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Harm Pralle, Sergey Shpectorov

The ovoidal hyperplanes of a dual polar space of rank 4

Keywords: Dual polar space, generalised quadrangle, hyperplane, ovoid, polar space, spread, symplectic spread

Let π be a thick polar space of rank 4 over a field 𝕂 such that the generalised quadrangle Res⁺(α) of a line α of π which consists of the planes and 3-spaces of π containing α, admits a spread. Let be a line-spread of π with the following property: Let be the set of 3-spaces of π in which induces spreads. For every point Σ of π, the 3-spaces of containing Σ all contain the spread line λ ∈ covering Σ and form a spread of the generalised quadrangle . Then is a generalised quadrangle. The polar spaces Sp₈ (𝕂) and admit such spreads for both finite and infinite fields 𝕂. They are the only finite classical polar spaces admitting such spreads. If π ≅ Sp₈(q), respectively , then Γ ≅ Sp₄(q²), respectively H₅(q²). If π ≅ Sp₈(𝕂) for some infinite field 𝕂, then Γ ≅ Sp₄(ℍ) for some field ℍ. In the infinite case, there exists an example of a spread in H₈(ℂ) over the complex numbers ℂ with over the quaternions.

Dualizing π, the point set of the dual polar space Δ dual to π is a hyperplane of Δ intersecting each symp Σ, i.e. an element of maximal type of Δ, in the set of neighbours of an ovoid of a quad of Σ.

Advances in Geometry, Walter de Gruyter

Print ISSN: 1615-715X
Volume: 7, 01/2007
Pages: 1 - 17

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