The hyperplanes of finite symplectic dual polar spaces which arise from projective embeddings
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.
scientific article; zbMATH DE number 5982483
scientific article