The Zariski-Lipman conjecture for complete intersections

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 $B_{V/k}^t=\varnothing$, then $V/k$ is smooth (=regular). We prove this conjecture when $V/k$ is a locally complete intersection. We prove also that $B_{V/k}^t=\varnothing$ implies ${\mathrm{codim}}_V{B}_{V/k}\le 1$ in positive characteristic, if $V/k$ is the fibre of a flat morphism satisfying generic smoothness.

17 pages. A mistake in the previous version was discovered. The new version is a rather thorough rewrite