The Zariski–Lipman conjecture for complete intersections

The tangential branch locus is the subset of points in the branch locus where the sheaf of relative vector fields TX/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 , then V/k is smooth (= regular). We prove this conjecture when V/k is a locally complete intersection. We prove also that implies in positive characteristic, if V/k is the fibre of a flat morphism satisfying generic smoothness.