An infinite family of hyperovals of \(Q^+(5,q)\), \(q\) even
A hyperoval of a point-line geometry is a nonempty set of points meeting each line in either \(0\) or \(2\) points. The author investigates hyperovals of the Klein quadric \(Q^+(5,q)\). It is known that hyperovals of \(Q^+(5,q)\) can exist only for \(q\) even, and that the size of such a hyperoval is divisible by \(q+2\) and lies between \(\frac{(q^2+q+2)(q+2)}{2}\) and \((q^2 + 1)(q + 2)\). Let \(Q^+(5, q)\) be a hyperbolic quadric in PG\((5,q)\), \(q\) even, and let \(\Pi\) be a \(3\)-dimensional subspace of PG\((5,q)\) intersecting \(Q^+(5,q)\) in an elliptic quadric \(Q^-(3,q)\). The incidence structure having all points of \(\Pi\) as points and all tangent lines of \(Q^-(3,q)\) as lines gives an embedding of the generalized quadrangle \(W(q)\) in PG\((5,q)\). With each ovoid \(O\) of \(W(q)\) distinct from \(Q^-(3,q)\) the author associates a hyperoval of \(Q^+(5,q)\) of size \(((q^2+1) - \vert O \cap Q^-(3,q)\vert ) (q+2)\). He also determines necessary and sufficient conditions for two hyperovals arising from the construction to be isomorphic.
scientific article; zbMATH DE number 7920045
scientific article