Reguli and regular spreads

Let ${\displaystyle \Sigma }$ be the projective space ${\displaystyle PG(2n+1,K)}$ for ${\displaystyle n\geq 1}$ an integer, and ${\displaystyle K}$ a division ring. A regulus^[4] ${\displaystyle R}$ in ${\displaystyle \Sigma }$ is a collection of pairwise disjoint ${\displaystyle n}$-dimensional subspaces with the following properties:

1. ${\displaystyle R}$ contains at least 3 elements
2. Every line meeting three elements of ${\displaystyle R}$, called a transversal, meets every element of ${\displaystyle R}$
3. Every point of a transversal to ${\displaystyle R}$ lies on some element of ${\displaystyle R}$

Any three pairwise disjoint ${\displaystyle n}$-dimensional subspaces in ${\displaystyle \Sigma }$ lie in a unique regulus.^[5] A spread ${\displaystyle S}$ of ${\displaystyle \Sigma }$ is regular if for any three distinct ${\displaystyle n}$-dimensional subspaces of ${\displaystyle S}$, all the members of the unique regulus determined by them are contained in ${\displaystyle S}$. Regular spreads are significant in the theory of translation planes, in that they generate Moufang planes in general, and Desarguesian planes in the finite case when the order of the ambient field is greater than ${\displaystyle 2}$. All spreads of ${\displaystyle PG(2n+1,2)}$ are trivially regular, since a regulus only contains three elements.

Constructing a regular spread

Construction of a regular spread is most easily seen using an algebraic model. Letting ${\displaystyle V}$ be a ${\displaystyle (2n+2)}$-dimensional vector space over a field ${\displaystyle F}$, one can model the ${\displaystyle k}$-dimensional subspaces of ${\displaystyle PG(2n+1,F)}$ using the ${\displaystyle (k+1)}$-dimensional subspaces of ${\displaystyle V}$; this model uses homogeneous coordinates to represent points and hyperplanes. Incidence is defined by intersection, with subspaces intersecting in only the zero vector considered disjoint; in this model, the zero vector of ${\displaystyle V}$ is effectively ignored.

Let ${\displaystyle F}$ be a field and ${\displaystyle E}$ an ${\displaystyle n}$-dimensional extension field of ${\displaystyle F}$. Consider ${\displaystyle V=E\oplus E}$ as a ${\displaystyle 2n}$-dimensional vector space over ${\displaystyle F}$, which provides a model for the projective space ${\displaystyle PG(2n-1,F)}$ as above. Each element of ${\displaystyle V}$ can be written uniquely as ${\displaystyle (x,y)}$ where ${\displaystyle x,y\in E}$. A regular spread is given by the set of ${\displaystyle n}$-dimensional projective spaces defined by ${\displaystyle J(k)=\{(x,kx):x\in E\}}$, for each ${\displaystyle k\in E}$, together with ${\displaystyle J(\infty )=\{(0,y):y\in E\}}$.^[6]

Spread sets

The construction of a regular spread above is an instance of a more general construction of spreads, which uses the fact that field multiplication is a linear transformation over ${\displaystyle E}$ when considered as a vector space. Since ${\displaystyle E}$ is a finite ${\displaystyle n}$-dimensional extension over ${\displaystyle F}$, a linear transformation from ${\displaystyle E}$ to itself can be represented by an ${\displaystyle n\times n}$ matrix with entries in ${\displaystyle F}$. A spread set is a set ${\displaystyle S}$ of ${\displaystyle n\times n}$ matrices over ${\displaystyle F}$ with the following properties:

• ${\displaystyle S}$ contains the zero matrix and the identity matrix
• For any two distinct matrices ${\displaystyle X}$ and ${\displaystyle Y}$ in ${\displaystyle S}$, ${\displaystyle X-Y}$ is nonsingular
• For each pair of elements ${\displaystyle a,b\in E}$, there is a unique ${\displaystyle X\in S}$ such that ${\displaystyle aX=b}$

In the finite case, where ${\displaystyle E}$ is the field of order ${\displaystyle q^{n}}$ for some prime power ${\displaystyle q}$, the last condition is equivalent to the spread set containing ${\displaystyle q^{n}}$ matrices. Given a spread set ${\displaystyle S}$, one can create a spread as the set of ${\displaystyle n}$-dimensional projective spaces defined by ${\displaystyle J(k)=\{(x,xM):x\in E\}}$, for each ${\displaystyle M\in S}$, together with ${\displaystyle J(\infty )=\{(0,y):y\in E\}}$,^[2] As a specific example, the following nine matrices represent ${\displaystyle GF(9)}$ as 2 × 2 matrices over ${\displaystyle GF(3)}$ and so provide a spread set of ${\displaystyle AG(2,9)}$.^[6]

${\displaystyle \left[{\begin{matrix}0&0\\0&0\end{matrix}}\right],\left[{\begin{matrix}1&0\\0&1\end{matrix}}\right],\left[{\begin{matrix}2&0\\0&2\end{matrix}}\right],\left[{\begin{matrix}0&1\\2&0\end{matrix}}\right],\left[{\begin{matrix}1&1\\2&1\end{matrix}}\right],\left[{\begin{matrix}2&1\\2&2\end{matrix}}\right],\left[{\begin{matrix}0&2\\1&0\end{matrix}}\right],\left[{\begin{matrix}1&2\\1&1\end{matrix}}\right],\left[{\begin{matrix}2&2\\2&1\end{matrix}}\right]}$

Another example of a spread set yields the Hall plane of order 9^[6]

${\displaystyle \left[{\begin{matrix}0&0\\0&0\end{matrix}}\right],\left[{\begin{matrix}1&0\\0&1\end{matrix}}\right],\left[{\begin{matrix}2&0\\0&2\end{matrix}}\right],\left[{\begin{matrix}1&1\\1&2\end{matrix}}\right],\left[{\begin{matrix}2&2\\2&1\end{matrix}}\right],\left[{\begin{matrix}0&1\\2&0\end{matrix}}\right],\left[{\begin{matrix}0&2\\1&0\end{matrix}}\right],\left[{\begin{matrix}1&2\\2&2\end{matrix}}\right],\left[{\begin{matrix}2&1\\1&1\end{matrix}}\right]}$

Modifying spreads

One common approach to creating new spreads is to start with a regular spread and modify it in some way. The techniques presented here are some of the more elementary examples of this approach.

Geometric techniques

There are several known ways to construct spreads of ${\displaystyle PG(3,q)}$ from other geometrical objects without reference to an initial regular spread. Some well-studied approaches to this are given below.