Gauss maps with nontrivial separable degree in positive characteristic

For a projective variety of dimension n in a projective space image defined over an algebraically closed field, the Gauss map is the rational map of the variety to the Grassmannian of n-planes in image, mapping a smooth point to the embedded tangent space to the variety at the point. The purpose here is to give three examples of Gauss maps with separable degrees greater than one onto their images in positive characteristic: (1) a smooth variety with Kodaira dimension κ