Primitive ovoids in \(O_8^+ (q)\)
An orthogonal space \(O^+_8(q)\) is a pair \((V,Q)\) such that \(V\) is a vector space of dimension 8 over \(GF(q)\) and \(Q:V\to GF(q)\) is a nondegenerate quadratic form of hyperbolic type. An ovoid \(O\) of \(O^+_8(q)\) is a set of \(q^3+1\) singular points of \(O_8^+(q)\) such that every maximal totally singular subspace contains just one point of \(O\). A primitive (2-transitive) ovoid \(O\) is an ovoid whose automorphism group acts primitively (2-transitively) on \(O\). The 2-transitive ovoids have been classified by Kleidman [\textit{P. B. Kleidman}, J. Algebra 117, No. 1, 117-135 (1988; Zbl 0652.51013)], proving that they are exactly the ovoids constructed by Kantor [\textit{W. M. Kantor}, Can. J. Math. 34, 1195-1207 (1982; Zbl 0493.51006)]. In this paper it has been proved that a primitive ovoid of \(O^+_8(q)\) either is 2-transitive or it is isomorphic to the ovoid of \(O_8^+(5)\) due to Cooperstein [\textit{J. H. Conway}, \textit{P. B. Kleidman}, \textit{R. A. Wilson}, Geom. Dedicata 26, No. 2, 157-170 (1988; Zbl 0643.51015)].
scientific article; zbMATH DE number 1421331
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