Cubic surface
A central feature of smooth cubic surfaces X over an algebraically closed field is that they are all rational, as shown by Alfred Clebsch in 1866.
[1] That is, there is a one-to-one correspondence defined by rational functions between the projective plane
[4] As a result, every smooth cubic surface over the complex numbers is diffeomorphic to the connected sum
As a complex manifold (or an algebraic variety), the surface depends on the arrangement of those 6 points.
More precisely, Arthur Cayley and George Salmon showed in 1849 that every smooth cubic surface over an algebraically closed field contains exactly 27 lines.
As the coefficients of a smooth complex cubic surface are varied, the 27 lines move continuously.
As a result, a closed loop in the family of smooth cubic surfaces determines a permutation of the 27 lines.
[6] This graph was analyzed in the 19th century using subgraphs such as the Schläfli double six configuration.
For example, the 27 lines can be identified with the weights of the fundamental representation of the Lie group
The possible sets of singularities that can occur on a cubic surface can be described in terms of subsystems of the
For a smooth complex cubic surface, the Picard lattice can also be identified with the cohomology group
Pursuing these analogies, Vera Serganova and Alexei Skorobogatov gave a direct geometric relation between cubic surfaces and the Lie group
[10] In physics, 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.
After a complex linear change of coordinates, the Clebsch surface can also be defined by the equation in
Its connected components (in other words, the classification of smooth real cubic surfaces up to isotopy) were determined by Ludwig Schläfli (1863), Felix Klein (1865), and H. G. Zeuthen (1875).
denotes the connected sum of r copies of the real projective plane
[14] when the defining polynomial for X is sampled at random from the Gaussian ensemble induced by the Bombieri inner product.
Two smooth cubic surfaces are isomorphic as algebraic varieties if and only if they are equivalent by some linear automorphism of
More precisely, it is an open subset of the weighted projective space P(12345), by Salmon and Clebsch (1860).
[15] The lines on a cubic surface X over an algebraically closed field can be described intrinsically, without reference to the embedding of X in
There is a similar description of the cone of curves for any del Pezzo surface.
A smooth cubic surface X over a field k which is not algebraically closed need not be rational over k. As an extreme case, there are smooth cubic surfaces over the rational numbers Q (or the p-adic numbers
[18] For k infinite, unirationality implies that the set of k-rational points is Zariski dense in X.
The absolute Galois group of k permutes the 27 lines of X over the algebraic closure
If some orbit of this action consists of disjoint lines, then X is the blow-up of a "simpler" del Pezzo surface over k at a closed point.
More strongly, Yuri Manin proved a birational rigidity statement: two smooth cubic surfaces with Picard number 1 over a perfect field k are birational if and only if they are isomorphic.
[19] For example, these results give many cubic surfaces over Q that are unirational but not rational.
If the surface contains two singularities of the same type, the automorphism may permute them.
The following table shows all automorphism groups of singular cubic surfaces with no parameters.
