Given an ideal $\mathcal I$ on a variety $X$ with toroidal singularities, we produce a modification $X' \to X$, functorial for toroidal morphisms, making the ideal monomial on a toroidal stack $X'$. We do this by adapting the methods of [W{\l}o05], discarding steps which become redundant. We deduce functorial resolution of singularities for varieties with logarithmic structures. This is the first step in our program to apply logarithmic desingularization to a morphism $Z \to B$, aiming to prove functorial semistable reduction theorems.

中文翻译：

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环形轨道上的理想公化

给定一个理想的 $\mathcal I$ 在具有环形奇点的变量 $X$ 上，我们产生一个修改 $X' \to X$，环形态射的函子，在环形堆栈 $X'$ 上形成理想的单项式。我们通过调整 [W{\l}o05] 的方法来做到这一点，丢弃变得多余的步骤。我们为具有对数结构的变体推导出奇点的函解。这是我们程序的第一步，将对数去奇异化应用于态射 $Z \to B$，旨在证明函子半稳定约简定理。