Log canonical thresholds in positive characteristic

TL;DR

This paper extends the theory of log canonical thresholds and their relation to jet schemes from characteristic zero to positive characteristic, providing new tools for studying singularities without log resolutions.

Contribution

It generalizes the correspondence between cylinders and valuations to positive characteristic and proves Mustata's formula without log resolutions.

Findings

Established the correspondence in positive characteristic.

Proved Mustata's log canonical threshold formula without log resolutions.

Derived a comparison theorem via reduction modulo p.

Abstract

In this paper, we study the singularities of a pair (X,Y) in arbitrary characteristic via jet schemes. For a smooth variety X in characteristic 0, Ein, Lazarsfeld and Mustata showed that there is a correspondence between irreducible closed cylinders and divisorial valuations on X. Via this correspondence, one can relate the codimension of a cylinder to the log discrepancy of the corresponding divisorial valuation. We now extend this result to positive characteristic. In particular, we prove Mustata's log canonical threshold formula avoiding the use of log resolutions, making the formula available also in positive characteristic. As a consequence, we get a comparison theorem via reduction modulo p and a version of Inversion of Adjunction in positive characteristic.