# Cubic surface

Cubic surface (Redirected from Cayley–Salmon theorem) Ir para a navegação Ir para a pesquisa Em matemática, 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 {estilo de exibição mathbf {P} ^{3}} . 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 {estilo de exibição x^{3}+^{3}+z^{3}+w^{3}=0} dentro {estilo de exibição mathbf {P} ^{3}} . Many properties of cubic surfaces hold more generally for del Pezzo surfaces.

A smooth cubic surface (the Clebsch surface) Conteúdo 1 Rationality of cubic surfaces 2 27 lines on a cubic surface 3 Special cubic surfaces 4 Real cubic surfaces 5 The moduli space of cubic surfaces 6 The cone of curves 7 Cubic surfaces over a field 8 Singular cubic surfaces 8.1 Classification 8.2 Lines on singular cubic surfaces 8.3 Automorphism groups of singular cubic surfaces with no parameters 9 Veja também 10 Notas 11 Referências 12 External links 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.[1] Aquilo é, there is a one-to-one correspondence defined by rational functions between the projective plane {estilo de exibição mathbf {P} ^{2}} minus a lower-dimensional subset and X minus a lower-dimensional subset. De forma geral, every irreducible cubic surface (possibly singular) over an algebraically closed field is rational unless it is the projective cone over a cubic curve.[2] In this respect, cubic surfaces are much simpler than smooth surfaces of degree at least 4 dentro {estilo de exibição mathbf {P} ^{3}} , which are never rational. In characteristic zero, smooth surfaces of degree at least 4 dentro {estilo de exibição mathbf {P} ^{3}} are not even uniruled.[3] Mais fortemente, Clebsch showed that every smooth cubic surface in {estilo de exibição mathbf {P} ^{3}} over an algebraically closed field is isomorphic to the blow-up of {estilo de exibição mathbf {P} ^{2}} no 6 points.[4] Como resultado, every smooth cubic surface over the complex numbers is diffeomorphic to the connected sum {estilo de exibição mathbf {PC} ^{2}#6(-mathbf {PC} ^{2})} , where the minus sign refers to a change of orientation. Por outro lado, the blow-up of {estilo de exibição mathbf {P} ^{2}} no 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 pontos.

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 {estilo de exibição mathbf {P} ^{3}} é isomórfico a {estilo de exibição mathbf {P} ^{1}} .) Mais precisamente, Arthur Cayley and George Salmon showed in 1849 that every smooth cubic surface over an algebraically closed field contains exactly 27 lines.[5] 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 dentro {estilo de exibição mathbf {P} ^{3}} 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 {estilo de exibição mathbf {P} ^{3}} .

As the coefficients of a smooth complex cubic surface are varied, a 27 lines move continuously. Como resultado, 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 {estilo de exibição S_{27}} ; it is a group of order 51840, acting transitively on the set of lines.[4] This group was gradually recognized (by Élie Cartan (1896), Arthur Coble (1915-17), and Patrick du Val (1936)) as the Weyl group of type {estilo de exibição E_{6}} , a group generated by reflections on a 6-dimensional real vector space, related to the Lie group {estilo de exibição E_{6}} of dimension 78.[4] 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.[6] 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.

The Schläfli graph Many problems about cubic surfaces can be solved using the combinatorics of the {estilo de exibição E_{6}} root system. Por exemplo, a 27 lines can be identified with the weights of the fundamental representation of the Lie group {estilo de exibição E_{6}} . The possible sets of singularities that can occur on a cubic surface can be described in terms of subsystems of the {estilo de exibição E_{6}} root system.[7] One explanation for this connection is that the {estilo de exibição E_{6}} lattice arises as the orthogonal complement to the anticanonical class {displaystyle -K_{X}} in the Picard group {nome do operador de estilo de exibição {Pic} (X)cong mathbf {Z} ^{7}} , 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 {estilo de exibição H^{2}(X,mathbf {Z} )} .

