 Klein quadric

The Plücker embedding of $\mathrm{Gr}(2,{\mathbb{C}}^4)$ is a smooth projective quadric in ${\mathbb{CP}}^5$. An antisymmetric $4\times 4$ 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.

The Klein quadric, also known as the Klein surface, is a remarkable geometric object in the field of algebraic geometry and topology. It is represented as a certain kind of algebraic variety, specifically a projective quadric surface in projective 3-space.