Group–theoretic characterizations of classical ovoids

An ovoid of PG(3, q), q > 2, is a set of q + 1 points of PG(3, q), no three of which are collinear. The only known ovoids of PG(3, q) are the elliptic quadrics, which exist for all q, and the Suzuki-Tits ovoids, which exist for q = 2, e ≥ 3 odd, [10]. It is well known that for odd q, the only ovoids are the elliptic quadrics. For even q, the ovoids have been classified only for q up to and including 32. Elliptic ovoids and Suzuki-Tits ovoids are usually called “classical ovoids”. There are several results characterizing ovoids in PG(3, q), some involving geometry, some involving group-theory. Since a plane of PG(3, q) meets an ovoid either in a single point or in an oval, a successful technique in studying ovoids has involved examining their plane sections. The plane sections of an elliptic ovoid are all conics, while those of the SuzukiTits ovoid are all translation ovals, namely, ovals invariant under a group of elations of order q having a common axis, which are not conics. Conversely, it has been shown recently [14], that an ovoid admitting a pencil of translation ovals must be either an elliptic quadric or a Suzuki-Tits ovoid (by a pencil of an ovoid with carrier L is meant the set of ovals occurring as secant plane sections for the planes on a fixed tangent line L). The result by O’Keefe and Penttila is a refinement of a previous result by D.G. Glynn [9]. For other results in this direction see also [15], [19] and [13] for a survey on ovoids in PG(3, q). On the other hand, in 1966 Lüneburg [17], proved that if an ovoid O in PG(3, q), q even, admits an automorphism group containing a subgroup of even order which is transitive on the points of the ovoid, then O is a classical ovoid. Using linear codes, in 1987 Bagchi and Sastry [2] extended Luneburg’s result proving that if an ovoid of PG(3, q), q even, admits a point-transitive automorphism group, the the ovoid must be classical.