On line covers of finite projective and polar spaces

Abstract

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