Triads, flocks of conics and \(\text{Q}^-(5,q)\)
Let \(\mathcal O\) be an ovoid in \(PG(3,q)\), \(q\) even. Embed \(PG(3,q)= \Sigma_\infty\) as a hyperplane in \(PG(4,q)\) and consider the GQ \(T_3({\mathcal O})\). We will call the points of \(T_3({\mathcal O})\) which are not collinear with \((\infty)\) (that is the points of \(PG(4,q)-\Sigma_\infty\)) the affine points of \(T_3({\mathcal O})\). A triad \(\{P,Q,R\}\) of non-collinear affine points of \(T_3({\mathcal O})\) has the distinct projections property if the projections of the plane \(\langle P,Q,R\rangle\) from each of the affine centers of \(\{P,Q,R\}\) onto \(\Sigma_\infty\) are pairwise distinct. The main result of this paper is Theorem 6. Suppose there is a plane \(\Pi\) of \(PG(4,q)\) meeting \(\Sigma_\infty\) in a line external to \(\mathcal O\) and such that each triad \(\{P,Q,R\}\) of non-collinear affine points of \(\Pi\) has the distinct projections property. Then \(T_3({\mathcal O})\) is isomorphic to \(Q^-(5,q)\).
scientific article; zbMATH DE number 1548358