Number of Affine Equivalence Classes of (k, n − k) -Spherical Tube Hypersurfaces for k ≤ n − 2

Abstract

It follows from the results of Chapters 5, 6, 7 that the number of affine equivalence classes of closed (k, n-k)-spherical tube hypersurfaces in ℂⁿ⁺¹, with n ≤ 2k, is finite in the cases: (a) k = n, (b) k = n - 1, and (c) k = n - 2 with n ≤ 6. The first result of this short chapter states that this number is infinite (in fact uncountable) in the following situations: (i) k = n - 2 with n ≥ 7, (ii) k = n - 3 with n ≥ 7, and (iii) k ≤ n - 4. The question about the number of affine equivalence classes in the only remaining case k = 3, n = 6 had been open since 1989 until it was resolved by Fels and Kaup in 2009. They gave an example of a family of (3, 3)-spherical tube hy- persurfaces in ℂ⁷ that contains uncountably many pairwise affinely non-equivalent elements. The original approach due to Fels and Kaup is explained in Chapter 9. In this chapter we present the Fels-Kaup family but deal with it by different methods. Namely, we give a direct proof of the sphericity of the hypersurfaces in the fam- ily and use the j-invariant to show that the family indeed contains an uncountable subfamily of pairwise affinely non-equivalent hypersurfaces.