On Separable Higher Gauss Maps

Abstract

We study the mth Gauss map in the sense of F. L. Zak of a projective variety X⊂PN over an algebraically closed field in any characteristic. For all integers m with n:=dim(X)⩽m<N, we show that the contact locus on X of a general tangent m-plane is a linear variety if the mth Gauss map is separable. We also show that for smooth X with n<N−2, the (n+1)th Gauss map is birational if it is separable, unless X is the Segre embedding P1×Pn⊂P2n−1. This is related to Ein’s classification of varieties with small dual varieties in characteristic zero.