On line covers of finite projective and polar spaces

An $m$-$cover$ of lines of a finite projective space ${\rm PG}(r,q)$ (of a finite polar space $\cal P$) is a set of lines $\cal L$ of ${\rm PG}(r,q)$ (of $\cal P$) such that every point of ${\rm PG}(r,q)$ (of $\cal P$) contains $m$ lines of $\cal L$, for some $m$. Embed ${\rm PG}(r,q)$ in ${\rm PG}(r,q^2)$. Let $\bar{\cal L}$ denote the set of points of ${\rm PG}(r,q^2)$ lying on the extended lines of $\cal L$. An $m$-cover $\cal L$ of ${\rm PG}(r,q)$ is an $(r-2)$-dual $m$-cover if there are two possibilities for the number of lines of $\cal L$ contained in an $(r-2)$-space of ${\rm PG}(r,q)$. Basing on this notion, we characterize $m$-covers $\cal L$ of ${\rm PG}(r,q)$ such that $\bar{\cal L}$ is a two-character set of ${\rm PG}(r,q^2)$. In particular, we show that if $\cal L$ is invariant under a Singer cyclic group of ${\rm PG}(r,q)$ then it is an $(r-2)$-dual $m$-cover. Assuming that the lines of $\cal L$ are lines of a symplectic polar space ${\cal W}(r,q)$ (of an orthogonal polar space ${\cal Q}(r,q)$ of parabolic type), similarly to the projective case we introduce the notion of an $(r-2)$-dual $m$-cover of symplectic type (of parabolic type). We prove that an $m$-cover $\cal L$ of ${\cal W}(r,q)$ (of ${\cal Q}(r,q)$) has this dual property if and only if $\bar{\cal L}$ is a tight set of an Hermitian variety ${\cal H}(r,q^2)$ or of ${\cal W}(r,q^2)$ (of ${\cal H}(r,q^2)$ or of ${\cal Q}(r,q^2)$). We also provide some interesting examples of $(4n-3)$-dual $m$-covers of symplectic type of ${\cal W}(4n-1,q)$.Comment: 20 page