# On separable higher Gauss maps

## TL;DR

This paper investigates the properties of the $m$-th Gauss map of projective varieties, establishing conditions under which contact loci are linear and when the map is birational, extending classical results to arbitrary characteristic.

## Contribution

It generalizes the understanding of higher Gauss maps, proving linearity of contact loci under separability and characterizing birationality for smooth varieties, including in positive characteristic.

## Key findings

- Contact locus on $X$ is linear if the $m$-th Gauss map is separable.
- The $(n+1)$-th Gauss map is birational for smooth $X$ with $n < N-2$, unless $X$ is a specific Segre embedding.
- Extension of classical results to varieties over fields of arbitrary characteristic.

## 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.