On Separable Higher Gauss Maps

We study the mth Gauss map in the sense of F. L. Zak of a projective variety X subset of P-N 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 P-1 x P-n subset of P2n-1.This is related to Ein's classification of varieties with small dual varieties in characteristic zero.