Duality with expanding maps and shrinking maps, and its applications to Gauss maps

TL;DR

This paper explores duality between expanding and shrinking maps on Grassmann varieties, providing new characterizations of Gauss maps and their images, with applications to linear fibers and developable varieties.

Contribution

It introduces the expanding map as a dual to the shrinking map, generalizing the Gauss map and advancing the understanding of Gauss images in algebraic geometry.

Findings

Characterization of separable Gauss maps and their images

Linearity of general fibers of separable Gauss maps

Duality on parameter spaces of linear subvarieties

Abstract

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.