---
tag: H8J52N0P9A
kind: definition
name: Spread
msc: ["51E23", 51A40]
defines: H1AT0G82ZD
uses: [ZT9A6JK8QF, 2RD6T5KG59, 0ZXZ82FTX2, JZ5J6REAMD]
sources:
  - url: https://en.wikipedia.org/wiki/Spread_(projective_geometry)
---

## Definition

Let $\mathcal{P}$ be a [[projective space|ZT9A6JK8QF]] and let $k \ge 0$ be an integer.
A *$k$-spread* of $\mathcal{P}$ is a set $\mathcal{S}$ of $k$-dimensional [[subspaces|2RD6T5KG59]] such that every point lies in exactly one member of $\mathcal{S}$.

## Remarks

A $1$-spread is a *line spread*.
In the finite space $\mathrm{PG}(n, q)$ over the [[finite field|0ZXZ82FTX2]] $\F_q$, 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|JZ5J6REAMD]] of order $q^2$.