Let n and d be natural integers satisfying n greater than or equal to 3 and d greater than or equal to 10. Let X be an irreducible real hypersurface X in P-n 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 P-n(R) of dimension n - 2 such that L subset of H for all H e As a consequence, the normalization of X is fibered over P-1 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 hypersurfaccs in P-n, for n greater than or equal to 3. [References: 7]
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