On separable higher Gauss maps

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We study the m-th 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 integer m with n:=(X) ≤ m < N, we show that the contact locus on X of a general tangent m-plane is a linear variety if the m-th 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 L. Ein's classification of varieties with small dual varieties in characteristic zero.