# Cayley's nodal cubic surface

Real points of the Cayley surface
3D model of Cayley surface

In algebraic geometry, the Cayley surface, named after Arthur Cayley, is a cubic nodal surface in 3-dimensional projective space with four conical points. It can be given by the equation

${\displaystyle wxy+xyz+yzw+zwx=0\ }$

when the four singular points are those with three vanishing coordinates. Changing variables gives several other simple equations defining the Cayley surface.

As a del Pezzo surface of degree 3, the Cayley surface is given by the linear system of cubics in the projective plane passing through the 6 vertices of the complete quadrilateral. This contracts the 4 sides of the complete quadrilateral to the 4 nodes of the Cayley surface, while blowing up its 6 vertices to the lines through two of them. The surface is a section through the Segre cubic.[1]

The surface contains nine lines, 11 tritangents and no double-sixes.[1]

A number of affine forms of the surface have been presented. Hunt uses

${\displaystyle (1-3x-3y-3z)(xy+xz+yz)+6xyz=0}$
by transforming coordinates ${\displaystyle (u_{0},u_{1},u_{2},u_{3})}$ to ${\displaystyle (u_{0},u_{1},u_{2},v=3(u_{0}+u_{1}+u_{2}+2u_{3}))}$ and dehomogenizing by setting ${\displaystyle x=u_{0}/v,y=u_{1}/v,z=u_{2}/v}$.[1] A more symmetrical form is

${\displaystyle x^{2}+y^{2}+z^{2}+x^{2}z-y^{2}z-1=0.}$[2]

## References

1. ^ a b c Hunt, Bruce (1996). The Geometry of Some Special Arithmetic Quotients. Springer-Verlag. pp. 115–122. ISBN 3-540-61795-7.
2. ^