On Separable Higher Gauss Maps

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 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 $m$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 Ein’s classification of varieties with small dual varieties in characteristic zero.