Toroidal orbifolds, destackification, and Kummer blowings up

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 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 ${\widetilde{\mathcal{ℱ}}}_{X}Y\to X$ 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 ${\widetilde{\mathcal{ℱ}}}_X$ are functorial with respect to strict inertia preserving morphisms $X^{\prime }\to X$.

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.