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

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 $X$.

27 pages, Corollary 5.4 added

References in corpus (1)

• Gauss maps of toric varieties