Varieties in Projective Spaces and Their Gauss Maps

Abstract

In this chapter, after introducing in Sections 2.1 and 2.2 the basic notions (such as the tangent, osculating and normal subspaces, the second fundamental tensor and the second fundamental form, and the asymptotic lines and asymptotic cone) associated with a variety in a projective space ℙ^N, in Section 2.3, we define the rank of a variety and varieties with degenerate Gauss maps. In Section 2.4, we consider the main examples of varieties with degenerate Gauss maps (cones, torses, hypersurfaces, joins, etc.). In Section 2.5, we study the duality principle and its applications, consider another example of varieties with degenerate Gauss maps (the cubic symmetroid) and correlative transformations, and in Section 2.6, we investigate a hypersurface with a degenerate Gauss map associated with a Veronese variety and find its singular points.