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Canonical reduction of stabilizers for Artin stacks with good moduli spaces. (English) Zbl 1479.14020

Summary: We present a complete generalization of Kirwan’s partial desingularization theorem on quotients of smooth varieties. Precisely, we prove that if \(\mathcal{X}\) is an irreducible Artin stack with stable good moduli space \(\mathcal{X}\stackrel{\pi }{\to }\mathbf{X} \), then there is a canonical sequence of birational morphisms of Artin stacks \({\mathcal{X}_n}\to{\mathcal{X}_{n-1}}\to \cdots \to{\mathcal{X}_0}=\mathcal{X}\) with the following properties: (1) the maximum dimension of a stabilizer of a point of \({\mathcal{X}_{k+1}}\) is strictly smaller than the maximum dimension of a stabilizer of \({\mathcal{X}_k}\) and the final stack \({\mathcal{X}_n}\) has constant stabilizer dimension; (2) the morphisms \({\mathcal{X}_{k+1}}\to{\mathcal{X}_k}\) induce projective and birational morphisms of good moduli spaces \({\mathbf{X}_{k+1}}\to{\mathbf{X}_k} \). If in addition the stack \(\mathcal{X}\) is smooth, then each of the intermediate stacks \({\mathcal{X}_k}\) is smooth and the final stack \({\mathcal{X}_n}\) is a gerbe over a tame stack. In this case the algebraic space \({\mathbf{X}_n}\) has tame quotient singularities and is a partial desingularization of the good moduli space \(\mathbf{X}\). When \(\mathcal{X}\) is smooth our result can be combined with D. Bergh’s recent destackification theorem for tame stacks to obtain a full desingularization of the algebraic space \(\mathbf{X}\).

MSC:

14D23 Stacks and moduli problems
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14L24 Geometric invariant theory
14A20 Generalizations (algebraic spaces, stacks)
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References:

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