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Clebsch Diagonal Cubic

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The Clebsch diagonal cubic is 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)
ClebschDiagonalCubic

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^2y+x^2z+y^2x+y^2z+z^2x+z^2y)+54xyz+126(xy+xz+yz)-9(x^2+y^2+z^2)-9(x+y+z)+1=0
(3)

(Hunt 1996), illustrated above.

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 1986ab, Hunt 1996), and the Clebsch diagonal surface is the unique cubic surface containing 10 such points (Hunt 1996).

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

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References

Fischer, G. (Ed.). Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband. Braunschweig, Germany: Vieweg, pp. 9-11, 1986a.Fischer, G. (Ed.). Plates 10-12 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 13-15, 1986b.Hunt, B. The Geometry of Some Special Arithmetic Quotients. New York: Springer-Verlag, pp. 122-128, 1996.Nordstrand, T. "Clebsch Diagonal Surface." http://jalape.no/math/clebtxt.Yode. "How to Plot Clebsch Surface with those 27 Lines?" Feb. 1, 2023. https://mathematica.stackexchange.com/questions/279405/how-to-plot-clebsch-surface-with-those-27-lines#comment697874_279420.

Cite this as:

Weisstein, Eric W. "Clebsch Diagonal Cubic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ClebschDiagonalCubic.html

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