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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   $X$  .

On Gauss Maps in Positive Characteristic in View of Images, Fibers, and Field Extensions