Segre variety and ovoids in the polar space \(Q^+(7,q)\)
In PG(n,q), denote the point with coordinate vector X by P(X). Let \(S_{1,1,1}(q)\) be the Segre product of three PG(1,q)'s; that is, if they separately have coordinates \((x_ 0,x_ 1)\), \((y_ 0,y_ 1)\), \((z_ 0,z_ 1)\), then \(S_{1,1,1}(q)=\{P(x_ 0y_ 0z_ 0\), \(x_ 0y_ 0z_ 1\), \(x_ 0y_ 1z_ 0\), \(x_ 0y_ 1z_ 1\), \(x_ 1y_ 0z_ 0\), \(x_ 1y_ 0z_ 1\), \(x_ 1y_ 1z_ 0\), \(x_ 1y_ 1z_ 1)\}\). Let \(f: PG(5,q^ 3)\to PG(5,q^ 3)\) be given by \(P(X)f=P(X^ q)\), let L be a given line none of whose points are fixed by f, let U be the set of all planes meeting each of L, Lf, \(Lf^ 2\) in a point and let F be the subset of U consisting of planes D such that \(D\cap Lf^ i=Pf^ i\), \(i=0,1,2\), for some point P. Let \(g: PG^{(2)}(5,q^ 3)\to PG(19,q^ 3)\) be the Grassmann map from the planes of \(PG(5,q^ 3)\) to the points of \(PG(19,q^ 3)\) whose image is the Grassmannian \(G_{2,5}(q^ 3)\). Then Ug lies in a \(PG(7,q^ 3)\) and \(0_ 3=Fg\) is precisely \(T\cap G_{2,5}(q)\), where T is a PG(7,q). In fact, \(0_ 3=P(t^{1+q+q^ 2}\), \(t^{1+q}\), \(t^{1+q^ 2}\), t, \(t^{q+q^ 2}\), \(t^ q\), \(t^{q^ 2}\), \(1| t\in GF(q)\cup \{\infty \}\}\). Also Ug is the Segre variety \(S_{1,1,1}(q^ 3)\), which lies on the hyperbolic quadric \(H_ 7=V(x_ 0x_ 7+x_ 1x_ 6+x_ 2x_ 5+x_ 3x_ 4)\) when q is even. In that case \(0_ 3\) is an ovoid of \(H_ 7\), that is, a set of \(q^ 3+1\) points no two conjugate and meeting each generator of \(H_ 7\). The ovoid is the same as that constructed by \textit{W. M. Kantor} [SIAM J. Algebraic Discrete Methods 3, 151-165 (1982; Zbl 0493.51008)].
scientific article; zbMATH DE number 3944681