Cayley's nodal cubic surface

Real points of the Cayley surface

File:3D model of Cayley surface.stl 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

$wxy+xyz+yzw+zwx=0$

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.

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

A number of affine forms of the surface have been presented. Hunt uses $(1-3x-3y-3z)(xy+xz+yz)+6xyz=0$ by transforming coordinates $(u_0,u_1,u_2,u_3)$ to $(u_0,u_1,u_2,v=3(u_0+u_1+u_2+2u_3))$ and dehomogenizing by setting $x=u_0/v,y=u_1/v,z=u_2/v$. A more symmetrical form is

$x^2+y^2+z^2+x^{2}z-y^{2}z-1=0.$

References

• Cayley, Arthur (1869), "A Memoir on Cubic Surfaces", Philosophical Transactions of the Royal Society of London (The Royal Society) 159: 231–326, doi:10.1098/rstl.1869.0010, ISSN 0080-4614 
• Heath-Brown, D. R. (2003), "The density of rational points on Cayley's cubic surface", Proceedings of the Session in Analytic Number Theory and Diophantine Equations, Bonner Math. Schriften, 360, Bonn: Univ. Bonn, pp. 33 
• Hunt, Bruce (2000), "Nice modular varieties", Experimental Mathematics 9 (4): 613–622, doi:10.1080/10586458.2000.10504664, ISSN 1058-6458, http://projecteuclid.org/getRecord?id=euclid.em/1045759526 

External links

• Cayley’s Nodal Cubic Surface, John Baez, Visual Insight, 15 August, 2016
• Cayley Surface on MathCurve.