The Zariski-Lipman conjecture for complete intersections

The tangential branch locus B-X/Y(t) subset of 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-V/k(t) = empty set. 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) = empty set implies codim(X) B-V/K <= 1 in positive characteristic, if V/k is the fibre of a flat morphism satisfying generic smoothness. (C) 2011 Elsevier Inc. All rights reserved.