特征零下的奇点阈值
Thresholds of singularities in characteristic zero
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- University of Utah（犹他大学）
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中文总结 AI 辅助
该研究在特征零情形下构造了Frobenius阈值的两个类似物Hironaka阈值与Koszul-Hironaka阈值，证明其对参数理想一致且为有理数，揭示了其与乘子理想跳跃数、模p约化Frobenius阈值极限的关联。
中文摘要 AI 辅助
受Epstein-McDonald-R.G.与第二作者工作的启发，我们研究了Frobenius阈值$c^J(\boldsymbol{\mathfrak a})$的特征零变体的性质。设$R$是特征相等零的优整环，具有对偶复形，$\boldsymbol{\mathfrak a}, J \subseteq R$是非零理想且满足$\boldsymbol{\mathfrak a} \subseteq \sqrt{J}$。通过利用$(R, \boldsymbol{\mathfrak a})$的奇点对数解消$Y$以及$Y$上线丛的导出整体截面，我们构造了Frobenius阈值的两个类似物：Hironaka阈值和Koszul-Hironaka阈值。我们证明这两个阈值对参数理想是一致的，且始终为有理数。我们还证明这些阈值具有经典Frobenius阈值的许多性质，尤其是在参数理想的情形下。若$R$是正则的，我们证明这类阈值的集合与所有可能的乘子理想跳跃数的集合重合。最后，我们证明对于川又对数终端（Kawamata log terminal）环中的参数理想，我们的阈值与模$p$约化的Frobenius阈值在$p$趋于无穷时的极限一致。甚至存在无需川又对数终端假设的相应版本。
英文摘要
We study properties of characteristic zero variants of Frobenius thresholds $c^J(\mathfrak a)$ inspired by the work of Epstein-McDonald-R.G. and the second author. Suppose $R$ is an excellent domain of equal characteristic zero with a dualizing complex and $\mathfrak a, J \subseteq R$ are nonzero ideals with $\mathfrak a \subseteq \sqrt{J}$. By using a log resolution of singularities $Y$ of $(R, \mathfrak a)$ as well as the derived global sections of a line bundle on $Y$, we construct two analogs of the Frobenius threshold, the Hironaka-threshold and the Koszul-Hironaka threshold. We show that these two thresholds agree for parameter ideals and are always rational numbers. We prove that these thresholds also share many of the properties of the classical Frobenius threshold, especially for parameter ideals. If $R$ is regular, we show that the set of thresholds coincides with the set of possible multiplier-ideal-jumping numbers. Finally, we show that for parameter ideals in a Kawamata log terminal ring, our thresholds coincide with the limit of the Frobenius thresholds of the mod-$p$-reductions as $p$ goes to infinity. There is even a version of this without that Kawamata log terminal hypothesis.
