Cayley's nodal cubic surface

Real points of the Cayley surface
3D model of Cayley surface

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

${\displaystyle wxy+xyz+yzw+zwx=0\ }$

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.

References

• Cayley, Arthur (1869), "A Memoir on Cubic Surfaces", Philosophical Transactions of the Royal Society of London, The Royal Society, 159: 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, 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