Berkovich log discrepancies in positive characteristic

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.

Keywords

log discrepancy, Berkovich spaces, multiplier ideals, graded sequences of ideals

2010 Mathematics Subject Classification

Primary 14F18. Secondary 12J25, 13A35, 14B05.

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The author was partially supported by DMS #1606414.

Received 18 November 2019

Accepted 20 November 2019

Published 17 February 2021