The $m$-ovoids of ${\cal W}(5,2)$

Abstract

In this paper we are concerned with $m$-ovoids of the symplectic polar space ${\cal W}(2n+1, q)$, $q$ even. In particular we show the existence of an elliptic quadric of ${\rm PG}(2n+1, q)$ not polarizing to ${\cal W}(2n+1, q)$ forming a $\left(\frac{q^n-1}{q-1}\right)$-ovoid of ${\cal W}(2n+1, q)$. A further class of $(q+1)$-ovoids of ${\cal W}(5, q)$ is exhibited. It arises by glueing together two orbits of a subgroup of ${\rm PSp}(6, q)$ isomorphic to ${\rm PSL}(2, q^2)$. We also show that the obtained $m$-ovoids do not fall in any of the examples known so far in the literature. Moreover, a computer classification of the $m$-ovoids of ${\cal W}(5, 2)$ is acquired. It turns out that ${\cal W}(5, 2)$ has $m$-ovoids if and only if $m = 3$ and that there are exactly three pairwise non-isomorphic examples. The first example comes from an elliptic quadric ${\cal Q}^-(5, 2)$ polarizing to ${\cal W}(5, 2)$, whereas the other two are the $3$-ovoids previously mentioned.