---
tag: 6WV71T5CK1
kind: object
name: Klein quadric
msc: ["51M30", "51A50", "51N15"]
instance_of: [Q3Q4FF23VG]
uses: [B8PAMSJ81T, Q3Q4FF23VG, ZT9A6JK8QF]
sources:
  - url: https://en.wikipedia.org/wiki/Klein_quadric
same_as:
  wikidata: Q6420195
  wikipedia: Klein quadric
---

## Description

Let $F$ be a field.
The *Klein quadric* is the [[quadric|Q3Q4FF23VG]] in $\mathbb{P}^5(F)$, with coordinates $[p_{01} : p_{02} : p_{03} : p_{12} : p_{13} : p_{23}]$, given by
$$p_{01} p_{23} - p_{02} p_{13} + p_{03} p_{12} = 0.$$
Sending a line of $\mathbb{P}^3(F)$ to its [[Plücker coordinates|B8PAMSJ81T]] is a bijection from the set of lines of $\mathbb{P}^3(F)$ onto the Klein quadric.
This bijection is the Klein correspondence.

Two lines of $\mathbb{P}^3(F)$ meet if and only if the line of $\mathbb{P}^5(F)$ joining their images lies entirely on the Klein quadric.
Check on an example: the lines $L_1$ through $e_0, e_1$ and $L_2$ through $e_0, e_2$ meet at $[e_0]$.
Their images are $[1 : 0 : 0 : 0 : 0 : 0]$ and $[0 : 1 : 0 : 0 : 0 : 0]$.
Every point $[s : t : 0 : 0 : 0 : 0]$ of the joining line satisfies the equation, since each product in it has a zero factor.

## Remarks

The quadratic form $p_{01} p_{23} - p_{02} p_{13} + p_{03} p_{12}$ is a sum of three hyperbolic planes, so the Klein quadric is a non-degenerate quadric of maximal Witt index in $\mathbb{P}^5(F)$.
Its largest linear subspaces are planes, and they come in two families.
The planes of one family correspond to the sets of lines through a fixed point of $\mathbb{P}^3(F)$.
The planes of the other family correspond to the sets of lines in a fixed plane of $\mathbb{P}^3(F)$.
Over a finite field $\F_q$ the Klein quadric has $(q^2 + 1)(q^2 + q + 1)$ points, the number of lines of $PG(3, q)$.

## Explanations

### L1 human

The lines of three-dimensional projective space form a four-dimensional family.
The Klein quadric packages this family as a single curved hypersurface in a five-dimensional projective space.
Each line becomes one point, and questions about lines, such as whether two lines meet, become questions about points on the quadric.
The coordinates are named after Julius Plücker and the quadric after Felix Klein.