We prove a version of Jonsson-Mustaţǎ's Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm Chi Li's conjecture that a minimizer of the normalized volume function is always quasi-monomial. Applying our techniques to a family of klt singularities, we show that the volume of klt singularities is a constructible function. As a corollary, we prove that in a family of klt log Fano pairs, the K-semistable ones form a Zariski open set. Together with [Jia17], we conclude that all K-semistable klt Fano varieties with a fixed dimension and volume are parametrized by an Artin stack of finite type, which then admits a separated good moduli space by [BX18, ABHLX19], whose geometric points parametrize K-polystable klt Fano varieties.

中文翻译：

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最小化估值是准单项式的

我们证明了 Jonsson-Mustaţǎ 猜想的一个版本，它说对于任何分级的理想序列，存在计算其对数规范阈值的准单项式估值。作为推论，我们证实了 Chi Li 的猜想，即归一化体积函数的极小值始终是拟单项式的。将我们的技术应用于 klt 奇点族，我们表明 klt 奇点的体积是一个可构造的函数。作为推论，我们证明在 klt log Fano 对家族中，K 半稳定对形成 Zariski 开集。与 [Jia17] 一起，我们得出结论，所有具有固定维数和体积的 K-半稳态 klt Fano 变体都由有限类型的 Artin 堆栈参数化，然后通过 [BX18, ABHLX19] 接纳一个分离的良好模空间，其几何点参数化 K-polystable klt Fano 品种。