Eggs in PG(4n−1,q), q even, containing a pseudo-pointed conic

Matthew R. Brown, Michel Lavrauw
2005 European journal of combinatorics (Print)  
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 ([5]) proved that if an ovoid of PG(3, q), q even, contains a pointed conic, then either q = 4 and the ovoid is an elliptic quadric, or q = 8 and the ovoid is a Tits ovoid. Generalising the definition of an ovoid to a set of (n − 1)-spaces of PG(4n − 1, q) J. A. Thas [24] introduced the notion of
more » ... s or eggs: a set of q 2n + 1 (n − 1)-spaces in PG(4n − 1, q), with the property that any three egg elements span a (3n − 1)-space and at every egg element there is a unique tangent (3n − 1)-space. We prove that an egg in PG(4n − 1, q), q even, contains a pseudo pointed conic, that is, a pseudo-oval arising from a pointed conic of PG(2, q n ), q even, if and only if the egg is elementary and the ovoid is either an elliptic quadric in PG(3, 4) or a Tits ovoid in PG(3, 8) .
doi:10.1016/j.ejc.2003.12.014 fatcat:mwzaaku6cnb7bbxouptpnjwlha