  We study expanding maps and shrinking maps of subvarieties of Grassmann
varieties in arbitrary characteristic. The shrinking map was studied
independently by Landsberg and Piontkowski in order to characterize Gauss
images. To develop their method, we introduce the expanding map, which is a
dual notion of the shrinking map and is a generalization of the Gauss map. Then
we give a characterization of separable Gauss maps and their images, which
yields results for the following topics: (1) Linearity of general fibers of
separable Gauss maps; (2) Generalization of the characterization of Gauss
images; (3) Duality on one-dimensional parameter spaces of linear subvarieties
lying in developable varieties.
