The hyperplanes of finite symplectic dual polar spaces which arise from projective embeddings
type: Item
instance of: scholarly article
title: The hyperplanes of finite symplectic dual polar spaces which arise from projective embeddings
zbMATH Open document ID: 1232.51003
author: Bart De Bruyn
published in: European Journal of Combinatorics
publication date: 2011-11-29T00:00:00Z
full work available at URL: LU-2027060
review text: A \textit{hyperplane} of a point-line geometry \(S\) is a proper subspace of \(S\) intersecting every line. The classical way to construct hyperplanes in a point-line geometry is to embed this point-line geometry fully in a projective space \(\Sigma\) and then intersect this point-line geometry with a hyperplane of \(\Sigma\). This leads to the following question: given an embeddable point-line geometry \(S\) and a class \(C\) of hyperplanes of \(S\). Does any hyperplane of \(C\) arise from a hyperplane of a projective space in which \(S\) is embedded? This problem has been investigated by many authors for the dual polar spaces. The author discusses the remaining case of the symplectic dual polar space \(DW(2n-1,q)\), \(n \geq 2\), associated with the polar space \(W(2n-1,q)\). His main theorem is: the hyperplanes of the dual polar space \(DW(2n-1,q)\), \(q\neq 2\), which arise from its Grassmann-embedding are precisely those hyperplanes \(H\) of \(DW(2n-1,q)\), which satisfy the following property: if \(Q\) is a quad of \(DW(2n-1,q)\) such that \(Q\cap H\) is an ovoid of \(Q\), then \(Q\cap H\) is a classical ovoid of \(Q\). The quads mentioned in the previous theorem are the generalized quadrangles \(Q(4,q)\) arising from the non-singular parabolic quadric in PG\((4,q)\). It is known from different authors that all ovoids of \(Q(4,q)\) are classical for \(q=4\), \(q=16\), and \(q\) prime. These latter results then lead to the following corollary of the main theorem of this article: For the dual polar spaces \(DW(2n-1,4), DW(2n-1,16), DW(2n-1,p)\), \(p >2\) prime, \(n\geq 2\), every hyperplane arises from its Grassmann-embedding.
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51A50
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51A45
zbMATH DE Number: 5982483
zbMATH Keywords: dual polar space
zbMATH Keywords: Grassmann-embedding
zbMATH Keywords: dual symplectic polar space
reviewed by: Leo Storme
MaRDI profile type: Publication
OpenAlex ID: W2001095195
cites work: On ovoids of parabolic quadrics
cites work: Q3227204
cites work: The universal embedding dimension of the binary symplectic dual polar space
cites work: Dual polar spaces
cites work: Ovoids in infinite incidence structures
cites work: The structure of full polarized embeddings of symplectic and Hermitian dual polar spaces
cites work: On the simple connectedness of hyperplane complements in dual polar spaces
cites work: On the generation of dual polar spaces of symplectic type over finite fields
cites work: Points and hyperplanes of the universal embedding space of the dual polar space DW\((5,q)\), \(q\) odd
cites work: The hyperplanes of \(DQ(2n, \mathbb K)\) and \(DQ^{ - }(2n+1,q)\) which arise from their spin-embeddings
cites work: A decomposition of the natural embedding spaces for the symplectic dual polar spaces
cites work: The hyperplanes of \(DW(5,2^h)\)which arise from embedding
cites work: On a class of hyperplanes of the symplectic and Hermitian dual polar spaces
cites work: The hyperplanes of DH(5, q 2)
cites work: THE HYPERPLANES OF $DW(5,q)$ WITH NO OVOIDAL QUAD
cites work: Valuations and hyperplanes of dual polar spaces
cites work: Ovoids and Translation Planes
cites work: Absolute embeddings of point-line geometries
cites work: On the universal embedding of the \(Sp_{2n}(2)\) dual polar space
cites work: Ovoids of PG(3,16) are elliptic quadrics
cites work: Ovoids of \(PG(3,16)\) are elliptic quadratics. II
cites work: Q3227205
cites work: Q3344217
cites work: Ovoids of parabolic spaces
cites work: The Hyperplanes ofDW(5, 2)
cites work: Embeddings and hyperplanes of discrete geometries
cites work: On Veldkamp lines
cites work: Hyperplanes of dual polar spaces and the spin module
cites work: Spreads and ovoids in finite generalized quadrangles
cites work: Buildings of spherical type and finite BN-pairs
cites work: Ovoides et groupes de Suzuki
DOI: 10.1016/J.EJC.2011.07.001
description: scientific article; zbMATH DE number 5982483