The 2-transitive ovoids

Abstract

An ovoid 0 in a classical polar space is a set of singular points such that every maximal totally singular subspace contains just one point in Lo. Ovoids are intimately connected with other combinatorial objects, including translation planes, spreads, partial geometries, codes, generalized hexagons, and Kerdock sets (see [S, 9, 163 for example). Interestingly enough, many of the known ovoids are in fact 2-transitive-that is, they admit a 2-transitive automorphism group (the notion of the automorphism group is made precise in (2.3)). For example, the Suzuki groups Sz(q) and the Ree groups *G,(q) act 2-transitively on ovoids in 4-dimensional symplectic geometry and 7-dimensional orthogonal geometry, respectively. Furthermore, the unitary groups PSU,(q) (for suitable prime powers q) and the linear groups PSL,(q3) (with q even) act 2-transitively on ovoids in 7or 8-dimensional orthogonal geometry. The occurrence of such a large number of 2-transitive ovoids suggests that a classification of them is worthwhile, much in the same spirit as Kantor’s classification of the finite linear spaces whose automorphism group acts 2-transitively on points [lo]. The classification of the 2-transitive ovoids appears as our Main Theorem in Section 2. We discover no new ovoids, however we obtain some new results concerning the number of isomorphism classes of 2-transitive ovoids (see Section 2). Our proof relies on the classification of the finite 2-transitive permutation groups, which in turn relies on the recent classification of finite simple groups. WC also draw upon several facts from the modular representation theory of finite groups.