An algebraic surface of order 3. Schläfli and Cayley classified the singular cubic surfaces. On the general
cubic, there exists a curious geometrical structure called double
sixes, and also a particular arrangement of 27 (possibly complex) lines, as discovered
by Schläfli (Salmon 1965, Fischer 1986) and sometimes called Solomon's
seal lines. A nonregular cubic surface can contain 3, 7, 15, or 27 real lines
(Segre 1942, Le Lionnais 1983). The Clebsch
diagonal cubic contains all possible 27. The maximum number of ordinary
double points on a cubic surface is four, and the unique cubic surface having
four ordinary double points is the Cayley
cubic.
Schoute (1910) showed that the 27 lines can be put into a one-to-one correspondence with the vertices of a particular polytope
in six-dimensional space in such a manner that all incidence relations between the
lines are mirrored in the connectivity of the polytope
and conversely (Du Val 1933). A similar correspondence can be made between the 28
bitangents of the general plane quartic curve and
a seven-dimensional polytope (Coxeter 1928) and between
the tritangent planes of the canonical curve of genus 4 and an eight-dimensional
polytope (Du Val 1933).
A smooth cubic surface contains 45 tritangents (Hunt). The Hessian of smooth cubic surface contains at least 10 ordinary
double points, although the Hessian of the Cayley
cubic contains 14 (Hunt).
Bruce, J. and Wall, C. T. C. "On the Classification of Cubic Surfaces." J. London Math. Soc.19, 245-256, 1979.Cayley,
A. "A Memoir on Cubic Surfaces." Phil. Trans. Roy. Soc.159,
231-326, 1869.Coxeter, H. S. M. "The Pure Archimedean
Polytopes in Six and Seven Dimensions." Proc. Cambridge Phil. Soc.24,
7-9, 1928.Du Val, P. "On the Directrices of a Set of Points in
a Plane." Proc. London Math. Soc. Ser. 235, 23-74, 1933.Fischer,
G. (Ed.). Mathematische
Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband.
Braunschweig, Germany: Vieweg, pp. 9-14, 1986.Fladt, K. and Baur,
A. Analytische Geometrie spezieler Flächen und Raumkurven. Braunschweig,
Germany: Vieweg, pp. 248-255, 1975.Hunt, B. "Algebraic Surfaces."
http://www.mathematik.uni-kl.de/~hunt/drawings.html.Hunt,
B. "The 27 Lines on a Cubic Surface" and "Cubic Surfaces." Ch. 4
and Appendix B.4 in The
Geometry of Some Special Arithmetic Quotients. New York: Springer-Verlag,
pp. 108-167 and 302-310, 1996.Klein, F. "Über Flächen
dritter Ordnung." Gesammelte Abhandlungen, Band II. Berlin: Springer-Verlag,
pp. 11-62, 1973.Le Lionnais, F. Les
nombres remarquables. Paris: Hermann, p. 49, 1983.Rodenberg,
C. "Zur Classification der Flächen dritter Ordnung." Math. Ann.14,
46-110, 1878.Salmon, G. Analytic
Geometry of Three Dimensions. New York: Chelsea, 1965.Schläfli,
L. "On the Distribution of Surfaces of Third Order into Species, in Reference
to the Absence or Presence of Singular Points, and the Reality of Their Lines."
Philos. Trans. Roy. Soc. London153, 193-241, 1863.Schoute,
P. H. "On the Relation Between the Vertices of a Definite Sixdimensional
Polytope and the Lines of a Cubic Surface." Proc. Roy. Acad. Amsterdam13,
375-383, 1910.Segre, B. The Nonsingular Cubic Surface. Oxford,
England: Clarendon Press, 1942.
