Maximal set of mutually skew lines in a finite projective space

Question

What is the maximum size of a set of mutually-skew lines in a finite projective space? The total number of lines in a finite projective space is well-known ($(q^2 + 1)(q^2 + q + 1)$) and mentioned in many places, including the Wikipedia page "Finite geometry". I am sure that the size of a set of skew lines has long been known, and I saw it on a web page yesterday, but today, realizing that I need the answer, I can't find any online reference. I am hoping that somebody knows the reference off the top of their head.

Answer 1

Indeed, in $\operatorname{PG}(3,q)$ there exist so-called spreads, which are sets of mutually skew lines covering all points. Hence, the number you asked for is $q^2 + 1$.

The standard construction is the Desarguesian spread. Take the set $S$ of all $q^2 + 1$ one-dimensional subspaces of the $\mathbb{F}_{q^2}$-vector space $V = (\mathbb{F}_{q^2})^2$ (i.e. all points of the projective line over $\mathbb{F}_{q^2}$). By the concept of field reduction, we consider $V$ as a 4-dimensional vector space over $\mathbb{F}_q$. Then $S$ is a set of $q^2 + 1$ two-dimensional subspaces of $V$ with pairwise trivial intersection. Hence $S$ is a spread in $\operatorname{PG}(V) \cong \operatorname{PG}(3,q)$.