The ovoidal hyperplanes of a dual polar space of rank 4
-
Harm Pralle
and
Sergey Shpectorov
Abstract
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.
© Walter de Gruyter 2007
Articles in the same Issue
- The ovoidal hyperplanes of a dual polar space of rank 4
- Jørgensen's inequality for metric spaces with application to the octonions
- On multiple blocking sets in Galois planes
- On twisted tensor product group embeddings and the spin representation of symplectic groups
- Projective ovoids and generalized quadrangles
- Multiple farthest points on Alexandrov surfaces
- Stiefel—Whitney classes for coherent real analytic sheaves
- On Weddle surfaces and their moduli
- Lower bounds for the first eigenvalue of the p-Laplace operator on compact manifolds with nonnegative Ricci curvature
Articles in the same Issue
- The ovoidal hyperplanes of a dual polar space of rank 4
- Jørgensen's inequality for metric spaces with application to the octonions
- On multiple blocking sets in Galois planes
- On twisted tensor product group embeddings and the spin representation of symplectic groups
- Projective ovoids and generalized quadrangles
- Multiple farthest points on Alexandrov surfaces
- Stiefel—Whitney classes for coherent real analytic sheaves
- On Weddle surfaces and their moduli
- Lower bounds for the first eigenvalue of the p-Laplace operator on compact manifolds with nonnegative Ricci curvature
