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Cayley Cubic


CayleyCubicCayleyCubic2

Cayley's cubic surface is the unique cubic surface having four ordinary double points (Hunt), the maximum possible for cubic surface (Endraß). The Cayley cubic is invariant under the tetrahedral group and contains exactly nine lines, six of which connect the four nodes pairwise and the other three of which are coplanar (Endraß).

If the ordinary double points in projective three-space are taken as (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), then the equation of the surface in projective coordinates is

 1/(x_0)+1/(x_1)+1/(x_2)+1/(x_3)=0
(1)

(Hunt). Defining "affine" coordinates with plane at infinity v=x_0+x_1+x_2+2x_3 and

x=(x_0)/v
(2)
y=(x_1)/v
(3)
z=(x_2)/v
(4)

then gives the equation

 -5(x^2y+x^2z+y^2x+y^2z+z^2y+z^2x)+2(xy+xz+yz)=0
(5)

plotted in the left figure above (Hunt). The slightly different form

 4(x^3+y^3+z^3+w^3)-(x+y+z+w)^3=0
(6)

is given by Endraß (2003) which, when rewritten in tetrahedral coordinates, becomes

 x^2+y^2-x^2z+y^2z+z^2-1=0,
(7)

plotted in the right figure above.

CayleyCubicHessian

The Hessian of the Cayley cubic is given by

 0=x_0^2(x_1x_2+x_1x_3+x_2x_3)+x_1^2(x_0x_2+x_0x_3+x_2x_3) 
 +x_2^2(x_0x_1+x_0x_3+x_1x_3)+x_3^2(x_0x_1+x_0x_2+x_1x_2)
(8)

in homogeneous coordinates x_0, x_1, x_2, and x_3. Taking the plane at infinity as v=5(x_0+x_1+x_2+2x_3)/2 and setting x, y, and z as above gives the equation

 25[x^3(y+z)+y^3(x+z)+z^3(x+y)]+50(x^2y^2+x^2z^2+y^2z^2) 
 -125(x^2yz+y^2xz+z^2xy)+60xyz-4(xy+xz+yz)=0,
(9)

plotted above (Hunt). The Hessian of the Cayley cubic has 14 ordinary double points, four more than the general Hessian of a smooth cubic surface (Hunt).


See also

Cayley Surface

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References

Endraß, S. "Flächen mit vielen Doppelpunkten." DMV-Mitteilungen 4, 17-20, Apr. 1995.Endraß, S. "The Cayley Cubic." Feb. 6, 2003. http://enriques.mathematik.uni-mainz.de/docs/Ecayley.shtml.Fischer, G. (Ed.). Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband. Braunschweig, Germany: Vieweg, p. 14, 1986.Fischer, G. (Ed.). Plate 33 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 33, 1986.Hunt, B. "Some Beautiful Algebraic Surfaces." http://www.mathematik.uni-kl.de/~hunt/drawings.html.Hunt, B. The Geometry of Some Special Arithmetic Quotients. New York: Springer-Verlag, pp. 115-122, 1996.Nordstrand, T. "The Cayley Cubic." http://jalape.no/math/cleytxt.

Cite this as:

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

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