Log canonical thresholds, F-pure thresholds, and non-standard extensions

TL;DR

This paper establishes a deep connection between log canonical thresholds in characteristic zero and F-pure thresholds in positive characteristic, showing their limit points coincide through non-standard methods.

Contribution

It introduces a novel relation between two singularity invariants and proves the set of their limit points coincide using non-standard analysis techniques.

Findings

Limit points of F-pure thresholds match log canonical thresholds.

The set of thresholds in positive characteristic converges to those in characteristic zero.

Combines results of Hara and Yoshida with non-standard constructions.

Abstract

We present a new relation between an invariant of singularities in characteristic zero (the log canonical threshold) and an invariant of singularities defined via the Frobenius morphism in positive characteristic (the F-pure threshold). We show that the set of limit points of sequences of the form (c_p), where c_p is the F-pure threshold of an ideal on an n-dimensional smooth variety in characteristic p, coincides with the set of log canonical thresholds of ideals on n-dimensional smooth varieties in characteristic zero. We prove this by combining results of Hara and Yoshida with non-standard constructions.