Definitions

# Cubic surface

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 $mathbb\left\{P\right\}^3$ has homogeneous co-ordinates $\left[X:Y:Z:W\right]$, and

$F\left(\left[X:Y:Z:W\right]\right) = X^3 + Y^3 + Z^3 + W^3,$

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, especially in the Italian school of algebraic geometry, and remain important examples to this day. They are examples of del Pezzo surfaces.

A smooth cubic surface over an algebraically closed field is well known to contain 27 lines. This was one of the most celebrated geometric results of the nineteenth century. A smooth cubic surface can also be described as a rational surface obtained by blowing up six points in the projective plane in general position (in this case, “general position” means no three points are aligned and no six are on a conic section).

With this description, the 27 lines can be listed out: the exceptional divisors above the 6 blown up points, the proper transforms of the 15 lines in $mathbb\left\{P\right\}^2$ which join two of the blown up points, and the proper transforms of the 6 conics in $mathbb\left\{P\right\}^2$ which contain all but one of the blown up points.

There is a lot of interest in trying to understand which of the lines can be found when the field in not algebraically closed. For example, Clebsch gave a model of a cubic surface where all the lines are defined over the field of reals.

There are other ways of thinking of these 27 lines. For example, if one projects the cubic from a point which is not on any line (most points of the cubic are like this) then we obtain a double cover of the plane branched along a smooth quartic curve. The 27 lines are mapped to 27 out of the 28 bi-tangents to this quartic curve; the 28th line is the image of the exceptional locus of the blow-up.

The 27 lines can also be identified with some objects arising in representation theory. In particular, these 27 lines can be thought of as forming the 27-dimensional fundamental representation of the group E6.

Of late there has been some interest in the cubic surface from physicists interested in String Theory. They have shown that the 27 lines 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

$F\left(\left[X:Y:Z:W\right]\right) = X^3 + Y^3 + Z^2 W$

gives an example of an irreducible singular cubic surface, with the singular point $\left[0:0:0:1\right]$. 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.