The Zariski-Lipman conjecture for complete intersections

The tangential branch locus $B_{X/Y}^t\subset B_{X/Y}$ is the subset of points in the branch 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 $\codim_X B_{V/k}\leq 1 $ in positive characteristic, if $V/k $ is the fibre of a flat morphism satisfying generic smoothness.