Clebsch Diagonal Cubic

A Cubic Algebraic Surface given by the equation

$${x_0}^3+{x_1}^3+{x_2}^3+{x_3}^3+{x_4}^3=0,$$ (1)

with the added constraint

$$x_0+x_1+x_2+x_3+x_4=0.$$ (2)

The implicit equation obtained by taking the plane at infinity as $x_0+x_1+x_2+x_3/2$ is

$81(x^3+y^3+z^3)-189(x^2 y+x^2 z+y^2 x+y^2 z+z^2 x+z^2 y)$
$+54x y z+126(x y+x z+y z)-9(x^2+y^2+z^2)-9(x+y+z)+1=0\quad$	(3)

(Hunt, Nordstrand). On Clebsch's diagonal surface, all 27 of the complex lines (Solomon's Seal Lines) present on a general smooth Cubic Surface are real. In addition, there are 10 points on the surface where 3 of the 27 lines meet. These points are called Eckardt Points (Fischer 1986, Hunt), and the Clebsch diagonal surface is the unique Cubic Surface containing 10 such points (Hunt).

If one of the variables describing Clebsch's diagonal surface is dropped, leaving the equations

$${x_0}^3+{x_1}^3+{x_2}^3+{x_3}^3=0,$$ (4)

$$x_0+x_1+x_2+x_3=0,$$ (5)

the equations degenerate into two intersecting Planes given by the equation

$$(x+y)(x+z)(y+z)=0.$$ (6)

See also Cubic Surface, Eckardt Point

References

Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 9-11, 1986.

Fischer, G. (Ed.). Plates 10-12 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 13-15, 1986.

Hunt, B. The Geometry of Some Special Arithmetic Quotients. New York: Springer-Verlag, pp. 122-128, 1996.

Nordstrand, T. ``Clebsch Diagonal Surface.'' http://www.uib.no/people/nfytn/clebtxt.htm.