Toroidal orbifolds, destackification, and kummer blowings up

Dan Abramovich, Michael Temkin, Jarosław Włodarczyk

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3 Scopus citations


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 ˜FX: Y → X such that Ytcs coincides with the usual coarse moduli space Ycs . In particular, this provides a toroidal resolution of the algebraic space Xcs . Both Xtcs and ˜FX 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.

Original languageAmerican English
Pages (from-to)2001-2035
Number of pages35
JournalAlgebra and Number Theory
Issue number8
StatePublished - 2020

Bibliographical note

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© 2020, Mathematical Science Publishers. All rights reserved.


  • Algebraic stacks
  • Birational geometry
  • Logarithmic schemes
  • Resolution of singularities
  • Toroidal geometry


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