The Violation of the Lipman-Zariski conjecture in positive characteristic

TL;DR

This paper investigates the failure of the Lipman-Zariski conjecture in positive characteristic, identifying specific exceptions and conditions under which it holds or fails for surface singularities.

Contribution

It provides a detailed analysis of the conjecture's validity in positive characteristic, including a classification of exceptions for rational double points and log canonical surface singularities.

Findings

The conjecture holds for all rational double points in characteristic p ≥ 7 except for Lipman's counterexample.

The conjecture remains valid for tame F-pure normal surface singularities under certain conditions.

Identifies specific finite exceptions where the conjecture fails in positive characteristic.

Abstract

We study the failure of the Lipman-Zariski conjecture in positive characteristic. For rational double points, the conjecture holds true except for a short finite list of exceptions. For log canonical surface singularities, the conjecture continues to hold with the same list of exceptions under an additional tameness hypothesis. In particular, among rational double points in characteristic $p \ge 7$ Lipman's counterexample is the only one, and the conjecture holds for all tame $F$-pure normal surface singularities.