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

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. Kleidmanʹs result was partially improved by Gunawardena (J. Combin. Theory Ser. A 89 (2000) 70) who determined the primitive ovoids of the quadric O8+(q). Transitive ovoids of the classical polar space arising from the Hermitian surface H(3,q2) of PG(3,q2) 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,q2) preserving the ovoid is still transitive on it. Furthermore, the full collineation group preserving either of them is a subgroup of PΓU(3,q2). 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 PΓU(3,q2).