Partial ovoids in classical finite polar spaces
A set \(O\) of points meeting exactly once every generator of a polar space \(P\) is called an ovoid of \(P\). A partial ovoid of \(P\) is a set meeting every generator of \(P\) at most once. There are several well-known results on the nonexistence of ovoids in polar spaces; for instance \(Q^{-}(2n+1,q)\), \(W(2n+1,q)\) and \(H(2n,q^2)\) for \(n\geq 2\) do not admit any ovoid. It is then most interesting to try to determine the maximum size of a partial ovoid contained in such spaces or, at least, to provide good upper bounds. In the first part of this paper, the author proves two inductive bounds on the size of an ovoid of \(Q^{-}(2n+1,q)\) and of \(W(2n+1,q)\) which improve (at least for large \(p\)) the results already known. In the second part of the paper a simple proof of the nonexistence of ovoids in \(H(2n+1,q^2)\) when \(n>q^3\) is presented.
scientific article; zbMATH DE number 2093317