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2020 Toroidal orbifolds, destackification, and Kummer blowings up
Dan Abramovich, Michael Temkin, Jarosław Włodarczyk
Algebra Number Theory 14(8): 2001-2035 (2020). DOI: 10.2140/ant.2020.14.2001


We show that any toroidal DM stack X with finite diagonalizable inertia possesses a maximal toroidal coarsening X tcs such that the morphism X X tcs is logarithmically smooth.

Further, we use torification results of Abramovich and Temkin (2017) to construct a destackification functor, a variant of the main result of Bergh (2017), on the category of such toroidal stacks X . Namely, we associate to X a sequence of blowings up of toroidal stacks ˜ X Y X such that Y tcs coincides with the usual coarse moduli space Y cs . In particular, this provides a toroidal resolution of the algebraic space  X cs .

Both X tcs and ˜ X are functorial with respect to strict inertia preserving morphisms X X .

Finally, we use coarsening morphisms to introduce a class of nonrepresentable birational modifications of toroidal stacks called Kummer blowings up.

These modifications, as well as our version of destackification, are used in our work on functorial toroidal resolution of singularities.


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Dan Abramovich. Michael Temkin. Jarosław Włodarczyk. "Toroidal orbifolds, destackification, and Kummer blowings up." Algebra Number Theory 14 (8) 2001 - 2035, 2020.


Received: 7 April 2018; Revised: 1 February 2020; Accepted: 25 March 2020; Published: 2020
First available in Project Euclid: 12 November 2020

MathSciNet: MR4172700
Digital Object Identifier: 10.2140/ant.2020.14.2001

Primary: 14A20
Secondary: 14E05 , 14E15

Keywords: algebraic stacks , birational geometry , logarithmic schemes , resolution of singularities , toroidal geometry

Rights: Copyright © 2020 Mathematical Sciences Publishers


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Vol.14 • No. 8 • 2020
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