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 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.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2017
- DOI:
- 10.48550/arXiv.1702.06010
- arXiv:
- arXiv:1702.06010
- Bibcode:
- 2017arXiv170206010F
- Keywords:
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- Mathematics - Algebraic Geometry;
- 14N05
- E-Print:
- 20 pages