Let Delta be a finite thick dual polar space of rank 3. We say that a hyperplane H of Delta is locally singular (respectively, quadrangular or ovoidal) if H boolean AND Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of Delta. If H is locally singular, quadrangular, or ovoidal, then we say that H is uniform. It is known that if H is locally singular, then either H is the set of points at non-maximal distance from a given point of Delta or Delta is the dual of L(6, q) and H arises from the generalized hexagon H(q). In this paper we prove that only two examples exist for the locally quadrangular case, arising in L(6, 2) and H (5, 4), respectively. We fail to rule out the locally ovoidal case, but we obtain some partial results on it, which imply that, in this case, the geometry Delta H induced by Delta on the complement of H cannot be flag-transitive. As a bi-product, the hyperplanes H with Delta H flag-transitive are classified. (C) 2001 Academic Press.

Pasini, A., & Shpectorov, S.V. (2001). Uniform hyperplanes of finite dual polar spaces of rank 3. JOURNAL OF COMBINATORIAL THEORY. SERIES A, 94(2), 276-288 [10.1006/jcta.2000.3136].

Uniform hyperplanes of finite dual polar spaces of rank 3

PASINI, ANTONIO;
2001

Abstract

Let Delta be a finite thick dual polar space of rank 3. We say that a hyperplane H of Delta is locally singular (respectively, quadrangular or ovoidal) if H boolean AND Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of Delta. If H is locally singular, quadrangular, or ovoidal, then we say that H is uniform. It is known that if H is locally singular, then either H is the set of points at non-maximal distance from a given point of Delta or Delta is the dual of L(6, q) and H arises from the generalized hexagon H(q). In this paper we prove that only two examples exist for the locally quadrangular case, arising in L(6, 2) and H (5, 4), respectively. We fail to rule out the locally ovoidal case, but we obtain some partial results on it, which imply that, in this case, the geometry Delta H induced by Delta on the complement of H cannot be flag-transitive. As a bi-product, the hyperplanes H with Delta H flag-transitive are classified. (C) 2001 Academic Press.
Pasini, A., & Shpectorov, S.V. (2001). Uniform hyperplanes of finite dual polar spaces of rank 3. JOURNAL OF COMBINATORIAL THEORY. SERIES A, 94(2), 276-288 [10.1006/jcta.2000.3136].
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11365/7073
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo