The Zariski–Lipman conjecture for complete intersections

Abstract

The tangential branch locus B X / Y t ⊂ 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 = ∅ , 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 = ∅ implies codim X B V / k ⩽ 1 in positive characteristic, if V / k is the fibre of a flat morphism satisfying generic smoothness.