An Eckardt point is a point where 3 do 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.[8] Given an identification between a cubic surface on X and the blow-up of {estilo de exibição mathbf {P} ^{2}} no 6 points in general position, a 27 lines on X can be viewed as: a 6 exceptional curves created by blowing up, the birational transforms of the 15 lines through pairs of the 6 points in {estilo de exibição mathbf {P} ^{2}} , and the birational transforms of the 6 conics containing all but one of the 6 points.[9] A given cubic surface can be viewed as a blow-up of {estilo de exibição mathbf {P} ^{2}} in more than one way (na verdade, dentro 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 {estilo de exibição E_{6}} 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 {estilo de exibição E_{6}} .[10] In physics, a 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.

Special cubic surfaces The smooth complex cubic surface in {estilo de exibição mathbf {P} ^{3}} with the largest automorphism group is the Fermat cubic surface, definido por {estilo de exibição x^{3}+^{3}+z^{3}+w^{3}=0.} Its automorphism group is an extension {estilo de exibição 3 ^{3}:S_{4}} , de ordem 648.[11] The next most symmetric smooth cubic surface is the Clebsch surface, which can be defined in {estilo de exibição mathbf {P} ^{4}} by the two equations {estilo de exibição x_{0}+x_{1}+x_{2}+x_{3}+x_{4}=x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}=0.} Its automorphism group is the symmetric group {estilo de exibição S_{5}} , de ordem 120. After a complex linear change of coordinates, the Clebsch surface can also be defined by the equation {estilo de exibição x^{2}y+y^{2}z+z^{2}w+w^{2}x=0} dentro {estilo de exibição mathbf {P} ^{3}} .

