Maximal Partial Spreads of Polar Spaces

Abstract

Some constructions of maximal partial spreads of finite classical polar spaces are provided. In particular we show that, for $n \ge 1$, $\mathcal{H}(4n-1,q^2)$ has a maximal partial spread of size $q^{2n}+1$, $\mathcal{H}(4n+1,q^2)$ has a maximal partial spread of size $q^{2n+1}+1$ and, for $n \ge 2$, $\mathcal{Q}^+(4n-1,q)$, $\mathcal{Q}(4n-2,q)$, $\mathcal{W}(4n-1,q)$, $q$ even, $\mathcal{W}(4n-3,q)$, $q$ even, have a maximal partial spread of size $q^n+1$.

Cameron-Liebler sets of generators in finite classical polar spaces

“…From [9, Proposition 4 and Remark 5] the analogous statement for the Hermitian varieties H(2d − 1, q 2 ) follows. The proof in [9] also uses Segre varieties, but it is probably possible to give a proof similar to the above. We do not go into detail because we do not need this result in this paper.…”