Relative m-ovoids of elliptic quadrics

Let Q − (2n+1,q) be an elliptic quadric of PG(2n+1,q). A relative m-ovoid of Q − (2n+1,q) (with respect to a parabolic section Q≔Q(2n,q)⊂Q − (2n+1,q)) is a subset R of points of Q − (2n+1,q)∖Q such that every generator of Q − (2n+1,q) not contained in Q meets R in precisely m points. A relative m-ovoid having the same size as its complement (in Q − (2n+1,q)∖Q) is called a relative hemisystem. We show that a nontrivial relative m-ovoid of Q − (2n+1,q) is necessarily a relative hemisystem, forcing q to be even. Also, we construct an infinite family of relative hemisystems of Q − (4n+1,q), n≥2, admitting PSp(2n,q 2 ) as an automorphism group. Finally, some applications are given.