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Eggs in PG(4 n − 1, q ), q Even, Containing a Pseudo‐Conic
Author(s) -
Brown Matthew R.,
Lavrauw Michel
Publication year - 2004
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609304003169
Subject(s) - ovoid , quadric , conic section , mathematics , tangent space , combinatorics , tangent , geometry , pure mathematics
An ovoid of PG(3, q ) can be defined as a set of q 2 + 1 points with the property that every three points span a plane, and at every point there is a unique tangent plane. In 2000, M. R. Brown proved that if an ovoid of PG(3, q ), q even, contains a conic, then the ovoid is an elliptic quadric. Generalising the definition of an ovoid to a set of ( n − 1)‐spaces of PG(4 n − 1, q ), J. A. Thas introduced the notion of pseudo‐ovoids or eggs : a set of q 2 n + 1 ( n − 1)‐spaces in PG(4 n − 1, q ), with the property that any three egg elements span a (3 n − 1)‐space and at every egg element there is a unique tangent (3 n − 1)‐space. In this paper, a proof is given that an egg in PG(4 n − 1, q ), q even, contains a pseudo‐conic (that is, a pseudo‐oval arising from a conic of PG(2, q n )) if and only if the egg is classical (that is, arising from an elliptic quadric in PG(3, q n )). 2000 Mathematics Subject Classification 51E20.