Cayley's nodal cubic surface Among singular complex cubic surfaces, Cayley's nodal cubic surface is the unique surface with the maximal number of nodes, 4: {displaystyle wxy+xyz+yzw+zwx=0.} Its automorphism group is {estilo de exibição S_{4}} , de ordem 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 (em outras palavras, the classification of smooth real cubic surfaces up to isotopy) were determined by Ludwig Schläfli (1863), Felix Klein (1865), e H. G. Zeuthen (1875).[12] Nomeadamente, existem 5 isotopy classes of smooth real cubic surfaces X in {estilo de exibição mathbf {P} ^{3}} , distinguished by the topology of the space of real points {estilo de exibição X(mathbf {R} )} . The space of real points is diffeomorphic to either {estilo de exibição W_{7},C_{5},C_{3},C_{1}} , or the disjoint union of {estilo de exibição W_{1}} and the 2-sphere, Onde {estilo de exibição W_{r}} denotes the connected sum of r copies of the real projective plane {estilo de exibição mathbf {PR} ^{2}} . Correspondingly, the number of real lines contained in X is 27, 15, 7, 3, ou 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.[13] The average number of real lines on X is {estilo de exibição 6{quadrado {2}}-3} [14] 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 {estilo de exibição mathbf {P} ^{3}} . 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. Mais precisamente, it is an open subset of the weighted projective space P(12345), by Salmon and Clebsch (1860). Em particular, it is a rational 4-fold.[15] 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 {estilo de exibição mathbf {P} ^{3}} : they are exactly the (−1)-curves on X, meaning the curves isomorphic to {estilo de exibição mathbf {P} ^{1}} that have self-intersection −1. Também, the classes of lines in the Picard lattice of X (or equivalently the divisor class group) are exactly the elements u of Pic(X) de tal modo que {displaystyle u^{2}=-1} e {displaystyle -K_{X}cdot u=1} . (This uses that the restriction of the hyperplane line bundle O(1) sobre {estilo de exibição mathbf {P} ^{3}} to X is the anticanonical line bundle {displaystyle -K_{X}} , 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 {estilo de exibição N_{1}(X)} of 1-cycles modulo numerical equivalence, or in the homology group {estilo de exibição H_{2}(X,mathbf {R} )} if the base field is the complex numbers). For a cubic surface, the cone of curves is spanned by the 27 lines.[16] Em particular, it is a rational polyhedral cone in {estilo de exibição N_{1}(X)cong mathbf {R} ^{7}} with a large symmetry group, the Weyl group of {estilo de exibição E_{6}} . 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 {estilo de exibição mathbf {Q} _{p}} ) with no rational points, in which case X is certainly not rational.[17] If X(k) is nonempty, then X is at least unirational over k, by Beniamino Segre and János Kollár.[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 {estilo de exibição {overline {k}}} of k (through some subgroup of the Weyl group of {estilo de exibição E_{6}} ). If some orbit of this action consists of disjoint lines, then X is the blow-up of a "mais simples" del Pezzo surface over k at a closed point. Por outro lado, X has Picard number 1. (The Picard group of X is a subgroup of the geometric Picard group {nome do operador de estilo de exibição {Pic} (X_{overline {k}})cong mathbf {Z} ^{7}} .) No último caso, Segre showed that X is never rational. Mais fortemente, 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] Por exemplo, 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. [20] Além disso, 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 {estilo de exibição X} dentro {estilo de exibição {textbf {P}}_{mathbb {C} }^{3}} with local coordinates {estilo de exibição [x_{0}:x_{1}:x_{2}:x_{3}]} is said to be in normal form if it is given by {displaystyle F=x_{3}f_{2}(x_{0},x_{1},x_{2})-f_{3}(x_{0},x_{1},x_{2})=0} . Depending on the type of singularity {estilo de exibição X} contains, it is isomorphic to the projective surface in {estilo de exibição {textbf {P}}^{3}} dado por {displaystyle F=x_{3}f_{2}(x_{0},x_{1},x_{2})-f_{3}(x_{0},x_{1},x_{2})=0} Onde {estilo de exibição f_{2},f_{3}} 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: {estilo de exibição a,b,c} are three distinct elements of {estilo de exibição mathbb {C} setminus {0,1}} , the parameters {estilo de exibição d,e} estão dentro {estilo de exibição mathbb {C} setminus {0,-1}} e {estilo de exibição você} é um elemento de {estilo de exibição mathbb {C} setminus {0}} . Notice that there are two different singular cubic surfaces with singularity {displaystyle D_{4}} . [21] Classification of singular cubic surfaces by singularity type [21] hide Singularity {estilo de exibição f_{2}(x_{0},x_{1},x_{2})} {estilo de exibição f_{3}(x_{0},x_{1},x_{2})} {estilo de exibição A_{1}} {estilo de exibição x_{0}x_{2}-x_{1}^{2}} {estilo de exibição (x_{0}-ax_{1})(-x_{0}+(b+1)x_{1}-bx_{2})(x_{1}-cx_{2})} {displaystyle 2A_{1}} {estilo de exibição x_{0}x_{2}-x_{1}^{2}} {estilo de exibição (x_{0}-2x_{1}+x_{2})(x_{0}-ax_{1})(x_{1}-bx_{2})} {estilo de exibição A_{1}UMA_{2}} {estilo de exibição x_{0}x_{2}-x_{1}^{2}} {estilo de exibição (x_{0}-x_{1})(-x_{1}+x_{2})(x_{0}-(a+1)x_{1}+ax_{2})} {displaystyle 3A_{1}} {estilo de exibição x_{0}x_{2}-x_{1}^{2}} {estilo de exibição x_{0}x_{2}(x_{0}-(a+1)x_{1}+ax_{2})} {estilo de exibição A_{1}UMA_{3}} {estilo de exibição x_{0}x_{2}-x_{1}^{2}} {estilo de exibição (x_{0}-x_{1})(-x_{1}+x_{2})(x_{0}-2x_{1}+x_{2})} {displaystyle 2A_{1}UMA_{2}} {estilo de exibição x_{0}x_{2}-x_{1}^{2}} {estilo de exibição x_{1}^{2}(x_{0}-x_{1})} {displaystyle 4A_{1}} {estilo de exibição x_{0}x_{2}-x_{1}^{2}} {estilo de exibição (x_{0}-x_{1})(x_{1}-x_{2})x_{1}} {estilo de exibição A_{1}UMA_{4}} {estilo de exibição x_{0}x_{2}-x_{1}^{2}} {estilo de exibição x_{0}^{2}x_{1}} {displaystyle 2A_{1}UMA_{3}} {estilo de exibição x_{0}x_{2}-x_{1}^{2}} {estilo de exibição x_{0}x_{1}^{2}} {estilo de exibição A_{1}2UMA_{2}} {estilo de exibição x_{0}x_{2}-x_{1}^{2}} {estilo de exibição x_{1}^{3}} {estilo de exibição A_{1}UMA_{5}} {estilo de exibição x_{0}x_{2}-x_{1}^{2}} {estilo de exibição x_{0}^{3}} {estilo de exibição A_{2}} {estilo de exibição x_{0}x_{1}} {estilo de exibição x_{2}(x_{0}+x_{1}+x_{2})(dx_{0}+ex_{1}+dex_{2})} {displaystyle 2A_{2}} {estilo de exibição x_{0}x_{1}} {estilo de exibição x_{2}(x_{1}+x_{2})(-x_{1}+dx_{2})} {displaystyle 3A_{2}} {estilo de exibição x_{0}x_{1}} {estilo de exibição x_{2}^{3}} {estilo de exibição A_{3}} {estilo de exibição x_{0}x_{1}} {estilo de exibição x_{2}(x_{0}+x_{1}+x_{2})(x_{0}-ux_{1})} {estilo de exibição A_{4}} {estilo de exibição x_{0}x_{1}} {estilo de exibição x_{0}^{2}x_{2}+x_{1}^{3}-x_{1}x_{2}^{2}} {estilo de exibição A_{5}} {estilo de exibição x_{0}x_{1}} {estilo de exibição x_{0}^{3}+x_{1}^{3}-x_{1}x_{2}^{2}} {displaystyle D_{4}(1)} {estilo de exibição x_{0}^{2}} {estilo de exibição x_{1}^{3}+x_{2}^{3}} {displaystyle D_{4}(2)} {estilo de exibição x_{0}^{2}} {estilo de exibição x_{1}^{3}+x_{2}^{3}+x_{0}x_{1}x_{2}} {displaystyle D_{5}} {estilo de exibição x_{0}^{2}} {estilo de exibição x_{0}x_{2}^{2}+x_{1}^{2}x_{2}} {estilo de exibição E_{6}} {estilo de exibição x_{0}^{2}} {estilo de exibição x_{0}x_{2}^{2}+x_{1}^{3}} {estilo de exibição {tilde {E}}_{6}} {estilo de exibição 0} {estilo de exibição x_{1}^{2}x_{2}-x_{0}(x_{0}-x_{2})(x_{0}-ax_{2})} In normal form, whenever a cubic surface {estilo de exibição X} contains at least one {estilo de exibição A_{1}} singularity, it will have an {estilo de exibição A_{1}} singularity at {estilo de exibição [0:0:0:1]} . [20] Lines on singular cubic surfaces According to the classification of singular cubic surfaces, the following table shows the number of lines each surface contains.

