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

Abstract

An ovoid of PG(3, q) can be defined as a set of q2 + 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(4n - 1, q), J. A. Thas introduced the notion of pseudo-ovoids 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. In this paper, a proof is given that an egg in PG(4n - 1, q), q even, contains a pseudo-conic (that is, a pseudo-oval arising from a conic of PG(2, qn)) if and only if the egg is classical (that is, arising from an elliptic quadric in PG(3, qn).