  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 $X$.
