On the number of affine equivalence classes of spherical tube hypersurfaces

We consider Levi non-degenerate tube hypersurfaces in Cⁿ⁺¹ that are (k, n - k)-spherical, i. e. locally CR-equivalent to the hyperquadric with Levi form of signature (k, n - k), with n ≤ 2k. We show that the number of affine equivalence classes of such hypersurfaces is infinite (in fact, uncountable) in the following cases: (i) k = n - 2, n ≥ 7; (ii) k = n - 3, n ≥ 7; (iii) k ≤ n - 4. For all other values of k and n, except for k = 3, n = 6, the number of affine classes was known to be finite. The exceptional case k = 3, n = 6 has been recently resolved by Fels and Kaup who gave an example of a family of (3, 3)-spherical tube hypersurfaces that contains uncountably many pairwise affinely non-equivalent elements. In this paper we deal with the Fels-Kaup example by different methods. We give a direct proof of the sphericity of the hypersurfaces in the Fels-Kaup family, and use the j-invariant to show that this family indeed contains an uncountable subfamily of pairwise affinely non-equivalent hypersurfaces.