Real Algebraic and Analytic Geometry |

Real hypersurfaces having many pseudo-hyperplanes.

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homepage: http://fraise.univ-brest.fr/~huisman/

Submission: 2002, October 16.

*Abstract:
Let n and d be natural integers
satisfying n > 2 and d > 9. Let X be an
irreducible real hypersurface X in P^{n}
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
P^{n}(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
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 hypersurfaces in
P^{n}, 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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