Cubic surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space, and so cubic surfaces are generally considered in projective 3-space formula_1. The theory also becomes more uniform by focusing on surfaces over the complex numbers rather than the real numbers; note that a complex surface has real dimension 4. A simple example is the Fermat cubic surface in formula_1. Many properties of cubic surfaces hold more  generally for del Pezzo surfaces. Rationality of cubic surfaces. 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. That is, there is a one-to-one correspondence defined by rational functions between the projective plane formula_4 minus a lower-dimensional subset and "X" minus a lower-dimensional subset. More generally, every irreducible cubic surface (possibly singular) over an algebraically closed field is rational unless it is the projective cone over a cubic curve. In this respect, cubic surfaces are much simpler than smooth  surfaces of degree at least 4 in formula_1, which are never rational. In characteristic zero, smooth surfaces of degree at least 4 in formula_1 are not even uniruled. More strongly, Clebsch showed that every smooth cubic surface in formula_1 over an algebraically closed field is isomorphic to the blow-up of formula_4 at 6 points. As a result, every smooth cubic surface over the complex numbers is diffeomorphic to the connected sum formula_9, where the minus sign refers to a change of orientation. Conversely, the blow-up of formula_4 at 6 points is isomorphic to a cubic surface if and  only if the points are in general position, meaning that no three points lie on a line and all 6 do not lie on a conic. As a complex manifold (or an algebraic variety), the surface depends on the arrangement of those 6 points. 27 lines on a cubic surface. Most proofs of rationality for cubic surfaces start by finding a line on the surface. (In the context of projective geometry, a line in formula_1 is isomorphic to formula_12.) More precisely, Arthur Cayley and George Salmon showed in 1849 that every smooth cubic surface over an algebraically closed field contains  exactly 27 lines. This is a distinctive feature of cubics: a smooth quadric (degree 2) surface is covered by a continuous family of lines, while most surfaces of degree at least 4 in formula_1 contain no lines. Another useful technique for finding the 27 lines involves Schubert calculus which computes the number of lines using the intersection theory of the Grassmannian of lines on formula_1. 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. The group of permutations of the 27 lines arising this way is called the monodromy group of the family of cubic surfaces. A remarkable 19th-century discovery was that the monodromy group is neither trivial nor the whole symmetric group formula_15; it is a group of order 51840, acting transitively on the set of lines. This group was gradually recognized (by Élie Cartan (1896), Arthur Coble (1915-17), and Patrick du Val (1936)) as the Weyl group of type formula_16, a group generated by reflections on a 6-dimensional real vector space, related to the Lie group formula_16 of  dimension 78. The same group of order 51840 can be described in combinatorial terms, as the automorphism group of the graph of the 27 lines, with a vertex for each line and an edge whenever two lines meet. This graph was analyzed in the 19th century using subgraphs such as the Schläfli double six configuration. The complementary graph (with an edge whenever two lines are disjoint) is known as the Schläfli graph. Many problems about cubic surfaces can be solved using the combinatorics of the formula_16 root system. For example, the 27 lines can be identified with the weights of  the fundamental representation of the Lie group formula_16. The possible sets of singularities that can occur on a cubic surface can be described in terms of subsystems of the formula_16 root system. One explanation for this connection is that the formula_16 lattice arises as the orthogonal complement to the anticanonical class formula_22 in the Picard group formula_23, with its intersection form (coming from the intersection theory of curves on a surface). For a smooth complex cubic surface, the Picard lattice can also be identified with the cohomology group formula_24. An Eckardt point is a point where 3 of the 27  lines meet. Most cubic surfaces have no Eckardt point, but such points occur on a codimension-1 subset of the family of all smooth cubic surfaces. Given an identification between a cubic surface on "X" and the blow-up of formula_4 at 6 points in general position, the 27 lines on "X" can be viewed as: the 6 exceptional curves created by blowing up, the birational transforms of the 15 lines through pairs of the 6 points in formula_4, and the birational transforms of the 6 conics containing all but one of the 6 points. A given cubic surface can be  viewed as a blow-up of formula_4 in more than one way (in fact, in 72 different ways), and so a description as a blow-up does not reveal the symmetry among all 27 of the lines. The relation between cubic surfaces and the formula_16 root system generalizes to a relation between all del Pezzo surfaces and root systems. This is one of many ADE classifications in mathematics. Pursuing these analogies, Vera Serganova and Alexei Skorobogatov gave a direct geometric relation between cubic surfaces and the Lie group formula_16. 