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}
-
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
- Hunt, Bruce (1996). The Geometry of Some Special Arithmetic Quotients. Springer-Verlag. pp. 115–122. ISBN 3-540-61795-7.
- 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
External links
- Cayley’s Nodal Cubic Surface, John Baez, Visual Insight, 15 August 2016
- Cayley Surface on MathCurve.