= Klein quadric
{c}
{title2=$p_{12}p_{34}-p_{13}p_{24}+p_{14}p_{23}=0$}
{wiki}
The <Plücker embedding> of $\operatorname{Gr}(2,\mathbb C^4)$ is a smooth projective quadric in $\mathbb{CP}^5$. An antisymmetric $4\times4$ matrix is nonzero and decomposable precisely when its Pfaffian vanishes, giving the displayed equation. The wedge pairing of two points on this quadric vanishes exactly when the corresponding two-planes intersect. Thus the <Klein quadric> carries the complex conformal null-separation relation of <complexified compactified Minkowski space>.
Back to article page
