Toric stacks II: Intrinsic characterization of toric stacks

Geraschenko, Anton; Satriano, Matthew

doi:10.1090/s0002-9947-2014-06064-9

Cited by 23 publications

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References 21 publications

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“…• a generalization of Luna'sétale slice theorem to non-affine schemes ( §2.2); • a generalization of Sumihiro's theorem on torus actions to Deligne-Mumford stacks ( §2.3), confirming an expectation of Oprea [Opr06, §2]; • Bia lynicki-Birula decompositions for smooth Deligne-Mumford stacks ( §2.4), generalizing Skowera [Sko13]; • a criterion for the existence of a good moduli space ( §2.9), generalizing [KM97,AFS17]; • a criterion forétale-local equivalence of algebraic stacks ( §2.8), extending Artin's corresponding results for schemes [Art69a, Cor. 2.6]; • the existence of equivariant miniversal deformation spaces for curves ( §2.5), generalizing [AK16]; • a characterization of toric Artin stacks in terms of stacky fans [GS15, Thm. 6.1] ( §2.12); • theétale-local quotient structure of a good moduli space ( §2.6); Date: Mar 18, 2019.…”

Section: Introductionmentioning

confidence: 99%

A Luna étale slice theorem for algebraic stacks

2020

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Section: Introductionmentioning

confidence: 99%

A Luna étale slice theorem for algebraic stacks

2020

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“…• 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.…”

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confidence: 84%

“…This follows from the main theorem of [GS11b], stated below. See [GS11b,Remark 6.2] for more details.…”

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confidence: 84%

“…This paper develops a rich dictionary between the combinatorics of stacky fans and the geometry of toric stacks. In contrast, [GS11b] focuses on how to show a given stack is toric (and so amenable to the combinatorial anlaysis of stacky fans). A classical result (see for example [CLS11, Corollary 3.1.8]) shows that if X is a finite type scheme with a dense open torus T whose action on itself extends to X, then X is a toric variety if and only if it is normal and separated.…”

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Toric stacks I: The theory of stacky fans

Geraschenko

Satriano

2014

Trans. Amer. Math. Soc.

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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....

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“…The notion which we will primarily use in this paper, was first introduced by Tyomkin [Tyo12] as a generalization of Borisov-Chen-Smith's construction. The work of Geraschenko and Satriano (see [GS15a] and [GS15b]) provides an extensive description of several notions of toric stacks. In addition, they develop a theory of toric stacks that encompasses, among other theories, the work of both Borisov-Chen-Smith as well as Tyomkin.…”

Section: Toric Stacksmentioning

confidence: 99%

Logarithmic stable toric varieties and their moduli

Ascher¹,

Molcho²

2016

Alg. Geom.

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The Chow quotient of a toric variety by a subtorus, as defined by Kapranov-Sturmfels-Zelevinsky, coarsely represents the main component of the moduli space of stable toric varieties with a map to a fixed projective toric variety, as constructed by Alexeev and Brion. We show that, after endowing both spaces with the structure of a logarithmic stack, the spaces are isomorphic. Along the way, we construct the Chow quotient stack and demonstrate several properties it satisfies.

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Toric stacks II: Intrinsic characterization of toric stacks

Cited by 23 publications

References 21 publications

A Luna étale slice theorem for algebraic stacks

A Luna étale slice theorem for algebraic stacks

Toric stacks I: The theory of stacky fans

Logarithmic stable toric varieties and their moduli

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