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- W2951570324 abstract "Let Δ be the dual of a thick polar space Π of rank 4 . The points, lines, quads, and hexes of Δ correspond with the singular 3 -spaces, planes, lines, respectively points of Π . Pralle and Shpectorov have investigated ovoidal hyperplanes of Δ which intersect every hex in the extension of an ovoid of a quad. With every ovoidal hyperplane there corresponds a unique generalized quadrangle Γ . In the finite case, Γ has been classified combinatorially, and it has been shown that only the symplectic and elliptic dual polar spaces D S p 8 ( q ) and D O 1 0 − ( q ) of Witt index 4 have ovoidal hyperplanes. For D S p 8 ( K ) over an arbitrary field K , it holds Γ ≅ S p 4 ( ℍ ) for some field ℍ . In this paper, we construct an embedding projective space for the generalized quadrangle Γ arising from an ovoidal hyperplane of the orthogonal dual polar space D O 1 0 − ( K ) for a field K . Assuming c h a r ( K ) ≠ 2 when K is infinite, we prove that Γ is a hermitian generalized quadrangle over some division ring ℍ . Moreover we show that an ovoidal hyperplane H arises from the universal embedding of Δ , if the ovoids Q ∩ H of all ovoidal quads Q are classical. This condition is satisfied for the finite dual polar spaces D S p 8 ( q ) and D O 1 0 − ( q ) by Pralle and Shpectorov." @default.
- W2951570324 created "2019-06-27" @default.
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- W2951570324 date "2006-01-01" @default.
- W2951570324 modified "2023-09-23" @default.
- W2951570324 title "Certain generalized quadrangles inside polar spaces of rank 4" @default.
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- W2951570324 doi "https://doi.org/10.2140/iig.2006.4.109" @default.
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