On Gauss maps in positive characteristic in view of images, fibers, and field extensions

TL;DR

This paper investigates the properties of Gauss maps of projective varieties in positive characteristic, demonstrating constructions of varieties with prescribed fibers and images, and characterizing inseparable field extensions via Gauss maps.

Contribution

It provides new methods to construct varieties with specific Gauss map fibers and images, and characterizes inseparable extensions in positive characteristic.

Findings

Constructed varieties with prescribed Gauss map fibers and images.

Shown that any inseparable extension arises from a Gauss map in characteristic not 2.

Abstract

The Gauss map of a projective variety $X \subset \mathbb{P}^N$ is a rational map from $X$ to a Grassmann variety. In positive characteristic, we show the following results. (1) For given projective varieties $F$ and $Y$, we construct a projective variety $X$ whose Gauss map has $F$ as its general fiber and has $Y$ as its image. More generally, we give such construction for families of varieties over $Y$ instead of fixed $F$. (2) At least in the case when the characteristic is not equal to $2$, any inseparable field extension appears as the extension induced from the Gauss map of some $X$.