On regular systems of finite classical polar spaces

Abstract

Let $\mathcal{P}$ be a finite classical polar space of rank $d$. An $m$-regular system with respect to $(k-1)$-dimensional projective spaces of $\mathcal{P}$, $1\le k\le d-1$, is a set $\mathcal{R}$ of generators of $\mathcal{P}$ with the property that every $(k-1)$-dimensional projective space of $\mathcal{P}$ lies on exactly $m$ generators of $\mathcal{R}$. Regular systems of polar spaces are investigated. Some non-existence results about certain 1-regular systems of polar spaces with low rank are proved and a procedure to obtain $m^{\prime }$-regular systems from a given $m$-regular system is described. Finally, three different construction methods of regular systems w.r.t. points of various polar spaces are discussed.