Transitive ovoids of the Hermitian surface of PG(3, q2), q even

Abstract

As it is well known, the transitive ovoids of PG(3, q) are the non-degenerate quadrics and the Suzuki–Tits ovoids (see in: A. Blokhuis, J.W.P. Hirschfeld, D. Jugnickel, J.A. Thas (Eds.), Finite Geometries, Proceedings of the Fourth Isle of Thorns Conference, Developments in Mathematics, Kluwer, Boston, 2001, pp. 121–131). Kleidman (J. Algebra 117 (1988) 117) classified the 2-transitive ovoids of finite classical polar spaces. Kleidman's result was partially improved by Gunawardena (J. Combin. Theory Ser. A 89 (2000) 70) who determined the primitive ovoids of the quadric O 8 +( q). Transitive ovoids of the classical polar space arising from the Hermitian surface H(3,q 2) of PG(3, q 2) with even q are investigated in this paper. There are known two such ovoids up to projectivity, namely the classical ovoid and the Singer-type ovoid. Both are linearly transitive in the sense that the subgroup of PGU(4, q 2) preserving the ovoid is still transitive on it. Furthermore, the full collineation group preserving either of them is a subgroup of P ΓU(3, q 2). Our main result states that for q even the only linearly transitive ovoids are the classical ovoids and the Singer-type ovoid. It remains open the problem of finding other (i.e. non-linearly) transitive ovoids, although we prove that the full collineation group of any transitive ovoid is a subgroup of P ΓU(3, q 2).