nLab Klein quadric

Idea

A projective variety that is moreover a quadric given by zeroes of the quadratic form

$$x_0x_5 + x_1 x_4 + x_2 x_3$$

where $[x_0:x_1:x_2:x_3:x_4:x_5]$ are homogeneous coordinates on $\mathbb{P}^5$. It was studied in (Klein 1868), together with what is now called the Klein correspondence?, by Felix Klein in his 1868 dissertation under Julius Plücker?.

Properties

The conformal compactification, of complexified Minkowski spacetime $\mathbb{R}^{3,1}$, is given by the Klein quadric. (eg, Fioresi-Lledo-Varadarajan 07, section 2).

Related concepts

• twistor correspondence

References

• Felix Klein, Über die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Koordinaten auf eine kanonische Form, dr. Phil. thesis (1868), published as Math. Ann 23 (1884), 539-578. doi:10.1007/BF01446603 (English translation pdf)

• The Klein quadric and triality (pdf)

• R. Fioresi, M. A. Lledo, Veeravalli Varadarajan, The Minkowski and conformal superspaces, J.Math.Phys.48:113505,2007 (arXiv:math/0609813)