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.
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