The Zariski-Lipman conjecture for complete intersections

TL;DR

This paper proves the Zariski-Lipman conjecture for locally complete intersection varieties over characteristic zero fields, showing that the absence of tangential ramification implies smoothness, and discusses related results in positive characteristic.

Contribution

It establishes the conjecture for locally complete intersections over characteristic zero fields and explores implications in positive characteristic settings.

Findings

Proves the Zariski-Lipman conjecture for locally complete intersections in characteristic zero.

Shows that absence of tangential ramification implies smoothness.

In positive characteristic, tangential ramification absence bounds the codimension of the ramification locus.

Abstract

The tangential ramification locus $B_{X/Y}^t\subset B_{X/Y}$ is the subset of points in the ramification locus where the sheaf of relative vector fields $T_{X/Y}$ fails to be locally free. It was conjectured by Zariski and Lipman that if $V/k$ is a variety over a field $k$ of characteristic 0 and $B^t_{V/k}= \emptyset$, then $V/k$ is smooth (=regular). We prove this conjecture when $V/k$ is a locally complete intersection. We prove also that $B_{V/k}^t= \emptyset$ implies $\operatorname {codim}_V B_{V/k}\leq 1 $ in positive characteristic, if $V/k $ is the fibre of a flat morphism satisfying generic smoothness.