A motivic change of variables formula for Artin stacks

Satriano, Matthew; Usatine, Jeremy

doi:10.48550/arxiv.2109.09800

Cited by 3 publications

(5 citation statements)

References 11 publications

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“…Namely, we prove a motivic change of variables formula for stacks that are: (i) singular, (ii) locally of finite type, and (iii) non-equidimensional, see Theorem 1.2. This generalizes our previous work [SU21] where we proved a motivic change of variables formula for smooth stacks. Ultimately, we prove our main theorem (Theorem 1.2) by reducing to the smooth case; thus, the current work relies on [SU21] rather than supersede it.…”

supporting

confidence: 87%

“…This generalizes our previous work [SU21] where we proved a motivic change of variables formula for smooth stacks. Ultimately, we prove our main theorem (Theorem 1.2) by reducing to the smooth case; thus, the current work relies on [SU21] rather than supersede it.…”

supporting

confidence: 87%

“…It is straightforward from the definitions and [AHR23] to show that if p : X → Y is any morphism to an algebraic space (e.g., a good moduli space map), then there is a natural map p : W (X ) → Y, see Section 5. By Theorem 1.8, we may immediately apply the motivic change of variables formula from [SU21,SU23b] to p. We now explain why the change of variables formula applied to p, as opposed to p, behaves particularly nicely in light of the non-properness of X . The map p induces a morphism of arc stacks…”

Section: Introductionmentioning

confidence: 97%

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Stringy invariants and toric Artin stacks

Satriano

Usatine

2022

Forum of Mathematics, Sigma

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We propose a conjectural framework for computing Gorenstein measures and stringy Hodge numbers in terms of motivic integration over arcs of smooth Artin stacks, and we verify this framework in the case of fantastacks, which are certain toric Artin stacks that provide (nonseparated) resolutions of singularities for toric varieties. Specifically, let $\mathcal {X}$ be a smooth Artin stack admitting a good moduli space $\pi : \mathcal {X} \to X$ , and assume that X is a variety with log-terminal singularities, $\pi $ induces an isomorphism over a nonempty open subset of X and the exceptional locus of $\pi $ has codimension at least $2$ . We conjecture a change-of-variables formula relating the motivic measure for $\mathcal {X}$ to the Gorenstein measure for X and functions measuring the degree to which $\pi $ is nonseparated. We also conjecture that if the stabilisers of $\mathcal {X}$ are special groups in the sense of Serre, then almost all arcs of X lift to arcs of $\mathcal {X}$ , and we explain how in this case (assuming a finiteness hypothesis satisfied by fantastacks) our conjectures imply a formula for the stringy Hodge numbers of X in terms of a certain motivic integral over the arcs of $\mathcal {X}$ . We prove these conjectures in the case where $\mathcal {X}$ is a fantastack.

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supporting

confidence: 87%

supporting

confidence: 87%

Section: Introductionmentioning

confidence: 97%

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Stringy invariants and toric Artin stacks

Satriano

Usatine

2022

Forum of Mathematics, Sigma

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“…For algebraically closed fields of characteristic zero, Satriano and Usatine initiated an investigation for a method to study stringy Hodge numbers of a singular variety using motivic integration for Artin stacks in [SU1,SU2]. To address p-adic integration, an analogous framework for p-adic integration on Artin stacks needs to be developed.…”

Section: Introductionmentioning

confidence: 99%

Desingularization of binomial varieties using toric Artin stacks

Abramovich¹,

Schober²

2022

Preprint

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We show how the notion of fantastacks can be used to effectively desingularize binomial varieties defined over algebraically closed fields. In contrast to a desingularization via blow-ups in smooth centers, we drastically reduce the number of steps and the number of charts appearing along the process. Furthermore, we discuss how our considerations extend to a partial simultaneous normal crossings desingularization of finitely many binomial hypersurfaces.

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“…Nevertheless, we expect Theorem A to suffice for many computations in algebraic geometry that necessitate logarithmic resolution. For example, a motivic change of variables formula for Artin stacks, that is applicable to the context of Theorem D, was very recently developed by Satriano-Usatine [SU21].…”

Section: Introductionmentioning

confidence: 99%

Logarithmic resolution via multi-weighted blow-ups

Abramovich¹,

Quek²

2021

Preprint

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We first introduce and study the notion of multi-weighted blow-ups, which is later used to systematically construct an explicit yet efficient algorithm for functorial logarithmic resolution in characteristic zero, in the sense of Hironaka.Specifically, for a singular, reduced closed subscheme X of a smooth scheme Y over a field of characteristic zero, we resolve the singularities of X by taking proper transforms Xi ⊂ Yi along a sequence of multiweighted blow-ups YN → YN−1 → • • • → Y0 = Y which satisfies the following properties:(i) the Yi are smooth Artin stacks with simple normal crossing exceptional loci; (ii) at each step we always blow up the worst singular locus of Xi, and witness on Xi+1 an immediate improvement in singularities; (iii) and finally, the singular locus of X is transformed into a simple normal crossing divisor on XN .

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A motivic change of variables formula for Artin stacks

Cited by 3 publications

References 11 publications

Stringy invariants and toric Artin stacks

Stringy invariants and toric Artin stacks

Desingularization of binomial varieties using toric Artin stacks

Logarithmic resolution via multi-weighted blow-ups

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