Relative $m$-ovoids of elliptic quadrics

Abstract

Let ${\mathcal{Q}}^{-}(2n+1,q)$ be an elliptic quadric of $\mathrm{PG}(2n+1,q)$. A relative $m$-ovoid of ${\mathcal{Q}}^{-}(2n+1,q)$ (with respect to a parablic section $\mathcal{Q}:=\mathcal{Q}(2n,q)\subset {\mathcal{Q}}^{-}(2n+1,q)$) is a subset $\mathcal{R}$ of points of ${\mathcal{Q}}^{-}(2n+1,q)\setminus \mathcal{Q}$ such that every generator of ${\mathcal{Q}}^{-}(2n+1,q)$ not contained in $\mathcal{Q}$ meets $\mathcal{R}$ in precisely $m$ points. A relative $m$-ovoid having the same size as its complement (in ${\mathcal{Q}}^{-}(2n+1,q)\setminus \mathcal{Q}$) is called a relative hemisystem. We show that a nontrivial relative $m$-ovoid of ${\mathcal{Q}}^{-}(2n+1,q)$ is necessarily a relative hemisystem, forcing $q$ to be even. Also, we construct an infinite family of relative hemisystems of ${\mathcal{Q}}^{-}(4n+1,q)$, $n\ge 2$, admitting $\mathrm{PSp}(2n,q^2)$ as an automorphism group. Finally, some applications are given.