Relative $m$-ovoids of elliptic quadrics

Abstract

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