Cayley Cubic

A Cubic Ruled Surface (Fischer 1986) in which the director line meets the director Conic Section. Cayley's 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 3-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

$${1\over x_0}+{1\over x_1}+{1\over x_2}+{1\over x_3}=0$$

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

$$\begin{eqnarray*} x&=&{x_0\over v}\\ y&=&{x_1\over v}\\ z&=&{x_2\over v} \end{eqnarray*}$$

then gives the equation

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

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

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

is given by Endraß which, when rewritten in Tetrahedral Coordinates, becomes

$$x^2+y^2-x^2z+y^2z+z^2-1=0,$$

plotted in the right figure above.

The Hessian of the Cayley cubic is given by

$0={x_0}^2(x_1x_2+x_1x_3+x_2x_3)+x_1^2(x_0x_2+x_0x_3+x_2x_3)$
$+x_2^2(x_0x_1+x_0x_3+x_1x_3)+x_3^2(x_0x_1+x_0x_2+x_1x_2).$

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

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

plotted above (Hunt). The Hessian of the Cayley cubic has 14 Ordinary Double Points, four more than a the general Hessian of a smooth Cubic Surface (Hunt).

References

Endraß, S. ``Flächen mit vielen Doppelpunkten.'' DMV-Mitteilungen 4, 17-20, Apr. 1995.

Endraß, S. ``The Cayley Cubic.'' http://www.mathematik.uni-mainz.de/AlgebraischeGeometrie/docs/Ecayley.shtml.

Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, p. 14, 1986.

Fischer, G. (Ed.). Plate 33 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, p. 33, 1986.

Hunt, B. ``Algebraic Surfaces.'' http://www.mathematik.uni-kl.de/~wwwagag/Galerie.html.

Hunt, B. The Geometry of Some Special Arithmetic Quotients. New York: Springer-Verlag, pp. 115-122, 1996.

Nordstrand, T. ``The Cayley Cubic.'' http://www.uib.no/people/nfytn/cleytxt.htm.