The ovoidal hyperplanes of a dual polar space of rank 4

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Let Pi be a thick polar space of rank 4 over a field K such that the generalised quadrangle Res+(alpha) of a line alpha of Pi which consists of the planes and 3-spaces of Pi containing alpha, admits a spread. Let L be a line-spread of Pi with the following property: Let D be the set of 3-spaces of Pi in which L induces spreads. For every point Sigma of Pi, the 3-spaces of D containing Sigma all contain the spread line lambda is an element of L covering Sigma and form a spread of the generalised quadrangle Res(Pi)(+)(lambda). Then Gamma = (L, D) is a generalised quadrangle. The polar spaces Sp(8)(K) and O-10(-)(K) admit such spreads for both finite and infinite fields K. They are the only finite classical polar spaces admitting such spreads. If Pi congruent to Sp(8)(q), respectively O-10(-)(q), then Gamma congruent to Sp(4)(q(2)), respectively H-5(q(2)). If Pi congruent to Sp(8)(K) for some infinite field K, then Gamma congruent to Sp(4)(H) for some field H. In the infinite case, there exists an example of a spread L in H-8 (C) over the complex numbers C with Gamma = (L, D) congruent to H-4 (Q) over the quaternions. Dualizing Pi, the point set boolean OR(X is an element of D) X perpendicular to of the dual polar space Delta dual to Pi is a hyperplane of Delta intersecting each symp Sigma, i.e. an element of maximal type of Delta, in the set of neighbours of an ovoid of a quad of Sigma.
Original languageEnglish
Pages (from-to)1-17
Number of pages17
JournalAdvances in Geometry
Issue number1
Publication statusPublished - 1 Jan 2007


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