Spread

Remarks

A $1$-spread is a line spread. In the finite space $\mathrm{PG}(n, q)$ over the finite field $\F_q$, a $k$-spread exists if and only if $k + 1$ divides $n + 1$.

Example: let $V = \F_{q^2}^2$, which is a vector space of dimension $4$ over $\F_q$, so its $1$-dimensional $\F_q$-subspaces are the points of $\mathrm{PG}(3, q)$. The $1$-dimensional $\F_{q^2}$-subspaces of $V$ are $2$-dimensional over $\F_q$, so they are lines of $\mathrm{PG}(3, q)$. Every nonzero vector of $V$ spans exactly one $1$-dimensional $\F_{q^2}$-subspace, so these $q^2 + 1$ lines form a line spread, called the regular (or Desarguesian) spread.

Non-example: two lines of $\mathrm{PG}(3, q)$ that meet in a point $P$ cannot both belong to a spread, since $P$ would lie in two members.

Given a line spread of $\mathrm{PG}(3, q)$, the André–Bruck–Bose construction produces a translation plane of order $q^2$.