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

The Gauss map of a projective variety X ⊂ 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.