On the F-purity of isolated log canonical singularities

TL;DR

This paper proves that isolated log canonical singularities with certain invariants are of dense F-pure type, establishing a connection between log canonicity and F-purity in characteristic p, especially in three dimensions.

Contribution

It demonstrates that isolated log canonical singularities with invariant μ ≤ 2 are of dense F-pure type, and confirms the equivalence of log canonicity and dense F-pure type in 3D cases.

Findings

Isolated log canonical singularities with μ ≤ 2 are of dense F-pure type.

In 3D, log canonical singularities are equivalent to being of dense F-pure type.

Abstract

A singularity in characteristic zero is said to be of dense F-pure type if its modulo p reduction is locally F-split for infinitely many p. We prove that if $x \in X$ is an isolated log canonical singularity with $\mu(x \in X) \le 2$ (see Definition 1.4 for the definition of the invariant $\mu$), then it is of dense F-pure type. As a corollary, we prove the equivalence of log canonicity and being of dense F-pure type in the case of three-dimensional isolated singularities.