Toroidal orbifolds, destackification, and Kummer blowings up

We show that any toroidal DM stack $X$ with finite diagonalizable inertia possesses a maximal toroidal coarsening $X_{tcs}$ such that the morphism $X \to X_{tcs}$ 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 ${\tilde{ℱ}}_{X} Y \to X$ 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 ${\tilde{ℱ}}_{X}$ are functorial with respect to strict inertia preserving morphisms $X^{'} \to X$ .

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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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

Vol. 14, No. 8, 2020

Dan Abramovich, Michael Temkin and Jarosław Włodarczyk

Appendix: David Rydh

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors