Cayley's nodal cubic surface

Cubic Nodal 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

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

x 2 + y 2 + z 2 + x 2 z − y 2 z − 1 = 0. {\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. ^ Weisstein, Eric W. "Cayley cubic". MathWorld.
  • 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
  • Cayley’s Nodal Cubic Surface, John Baez, Visual Insight, 15 August 2016
  • Cayley Surface on MathCurve.