Real hypersurfaces having many pseudo-hyperplanes

Let n and d be natural integers satisfying n ≥ 3 and d ≥ 10. Let X be an irreducible real hypersurface X in Pn of degree d having many pseudo-hyperplanes. Suppose that X is not a projective cone. We show that the arrangement H of all d − 2 pseudo-hyperplanes of X is trivial, i.e., there is a real projective linear subspace L of Pn(R) of dimension n−2 such that L ⊆ H for all H ∈ H. As a consequence, the normalization of X is fibered over P1 in quadrics. Both statements are in sharp contrast with the case n = 2; the first statement also shows that there is no Brusotti-type result for hypersurfaces in Pn, for n ≥ 3. MSC 2000: 14P25, 52C35