Vol. 4, No. 1, 2006

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Certain generalized quadrangles inside polar spaces of rank $4$

Harm Pralle

Vol. 4 (2006), No. 1, 109–130
DOI: 10.2140/iig.2006.4.109

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 DSp8(q) and DO10(q) of Witt index 4 have ovoidal hyperplanes. For DSp8(K) over an arbitrary field K, it holds ΓSp4() for some field .

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 DO10(K) for a field K. Assuming char(K)2 when K is infinite, we prove that Γ is a hermitian generalized quadrangle over some division ring .

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 DSp8(q) and DO10(q) by Pralle and Shpectorov.

Mathematical Subject Classification 2000
Primary: 51A50, 51E12, 51E23
Received: 12 July 2006
Accepted: 6 December 2006
Harm Pralle