In algebraic geometry, the Cayley surface, named after Arthur Cayley, is a cubic

nodal surface In algebraic geometry, a nodal surface is a surface in (usually complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex ...

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 In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general ...

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 In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six l ...

. 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 In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by . Definition The Segre cubic is the set of points (''x''0:''x''1:''x''2:''x''3:''x''4:''x''5) of ''P''5 satisfyin ...

. The surface contains nine lines, 11 tritangents and no double-sixes. A number of affine forms of the surface have been presented. Hunt uses

(1-3 x-3y-3z)(xy+xz+yz)+6xyz = 0

by transforming coordinates

(u_0, u_1, u_2, u_3)

(u_0, u_1, u_2, v=3(u_0+u_1+u_2+ 2 u_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

* * *{{Citation , last1=Hunt , first1=Bruce , title=Nice modular varieties , url=http://projecteuclid.org/getRecord?id=euclid.em/1045759526 , year=2000 , journal=Experimental Mathematics , issn=1058-6458 , volume=9 , issue=4 , pages=613–622 , mr=1806296 , doi=10.1080/10586458.2000.10504664

External links

Cayley’s Nodal Cubic Surface

John Baez John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...

, Visual Insight, 15 August, 2016
Cayley Surface
on MathCurve. Algebraic surfaces Complex surfaces