D ⊆ X can the corresponding divisor in X can . Let e i be the ramification degrees of π over the irreducible components D i of D, and let D/X can denote the root stack of X can with order e i along D i . Then D/X can has tame quotient singularities and π factors as follows:Moreover, if D is Cartier (so that D/X is defined), then D/X has tame quotient singularities and π factors as follows:Remark 2. For X with non-trivial generic stabilizer, Theorem 1 can be combined with [AOV08, Theorem A.1], which shows that X is a gerbe over a smooth stack with trivial generic stabilizer. * geraschenko@gmail.com † Supported by NSF grant DMS-0943832 and an NSF postdoctoral fellowship (DMS-1103788). Remark 4. The proof we present can be applied if k is an arbitrary regular excellent base, but we work over a field for clarity. See [Vis] for the construction of canonical stacks in this case. The remainder of the proof applies directly.
Abstract. 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.
This article is motivated by the following local‐to‐global question: is every variety with tame quotient singularities globally the quotient of a smooth variety by a finite group? We show that the answer is ‘yes’ for quasi‐projective varieties which are expressible as a quotient of a smooth variety by a split torus (for example, quasi‐projective simplicial toric varieties). Although simplicial toric varieties are rarely toric quotients of smooth varieties by finite groups, we give an explicit procedure for constructing the quotient structure using toric techniques. The main result follows from a characterization of varieties which are expressible as the quotient of a smooth variety by a split torus. As an additional application of this characterization, we show that a variety with abelian quotient singularities may fail to be a quotient of a smooth variety by a finite abelian group. Concretely, we show that double-struckP2/A5 is not expressible as a quotient of a smooth variety by a finite abelian group.
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