Functorial destackification of tame stacks with abelian stabilisers

Abstract

We give an algorithm for removing stackiness from smooth, tame Artin stacks with abelian stabilisers by repeatedly applying stacky blow-ups. The construction works over a general base and is functorial with respect to base change and compositions with gerbes and smooth, stabiliser preserving maps. As applications, we indicate how the result can be used for destackifying general Deligne-Mumford stacks in characteristic zero, and to obtain a weak factorisation theorem for such stacks. Over an arbitrary field, the method can be used to obtain a functorial algorithm for desingularising varieties with simplicial toric quotient singularities, without assuming the presence of a toroidal structure.

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Cited by in corpus (17)

• Equivariant Hodge theory and noncommutative geometry
• Functorial embedded resolution via weighted blowings up
• K-theory and the singularity category of quotient singularities
• Equivariant birational types and Burnside volume
• Canonical reduction of stabilizers for Artin stacks with good moduli spaces
• Toroidal orbifolds, destackification, and Kummer blowings up
• Fundamental gerbes
• Conic bundles and iterated root stacks
• Functorial destackification and weak factorization of orbifolds
• Symbols and equivariant birational geometry in small dimensions
• Weak factorization and the Grothendieck group of Deligne-Mumford stacks
• Pro-Chern-Schwartz-MacPherson class for DM stacks
• Compactification of Dolbeault moduli spaces
• Maps of Mori dream spaces
• Birational types of algebraic orbifolds
• Building Data for Stacky Covers
• Torification of diagonalizable group actions on toroidal schemes