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.
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The Plücker embedding of is a smooth projective quadric in . An antisymmetric 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.