On the Number of Affine Equivalence Classes of Spherical Tube Hypersurfaces
نویسنده
چکیده
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 the 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 is 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 FelsKaup 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.
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