We introduce and study a log discrepancy function on the space of semivaluations centered on an integral noetherian scheme of positive characteristic. Our definition shares many properties with the analogue in characteristic zero; we prove that if log resolutions exist in positive characteristic, then our definition agrees with previous approaches to log discrepancies of semivaluations that use these resolutions.We then apply this log discrepancy to a variety of topics in singularity theory over fields of positive characteristic. Strong $F$-regularity and sharp $F$-purity of Cartier subalgebras are detected using positivity and non-negativity of log discrepancies of semivaluations, just as Kawamata log terminal and log canonical singularities are defined using divisorial log discrepancies, making precise a long-standing heuristic. We prove, in positive characteristic, several theorems of Jonsson and Mustaţă in characteristic zero regarding log canonical thresholds of graded sequences of ideals. Along the way, we give a valuation-theoretic proof that asymptotic multiplier ideals are coherent on strongly $F$-regular schemes.

中文翻译：

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Berkovich对数差异具有积极特征

我们引入并研究以正特征的积分noether方案为中心的半评估空间的对数差异函数。我们的定义与特征为零的类似物具有许多共同的特性。我们证明如果对数分辨率存在正特征，则我们的定义与使用这些分辨率的半值对数差异的先前方法相符。然后，将对数差异应用于奇异性理论中正特征领域的各种主题。卡地亚子代数的强$ F $正则性和锋利的$ F $纯度使用半值的对数差异的正负性来检测，就像使用除数对数差异定义Kawamata对数终极和对数典范奇异性一样，精确长启发式 我们证明 在正特性中，关于理想的渐变序列的对数正则阈值，Jonsson和Mustaţă的几个定理在特性零中。在此过程中，我们给出了一个估值理论上的证明，即渐近乘子理想在强$ F $-常规方案上是一致的。