## Abstract

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 ˜F_{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 ˜F_{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.

Original language | American English |
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Pages (from-to) | 2001-2035 |

Number of pages | 35 |

Journal | Algebra and Number Theory |

Volume | 14 |

Issue number | 8 |

DOIs | |

State | Published - 2020 |

### Bibliographical note

Publisher Copyright:© 2020, Mathematical Science Publishers. All rights reserved.

## Keywords

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