Main Structure Theorems

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In this chapter, in the projective space ℙ N we consider the basic types of varieties with degenerate Gauss maps: torsal varieties, hypersurfaces, and cones. For each of these types of varieties, we consider the structure of their focal images and find sufficient conditions for a variety with a degenerate Gauss map to belong to one of these types (for torsal varieties our condition is also necessary). In Theorems 4.3, 4.4, 4.5, 4.15, and 4.16, we establish connections between the types and the structure of focal images of varieties with degenerate Gauss maps of rank r. In Section 4.3, we consider varieties with degenerate Gauss maps in the affine space A N and find a new affine analogue of the Hartman-Nirenberg cylinder theorem. In Section 4.4, we define and study new types of varieties with degenerate Gauss maps: varieties with multiple foci and their particular case, the so-called twisted cones. We also prove here existence theorems for some varieties with degenerate Gauss maps, for example, for twisted cones in P4 and A4 (Theorems 4.12 and 4.14) and establish a structure of twisted cones in ℙ4 (Theorems 4.13). This structure allows us to find a procedure for construction of twisted cylinders in A4. In Section 4.5, we prove that varieties with degenerate Gauss maps that do not belong to one of the basic types considered in Sections 4.1–4.2 are foliated into varieties of basic types (Theorem 4.16). A classification of varieties X with degenerate Gauss maps presented in this chapter is based on the structure of the focal images F L and Φ L of X. In Section 4.6, we prove an embedding theorem for varieties with degenerate Gauss maps and find sufficient conditions for such a variety to be a cone (Theorems 4.18 and 4.19 in Section 4.6).