On line-parallelisms of PG(3, q)

TL;DR

This paper characterizes and counts specific classes of parallelisms in three-dimensional projective space over finite fields, revealing a rich structure and large diversity depending on the parity of q.

Contribution

It provides a geometric characterization and enumeration of a class of parallelisms in PG(3, q) stabilized by a particular group, including explicit lower bounds on their number.

Findings

At least Θ(q^{q-1} q!) mutually inequivalent parallelisms for even q.

At least Θ(q^{2q-3}) mutually inequivalent parallelisms for odd q.

Geometric description of parallelisms admitting a stabilizing elementary abelian group.

Abstract

Let $\mathrm{PG}(3, q)$ denote the three-dimensional projective space over the finite field with $q$ elements. A line-spread of $\mathrm{PG}(3, q)$ is a collection $\mathcal{S}$ of mutually skew lines such that every point of $\mathrm{PG}(3, q)$ lies on exactly one line of $\mathcal{S}$. A parallelism of $\mathrm{PG}(3, q)$ is a set $\Pi$ of mutually skew line-spreads of $\mathrm{PG}(3, q)$ such that every line of $\mathrm{PG}(3, q)$ is contained in precisely one line-spread of $\Pi$. For a Desarguesian spread $\mathcal{D}$ and an elementary abelian group $E$ of order $q^2$ that stabilizes $\mathcal{D}$ and one of its lines, let $\mathcal{T}$ be the class of parallelisms of $\mathrm{PG}(3, q)$ admitting $E$, and comprising $\mathcal{D}$ and $q^2+q$ Hall spreads, each of which is obtained by switching one of the $q^2+q$ reguli of $\mathcal{D}$ through its $E$-fixed line. In this paper, the parallelisms in $\mathcal{T}$ are characterized geometrically and enumerated. Moreover, it is shown that $\mathcal{T}$ contains at least $\Theta(q^{q-1} q!)$ mutually inequivalent parallelisms for $q$ even, and at least $\Theta(q^{2q-3})$ mutually inequivalent parallelisms when $q$ is odd.