In this article, we study the iteration of Cox rings of klt singularities (and Fano varieties) from a topological perspective. Given a klt singularity , we define the iteration of Cox rings of . The first result of this article is that the iteration of Cox rings of a klt singularity stabilizes for large enough. The second result is a boundedness one, we prove that for an -dimensional klt singularity , the iteration of Cox rings stabilizes for , where only depends on . Then, we use Cox rings to establish the existence of a simply connected factorial canonical (or scfc) cover of a klt singularity, with general fiber being an extension of a finite group by an algebraic torus. The scfc cover generalizes both the universal cover and the iteration of Cox rings. We prove that the scfc cover dominates any sequence of quasi-étale finite covers and reductive abelian quasi-torsors of the singularity. We characterize when the iteration of Cox rings is smooth and when the scfc cover is smooth. We also characterize when the spectrum of the iteration coincides with the scfc cover. Finally, we give a complete description of the regional fundamental group, the iteration of Cox rings, and the scfc cover of klt singularities of complexity one. Analogous versions of all our theorems are also proved for Fano-type morphisms. To extend the results to this setting, we show that the Jordan property holds for the regional fundamental group of Fano-type morphisms.

中文翻译：

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klt 奇点的 Cox 环的迭代

在本文中，我们从拓扑角度研究 klt 奇点（和 Fano 簇）的 Cox 环的迭代。给定 klt 奇点，我们定义 Cox 环的迭代。本文第一个结果是Cox环的迭代klt 奇点稳定为足够大。第二个结果是有界的，我们证明对于维 klt 奇点，Cox 环的迭代稳定为， 在哪里仅取决于。然后，我们使用 Cox 环来建立 klt 奇点的简单连接阶乘正则（或scfc ）覆盖的存在性，其中一般纤维是代数环面的有限群的延伸。 scfc 覆盖概括了通用覆盖和 Cox 环的迭代。我们证明了 scfc 覆盖支配着任何拟étale有限覆盖序列和奇点的还原阿贝尔准torsors。我们描述了 Cox 环迭代何时平滑以及 SCFC 覆盖何时平滑。我们还描述了迭代频谱何时与 SCFC 覆盖层一致的情况。最后，我们给出了区域基本群、Cox环迭代以及复杂度为1的klt奇点的scfc覆盖的完整描述。我们所有定理的类似版本也针对 Fano 型态射得到了证明。为了将结果扩展到这种情况，我们证明 Jordan 性质对于 Fano 型态射的区域基本群成立。