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

An ovoid of PG(3, q) can be defined as a set of q + 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 pseudo-ovoids or eggs: a set of q + 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), 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). ∗This research has been supported by the Australian Research Council. †This research has been supported by a Marie Curie Fellowship of the European Community programme ”Improving the Human Research Potential and the Socio-Economic Knowledge Base” under the contract number HMPF-CT-2001-01386.