Non-classical hyperplanes of \(\mathrm{DW}(5,q)\)

Bart De Bruyn¹

¹Ghent University Department of Mathematics Krijgslaan 281 (S22) B-9000 Gent (Belgium)

The electronic journal of combinatorics, Tome 20 (2013) no. 2

Résumé

The hyperplanes of the symplectic dual polar space $DW(5,q)$ arising from embedding, the so-called classical hyperplanes of $DW(5,q)$, have been determined earlier in the literature. In the present paper, we classify non-classical hyperplanes of $DW(5,q)$. If $q$ is even, then we prove that every such hyperplane is the extension of a non-classical ovoid of a quad of $DW(5,q)$. If $q$ is odd, then we prove that every non-classical ovoid of $DW(5,q)$ is either a semi-singular hyperplane or the extension of a non-classical ovoid of a quad of $DW(5,q)$. If $DW(5,q)$, $q$ odd, has a semi-singular hyperplane, then $q$ is not a prime number.

Détail

DOI : 10.37236/2425

Classification : 51A45, 51A50
Mots-clés : symplectic dual polar space, hyperplane, projective embedding