Cayley's nodal cubic surface

Not to be confused with Cayley's ruled cubic 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

Real points of the Cayley surface

3D model of Cayley surface

$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, 159, The Royal Society: 231–326, doi:10.1098/rstl.1869.0010, ISSN 0080-4614, JSTOR 108997
• 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, vol. 360, Bonn: Univ. Bonn, p. 33, MR 2075628
• Hunt, Bruce (2000), "Nice modular varieties", Experimental Mathematics, 9 (4): 613–622, doi:10.1080/10586458.2000.10504664, ISSN 1058-6458, MR 1806296

External links

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