On Hermitian Spreads

Abstract

Let $\perp$ be the polarity of $PG(5,q)$ defined by the elliptic quadric $Q^-(5,q)$. A locally Hermitian spread ${\cal S}$ of $Q^-(5,q)$, with respect to a line $L$, is associated in a canonical way with a spread ${\cal S}_{\Lambda}$ of the $3$-dimensional projective space $L^{\perp} = \Lambda$, and conversely. In this paper we give a geometric characterization of the regular spreads of $\Lambda$ which induce Hermitian spreads of $Q^-(5,q)$.