Berkovich log discrepancies in positive characteristic

TL;DR

This paper introduces a new log discrepancy function for semivaluations in positive characteristic, linking it to singularity classifications and extending key theorems from characteristic zero.

Contribution

It defines a log discrepancy in positive characteristic that aligns with existing notions when resolutions exist and applies it to characterize singularities and prove foundational theorems.

Findings

Log discrepancy detects strong F-regularity and F-purity.

Proves theorems on log canonical thresholds in positive characteristic.

Shows asymptotic multiplier ideals are coherent on strongly F-regular schemes.

Abstract

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, then our definition agrees with previous approaches to log discrepancies of semivaluations that 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 Mustata 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.