Liftings of ideals in positive characteristic to those in characteristic zero : Low dimension

TL;DR

This paper investigates the behavior of log discrepancies and log canonical thresholds for ideals on smooth varieties over fields of positive characteristic, establishing finiteness and rationality results in low dimensions and conditions for higher dimensions.

Contribution

It proves finiteness of minimal log discrepancies and rationality of accumulation points of log canonical thresholds in dimensions up to three, linking positive characteristic and complex cases.

Findings

Finiteness of minimal log discrepancies in dim ≤ 3.

Log canonical thresholds in positive characteristic are contained in those over C in dim ≤ 3.

Accumulation points of log canonical thresholds are rational in dim ≤ 3.

Abstract

We study a pair consisting of a smooth variety over a field of positive characteristic and a multi-ideal with a real exponent. We prove the finiteness of the set of minimal log discrepancies for a fixed exponent if the dimension is less than or equal to three. We also prove that the set of log canonical thresholds (lct for short) of ideals on a smooth variety in positive characteristic is contained in the set of lct's of ideals on a smooth variety over C, assuming the dimension is less than or equal to three. Under the same dimension assumption, it follows that the accumulation points of log canonical thresholds are rational. Our proofs also show the same statements for the higher dimensional case if all such pairs admit log resolutions by a composite of blow-ups by smooth centers.