### On a class of hyperplanes of the symplectic and Hermitian dual polar spaces.

Bart De Bruyn UGent (2009) 16(1). p.1-20
abstract
Let $\Delta$ be a symplectic dual polar space (2n-1,K)$or a Hermitian dual polar space (2n-1,K,\theta)$, \geq 2$. We define a class of hyperplanes of$\Delta$arising from its Grassmann-embedding and discuss several properties of these hyperplanes. The construction of these hyperplanes allows us to prove that there exists an ovoid of the Hermitian dual polar space (2n-1,K,\theta)$ arising from its Grassmann-embedding if and only if there exists an empty $\theta variety in$\PG(n-1,K)$. Using this result we are able to give the first examples of ovoids in thick dual polar spaces of rank at least 3 which arise from some projective embedding. These are also the first examples of ovoids in thick dual polar spaces of rank at least 3 for which the construction does not make use of transfinite recursion. Please use this url to cite or link to this publication: author organization year type journalArticle (original) publication status published subject keyword symplectic/Hermitian dual polar space, ovoid, Grassmann embedding journal title Electronic Journal of Combinatorics Electron. J. Combin. volume 16 issue 1 pages 1 - 20 Web of Science type Article Web of Science id 000262254700001 JCR category MATHEMATICS JCR impact factor 0.605 (2009) JCR rank 138/251 (2009) JCR quartile 3 (2009) ISSN 1077-8926 language English UGent publication? yes classification A1 copyright statement I don't know the status of the copyright for this publication id 879105 handle http://hdl.handle.net/1854/LU-879105 date created 2010-02-24 14:12:53 date last changed 2010-03-05 09:09:52 @article{879105, abstract = {Let \${\textbackslash}Delta\$be a symplectic dual polar space (2n-1,K)\$ or a Hermitian dual polar space (2n-1,K,{\textbackslash}theta)\$, {\textbackslash}geq 2\$. We define a class of hyperplanes of \${\textbackslash}Delta\$ arising from its Grassmann-embedding and discuss several properties of these hyperplanes. The construction of these hyperplanes allows us to prove that there exists an ovoid of the Hermitian dual polar space (2n-1,K,{\textbackslash}theta)\$arising from its Grassmann-embedding if and only if there exists an empty \${\textbackslash}theta variety in \${\textbackslash}PG(n-1,K)\$. Using this result we are able to give the first examples of ovoids in thick dual polar spaces of rank at least 3 which arise from some projective embedding. These are also the first examples of ovoids in thick dual polar spaces of rank at least 3 for which the construction does not make use of transfinite recursion.},
author       = {De Bruyn, Bart},
issn         = {1077-8926},
journal      = {Electronic Journal of Combinatorics},
keyword      = {symplectic/Hermitian dual polar space,ovoid,Grassmann embedding},
language     = {eng},
number       = {1},
pages        = {1--20},
title        = {On a class of hyperplanes of the symplectic and Hermitian dual polar spaces.},
volume       = {16},
year         = {2009},
}


Chicago
De Bruyn, Bart. 2009. “On a Class of Hyperplanes of the Symplectic and Hermitian Dual Polar Spaces.” Electronic Journal of Combinatorics 16 (1): 1–20.
APA
De Bruyn, B. (2009). On a class of hyperplanes of the symplectic and Hermitian dual polar spaces. Electronic Journal of Combinatorics, 16(1), 1–20.
Vancouver
1.
De Bruyn B. On a class of hyperplanes of the symplectic and Hermitian dual polar spaces. Electronic Journal of Combinatorics. 2009;16(1):1–20.
MLA
De Bruyn, Bart. “On a Class of Hyperplanes of the Symplectic and Hermitian Dual Polar Spaces.” Electronic Journal of Combinatorics 16.1 (2009): 1–20. Print.