Canonical Artin stacks over log smooth schemes

Satriano, Matthew

doi:10.1007/s00209-012-1096-7

Cited by 8 publications

(12 citation statements)

References 15 publications

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“…Let N ′ = N/T where T is the torsion subgroup of N and let β ′ be the composite of β with the projection N → N ′ . As noted in the proof of [Sat,Proposition 5.4] there is an injection of abelian groups DG(β ′ ) → DG(β). Since both groups have the same rank the map also has maximal rank.…”

Section: 2mentioning

confidence: 91%

Partial desingularizations of good moduli spaces of Artin toric stacks

Edidin¹,

More²

2012

Michigan Math. J.

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Let X be an Artin stack with good moduli space X → M . We define the Reichstein transform of X relative to a closed substack C ⊂ X to be the complement of the strict transform of the saturation of C in the blowup of X along C. The main technical result of the paper is that the Reichstein transform of a toric Artin stack relative to a toric substack is again a toric stack. Precisely, the Reichstein transform relative to a cone in a stacky fan is the toric stack determined by stacky star subdivision. This leads to our main theorem which states that for toric Artin stacks there is a canonical sequence of Reichstein transforms that produces a toric Deligne-Mumford stack. When the good moduli space of the toric stack is a projective toric variety our procedure can be interpreted in terms of Kirwan's [Kir] partial desingularization of geometric invariant theory quotients.

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Section: 2mentioning

confidence: 91%

Partial desingularizations of good moduli spaces of Artin toric stacks

Edidin¹,

More²

2012

Michigan Math. J.

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“…This paper, together with its prequel [GS11a], introduces a theory of toric stacks which encompasses and extends the many pre-existing theories in the literature [Laf02,BCS05,FMN10,Iwa09,Sat12,Tyo12]. Recall from [GS11a, Definition 1.1] that a toric stack is defined to be the stack quotient [X/G] of a normal toric variety X by a subgroup G of the torus of X.…”

Section: Introductionmentioning

confidence: 99%

Toric stacks II: Intrinsic characterization of toric stacks

Geraschenko

Satriano

2014

Trans. Amer. Math. Soc.

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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.

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“…They are smooth and have simplicial toric varieties as their coarse moduli spaces. The second author generalized this approach in [Sat12] to include certain smooth toric Artin stacks which have toric varieties as their good moduli spaces. Toric Varieties and Singular Toric Stacks.…”

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confidence: 99%

“…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.…”

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confidence: 99%

“…Though non-strict toric stacks are considerably more general than fantastacks, it is sometimes easiest to understand a non-strict toric stack in terms of its relation to some fantastack. The stacks defined in [BCS05] and [Sat12] which have no generic stabilizer are fantastacks; those with non-trivial generic stabilizer are closed substacks of fantastacks. Section 5 is devoted to the construction of the canonical stack over a toric stack.…”

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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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Canonical Artin stacks over log smooth schemes

Cited by 8 publications

References 15 publications

Partial desingularizations of good moduli spaces of Artin toric stacks

Partial desingularizations of good moduli spaces of Artin toric stacks

Toric stacks II: Intrinsic characterization of toric stacks

Toric stacks I: The theory of stacky fans

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