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 E then naturally acts as the U-duality group. This map between del Pezzo surfaces and M-theory on tori is known as mysterious duality. Special cubic surfaces. The smooth complex cubic surface in formula_1 with the largest automorphism group is the Fermat cubic surface, defined by Its automorphism group is an extension formula_32, of order 648. The next most symmetric smooth cubic surface is the Clebsch surface, which can be defined in formula_33 by the two equations Its automorphism group  is the symmetric group formula_35, of order 120. After a complex linear change of coordinates, the Clebsch surface can also be defined by the equation in formula_1. Among singular complex cubic surfaces, Cayley's nodal cubic surface is the unique surface with the maximal number of nodes, 4: Its automorphism group is formula_39, of order 24. Real cubic surfaces. In contrast to the complex case, the space of smooth cubic surfaces over the real numbers is not connected in the classical topology (based on the topology of R). 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). Namely, there are 5 isotopy classes of smooth real cubic surfaces "X" in formula_1, distinguished by the topology of the space of real points formula_41. The space of real points is diffeomorphic to either formula_42, or the disjoint union of formula_43 and the 2-sphere, where formula_44 denotes the connected sum of "r" copies of the real projective plane formula_45. Correspondingly, the number of real lines contained in "X" is 27, 15, 7, 3, or 3. A smooth real cubic surface is  rational over R if and only if its space of real points is connected, hence in the first four of the previous five cases. The average number of real lines on "X" is formula_46 when the defining polynomial for "X" is sampled at random from the Gaussian ensemble induced by the Bombieri inner product. The moduli space of cubic surfaces. Two smooth cubic surfaces are isomorphic as algebraic varieties if and only if they are equivalent by some linear automorphism of formula_1. Geometric invariant theory gives a moduli space of cubic surfaces, with one point for each isomorphism class of  smooth cubic surfaces. This moduli space has dimension 4. More precisely, it is an open subset of the weighted projective space P(12345), by Salmon and Clebsch (1860). In particular, it is a rational 4-fold. The cone of curves. The lines on a cubic surface "X" over an algebraically closed field can be described intrinsically, without reference to the embedding of "X" in formula_1: they are exactly the (−1)-curves on "X", meaning the curves isomorphic to formula_12 that have self-intersection −1. Also, the classes of lines in the Picard lattice of "X" (or equivalently the divisor class group) are  exactly the elements "u" of Pic("X") such that formula_50 and formula_51. (This uses that the restriction of the hyperplane line bundle O(1) on formula_1 to "X" is the anticanonical line bundle formula_22, by the adjunction formula.) For any projective variety "X", the cone of curves means the convex cone spanned by all curves in "X" (in the real vector space formula_54 of 1-cycles modulo numerical equivalence, or in the homology group formula_55 if the base field is the complex numbers). For a cubic surface, the cone of curves is spanned by the 27 lines. In particular, it is a  rational polyhedral cone in formula_56 with a large symmetry group, the Weyl group of formula_16. There is a similar description of the cone of curves for any del Pezzo surface. Cubic surfaces over a field. 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 formula_58) with no rational points, in which case "X" is certainly not rational. If "X"("k") is nonempty, then "X" is at least unirational over "k", by  Beniamino Segre and János Kollár. 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 formula_59 of "k" (through some subgroup of the Weyl group of formula_16). 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. Otherwise, "X" has Picard number 1. (The Picard group of "X" is a subgroup of the geometric Picard group formula_61.) In the latter  case, Segre showed that "X" is never rational. 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. For example, these results give many cubic surfaces over Q that are unirational but not rational. Singular cubic surfaces. In contrast to smooth cubic surfaces which contain 27 lines, singular cubic surfaces contain fewer lines. Moreover, they can be classified by the type of singularity which arises in their normal form. These singularities are classified using Dynkin diagrams. Classification. A normal  singular cubic surface formula_62 in formula_63 with local coordinates formula_64 is said to be in normal form if it is given by formula_65. Depending on the type of singularity formula_62 contains, it is isomorphic to the projective surface in formula_67 given by formula_65 where formula_69 are as in the table below. That means we can obtain a classification of all singular cubic surfaces. The parameters of the following table are as follows: formula_70 are three distinct elements of formula_71, the parameters formula_72 are in formula_73 and formula_74 is an element of formula_75. Notice that there are two different singular cubic  surfaces with singularity formula_76. In normal form, whenever a cubic surface formula_62 contains at least one formula_78 singularity, it will have an formula_78 singularity at formula_80. Lines on singular cubic surfaces. According to the classification of singular cubic surfaces, the following table shows the number of lines each surface contains. Automorphism groups of singular cubic surfaces with no parameters. An automorphism of a normal singular cubic surface formula_62 is the restriction of an automorphism of the projective space formula_67 to formula_62. Such automorphisms preserve singular points. Moreover, they do not permute singularities of different types. If the surface contains two  singularities of the same type, the automorphism may permute them. The collection of automorphisms on a cubic surface forms a group, the so-called automorphism group. The following table shows all automorphism groups of singular cubic surfaces with no parameters.