Surface singularities dominated by smooth varieties

TL;DR

This paper extends Mumford's theorem to characteristic p>0, providing a criterion for surface smoothness based on the fundamental group of the complement of a point, with historical and personal context.

Contribution

It offers a characteristic p>0 version of Mumford's theorem, linking surface smoothness to fundamental group triviality, expanding the theorem's applicability.

Findings

Characteristic p>0 version of Mumford's theorem established

Smoothness characterized by trivial fundamental group in positive characteristic

Provides new insights into surface singularities in algebraic geometry

Abstract

We give a version in characteristic $p>0$ of Mumford's theorem characterizing a smooth complex germ of surface $(X,x)$ by the triviality of the topological fundamental group of $U=X\setminus \{x\}$. This note relies on discussions the authors had during the Christmas break 2009/10 in Ivry. They have been written down by H\'el\`ene in the night when Eckart died, as a despaired sign of love.