Real Algebraic and Analytic Geometry
Submission: 2002, October 16.
Let n and d be natural integers satisfying n > 2 and d > 9. Let X be an irreducible real hypersurface X in Pn of degree d having many pseudo-hyperplanes, i.e., X has exactly d-2 pseudo-hyperplanes. Suppose that X is not a projective cone. We show that the arrangement A 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 is contained in each element of A. 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 > 2.
Mathematics Subject Classification (2000): 14P25, 52C35.
Keywords and Phrases: real hypersurface, quasi-hyperplane, pseudo-hyperplane, pseudo-line, arrangement of pseudo-lines, arrangement of pseudo-hyperplanes, hyperelliptic curve, fibration in quadrics.
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