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.

中文翻译：

---

环形轨道、解叠和库默爆炸

我们表明，任何具有有限对角化惯性的环形 DM 堆栈 $X$ 都具有最大环形粗化 $X_{tcs}$，使得态射 $X\to X_{tcs}$ 是对数平滑的。此外，我们使用 [AT17] 的 torification 结果来构造一个 destackification functor，这是 Bergh [Ber17] 主要结果的变体，在这种环形堆栈 $X$ 的类别上。也就是说，我们与 $X$ 关联一系列环形堆栈 $\widetilde{\mathcal{F}}_X\:Y\longrightarrow X$，使得 $Y_{tc}$ 与通常的粗模空间 $ Y_{cs}$。特别是，这提供了代数空间 $X_{cs}$ 的环形分辨率。$X_{tcs}$ 和 $\widetilde{\mathcal{F}}_X$ 都是关于严格惯性保持态射 $X'\to X$ 的函子。最后，我们使用粗化态射来引入一类不可表示的环形堆栈双有理修改，称为库默爆炸。这些修改以及我们的解叠版本用于我们关于奇异点的函式环形分辨率的工作。