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On separable higher Gauss maps (1702.06010v1)
Published 20 Feb 2017 in math.AG
Abstract: We study the $m$-th Gauss map in the sense of F.~L.~Zak of a projective variety $X \subset \mathbb{P}N$ over an algebraically closed field in any characteristic. For all integer $m$ with $n:=\dim(X) \leq m < N$, 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 $\mathbb{P}1 \times \mathbb{P}n \subset \mathbb{P}{2n-1}$. This is related to L. Ein's classification of varieties with small dual varieties in characteristic zero.