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Toroidal orbifolds, destackification, and Kummer blowings up

Dan Abramovich, Michael Temkin and Jarosław Włodarczyk

Appendix: David Rydh

Vol. 14 (2020), No. 8, 2001–2035
DOI: 10.2140/ant.2020.14.2001
Abstract

We show that any toroidal DM stack X with finite diagonalizable inertia possesses a maximal toroidal coarsening Xtcs such that the morphism X Xtcs 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 ˜XY 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 Xcs.

Both Xtcs 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.

Keywords
algebraic stacks, toroidal geometry, logarithmic schemes, birational geometry, resolution of singularities
Mathematical Subject Classification 2010
Primary: 14A20
Secondary: 14E05, 14E15
Milestones
Received: 7 April 2018
Revised: 1 February 2020
Accepted: 25 March 2020
Published: 18 September 2020
Authors
Dan Abramovich
Brown University
Providence, RI
United States
Michael Temkin
The Hebrew University of Jerusalem
Jerusalem
Israel
Jarosław Włodarczyk
Purdue University
West Lafayette, IN
United States
David Rydh
KTH Royal Institute of Technology
Stockholm
Sweden