## The Michigan Mathematical Journal

### 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 $n:=\dim(X)\leqslant m\lt N$, 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 $n\lt N-2$, 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.

#### Article information

Source
Michigan Math. J., Volume 68, Issue 3 (2019), 483-503.

Dates
Revised: 30 October 2017
First available in Project Euclid: 18 April 2019

https://projecteuclid.org/euclid.mmj/1555574416

Digital Object Identifier
doi:10.1307/mmj/1555574416

Mathematical Reviews number (MathSciNet)
MR3990168

Zentralblatt MATH identifier
07130696

Subjects

#### Citation

Furukawa, Katsuhisa; Ito, Atsushi. On Separable Higher Gauss Maps. Michigan Math. J. 68 (2019), no. 3, 483--503. doi:10.1307/mmj/1555574416. https://projecteuclid.org/euclid.mmj/1555574416

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