Lines on singular cubic surfaces [21] hide Singularity {estilo de exibição A_{1}} {displaystyle 2A_{1}} {estilo de exibição A_{1}UMA_{2}} {displaystyle 3A_{1}} {estilo de exibição A_{1}UMA_{3}} {displaystyle 2A_{1}UMA_{2}} {displaystyle 4A_{1}} {estilo de exibição A_{1}UMA_{4}} {displaystyle 2A_{1}UMA_{3}} {estilo de exibição A_{1}2UMA_{2}} {estilo de exibição A_{1}UMA_{5}} {estilo de exibição A_{2}} {displaystyle 2A_{2}} {displaystyle 3A_{2}} {estilo de exibição A_{3}} {estilo de exibição A_{4}} {estilo de exibição A_{5}} {displaystyle D_{4}} {displaystyle D_{5}} {estilo de exibição E_{6}} {estilo de exibição {tilde {E}}_{6}} Não. of lines 21 16 11 12 7 8 9 4 5 5 2 15 7 3 10 6 3 6 3 1 {displaystyle infty } Automorphism groups of singular cubic surfaces with no parameters An automorphism of a normal singular cubic surface {estilo de exibição X} is the restriction of an automorphism of the projective space {estilo de exibição {textbf {P}}^{3}} para {estilo de exibição X} . Such automorphisms preserve singular points. Além disso, 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.