Cubic surface
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A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single polynomial which is homogeneous of degree 3 (hence, cubic). For example, if <math>\mathbb{P}^3<math> has homogeneous co-ordinates <math>[X:Y:Z:W]<math>, and <math>F([X:Y:Z:W]) = X^3 + Y^3 + Z^3 + W^3<math>, then the set of points where F equals zero is a cubic surface.
Cubic surfaces are among the most famous examples of varieties studied by the classical algebraic geometers (before the introduction by Grothendieck of schemes), especially the Italian school of algebraic geometry, and remain important examples to this day. They are examples of del Pezzo surfaces.
The polynomial <math>F([X:Y:Z:W]) = X^3 + Y^3 + Z^3 + W^3<math> gives an example of a smooth cubic surface. A smooth cubic surface over an algebraically closed field famously contains 27 lines. The arrangement of these lines (e.g., which lines meet which other ones) is very interesting, and has links to Lie theory. The smooth cubic surfaces can also be described as rational surfaces obtained by blowing up six points in the projective plane in general position (in this case, general position means not all on a conic).
With this description, the 27 lines can be listed out: the 6 exceptional divisors, the proper transforms of the 15 lines in <math>\mathbb{P}^2<math> which join two of the blown up points, and the proper transforms of the 6 conics in <math>\mathbb{P}^2<math> which contain all but one of the blown up points. These 27 lines can be thought of as forming the 27-dimensional fundamental representation of the group E6. They can be identified with the 27 possible charges of M-theory on a six-dimensional torus (6 momenta; 15 membranes; 6 fivebranes) and the group E6 then naturally acts as the U-duality group. This map between del Pezzo surfaces and M-theory on tori is known as Mysterious duality.
The polynomial <math>F([X:Y:Z:W]) = X^3 + Y^3 + Z^2 W<math> gives an example of an irreducible singular cubic surface, with singular point <math>[0:0:0:1]<math>. The singular cubic surfaces also have interesting properties: they contain lines, and the number and arrangement of the lines is related to the type of the singularity.