Locally singular hyperplanes in thick dual polar spaces of rank 4

2017

Discrete Mathematics

A hyperplane of the symplectic dual polar space DW (2n−1, F), n ≥ 2, is said to be of subspace-type if it consists of all maximal singular subspaces of W (2n − 1, F) meeting a given (n − 1)-dimensional subspace of PG(2n − 1, F). We show that a hyperplane of DW (2n − 1, F) is of subspace-type if and only if every hex F of DW (2n − 1, F) intersects it in either F , a singular hyperplane of F or the extension of a full subgrid of a quad. In the case F is a perfect field of characteristic 2, a stronger result can be proved, namely a hyperplane H of DW (2n − 1, F) is of subspace-type or arises from the spin-embedding of DW (2n − 1, F) ∼ = DQ(2n, F) if and only if every hex F intersects it in either F , a singular hyperplane of F , a hexagonal hyperplane of F or the extension of a full subgrid of a quad.

Section: Lemma 42 ([6 9])mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A characterization of a class of hyperplanes of DW(2n−1,F)

2017

Discrete Mathematics

“…Moreover, they were able to show the following. Proposition 1.4 (Cardinali et al [4]). Let H be a locally singular hyperplane of DQ (8, q).…”

Section: Proposition 12 (De Bruyn and Prallementioning

confidence: 99%

“…Hence, admits a hyperplane whose points and deep lines determine a polar space Q (6, q). Now, let H be one of these Q(6, q)-hyperplanes of and define the following set H of points of DQ (8, q Then Cardinali et al [4] showed that H is a locally singular hyperplane of DQ (8, q). Moreover, they were able to show the following.…”

Section: Proposition 12 (De Bruyn and Prallementioning

confidence: 99%

Hyperplanes of DW(2n-1,q), q≠2, without ovoida

2007

Discrete Mathematics

Direct Constructions of Hyperplanes of Dual Polar Spaces Arising from Embeddings

2010

Ann. Comb.

Let e be one of the following full projective embeddings of a finite dual polar space ∆ of rank n ≥ 2: (i) the Grassmann-embedding of the symplectic dual polar space ∆ ∼ = DW (2n − 1, q); (ii) the Grassmannembedding of the Hermitian dual polar space ∆ ∼ = DH(2n − 1, q 2 ); (iii) the spin-embedding of the orthogonal dual polar space ∆ ∼ = DQ(2n, q); (iv) the spin-embedding of the orthogonal dual polar space ∆ ∼ = DQ − (2n + 1, q). Let H e denote the set of all hyperplanes of ∆ arising from the embedding e. We give a method for constructing the hyperplanes of H e without implementing the embedding e and discuss (possible) applications of the given construction.