Toroidal orbifolds, destackification, and Kummer blowings up

TL;DR

This paper develops a functorial destackification process for toroidal DM stacks, introduces Kummer blowings up as a new class of modifications, and applies these to achieve functorial toroidal resolution of singularities.

Contribution

It constructs a maximal toroidal coarsening and a destackification functor for toroidal DM stacks, introducing Kummer blowings up for non-representable modifications.

Findings

Existence of a maximal toroidal coarsening with logarithmically smooth morphism.

Construction of a functorial destackification sequence leading to the coarse moduli space.

Introduction of Kummer blowings up as non-representable birational modifications.

Abstract

We show that any toroidal DM stack $X$ with finite diagonalizable inertia possesses a maximal toroidal coarsening $X_{tcs}$ such that the morphism $X\to X_{tcs}$ is logarithmically smooth. Further, we use torification results of [AT17] to construct a destackification functor, a variant of the main result of Bergh [Ber17], on the category of such toroidal stacks $X$. Namely, we associate to $X$ a sequence of blowings up of toroidal stacks $\widetilde{\mathcal{F}}_X\:Y\longrightarrow X$ such that $Y_{tc}$ 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 $\widetilde{\mathcal{F}}_X$ are functorial with respect to strict inertia preserving morphisms $X'\to X$. Finally, we use coarsening morphisms to introduce a class of non-representable 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.