In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a cubic surface is a surface in 3-dimensional space defined by one
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
equation of degree 3. Cubic surfaces are fundamental examples in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. The theory is simplified by working in
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
rather than
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
, and so cubic surfaces are generally considered in projective 3-space
. The theory also becomes more uniform by focusing on surfaces over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s rather than the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s; note that a complex surface has real dimension 4. A simple example is the
Fermat cubic surface
In geometry, the Fermat cubic, named after Pierre de Fermat, is a surface defined by
: x^3 + y^3 + z^3 = 1. \
Methods of algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynom ...
:
in
. Many properties of cubic surfaces hold more generally for
del Pezzo surface
In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general ...
s.
Rationality of cubic surfaces
A central feature of
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
cubic surfaces ''X'' over an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
is that they are all
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
, as shown by
Alfred Clebsch
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. ...
in 1866. That is, there is a one-to-one correspondence defined by
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s between the projective plane
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
, which are never rational. In
characteristic zero, smooth surfaces of degree at least 4 in
are not even
uniruled.
More strongly, Clebsch showed that every smooth cubic surface in
over an algebraically closed field is isomorphic to the
blow-up
''Blowup'' (sometimes styled as ''Blow-up'' or ''Blow Up'') is a 1966 mystery drama thriller film directed by Michelangelo Antonioni and produced by Carlo Ponti. It was Antonioni's first entirely English-language film, and stars David Hemming ...
of
at 6 points.
[Dolgachev (2012), Chapter 9, Historical notes.] As a result, every smooth cubic surface over the complex numbers is
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two man ...
to the
connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
, where the minus sign refers to a change of
orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
. Conversely, the blow-up of
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
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
. As a
complex manifold (or an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
), 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
is isomorphic to
.) More precisely,
Arthur Cayley and
George Salmon
George Salmon FBA FRS FRSE (25 September 1819 – 22 January 1904) was a distinguished and influential Irish mathematician and Anglican theologian. After working in algebraic geometry for two decades, Salmon devoted the last forty years of his ...
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
contain no lines. Another useful technique for finding the 27 lines involves
Schubert calculus
In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of ...
which computes the number of lines using the intersection theory of the
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...
of lines on
.
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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of permutations of the 27 lines arising this way is called the
monodromy group
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...
of the family of cubic surfaces. A remarkable 19th-century discovery was that the monodromy group is neither trivial nor the whole
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
; it is a
group of order 51840, acting
transitively on the set of lines.
This group was gradually recognized (by
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
(1896),
Arthur Coble (1915-17), and
Patrick du Val
Patrick du Val (March 26, 1903 – January 22, 1987) was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity. The concept of Du Val singularity of an algebraic surface is named aft ...
(1936)) as the
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...
of type
, a group generated by reflections on a 6-dimensional real vector space, related to the
Lie group of dimension 78.
The same group of order 51840 can be described in combinatorial terms, as the
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of the
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
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
In geometry, the Schläfli double six is a configuration of 30 points and 12 lines, introduced by . The lines of the configuration can be partitioned into two subsets of six lines: each line is disjoint from ( skew with) the lines in its own subse ...
configuration. The complementary graph (with an edge whenever two lines are disjoint) is known as the
Schläfli graph
In the mathematical field of graph theory, the Schläfli graph, named after Ludwig Schläfli, is a 16- regular undirected graph with 27 vertices and 216 edges. It is a strongly regular graph with parameters srg(27, 16, 10, 8).
...
.
![Schläfli graph](https://upload.wikimedia.org/wikipedia/commons/f/f0/Schl%C3%A4fli_graph.svg)
Many problems about cubic surfaces can be solved using the combinatorics of the
root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representati ...
. 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
root system. One explanation for this connection is that the
lattice arises as the orthogonal complement to the
anticanonical class
in the
Picard group
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
, with its intersection form (coming from the
intersection theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
of curves on a surface). For a smooth complex cubic surface, the Picard lattice can also be identified with the
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
group
.
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
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
, 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
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
root system generalizes to a relation between all del Pezzo surfaces and root systems. This is one of many
ADE classification
In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, r ...
s in mathematics. Pursuing these analogies,
Vera Serganova and
Alexei Skorobogatov
Alexei Nikolaievich Skorobogatov (russian: Алексе́й Никола́евич Скоробога́тов) is a British-Russian mathematician and Professor in Pure Mathematics at Imperial College London specialising in algebraic geometry. His ...
gave a direct geometric relation between cubic surfaces and the Lie group
.
In physics, the 27 lines can be identified with the 27 possible charges of
M-theory
M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten's ...
on a six-dimensional
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
(6 momenta; 15
membrane
A membrane is a selective barrier; it allows some things to pass through but stops others. Such things may be molecules, ions, or other small particles. Membranes can be generally classified into synthetic membranes and biological membranes. B ...
s; 6
fivebranes) and the group E
6 then naturally acts as the
U-duality
In physics, U-duality (short for unified duality)S. Mizoguchi,On discrete U-duality in M-theory, 2000. is a symmetry of string theory or M-theory
M-theory is a theory in physics that unifies all consistent versions of superstring theory. ...
group. This map between
del Pezzo surface
In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general ...
s and
M-theory
M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten's ...
on tori is known as
mysterious duality
M-theory is a theory in physics that unifies all consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten' ...
.
Special cubic surfaces
The smooth complex cubic surface in
with the largest automorphism group is the Fermat cubic surface, defined by
:
Its automorphism group is an extension
, of order 648.
The next most symmetric smooth cubic surface is the
Clebsch surface, which
can be defined in
by the two equations
:
Its automorphism group is the symmetric group
, of order 120. After a complex linear change of coordinates, the Clebsch surface can also be defined by the equation
:
in
.
![Cayley_cubic_2](https://upload.wikimedia.org/wikipedia/commons/a/a8/Cayley_cubic_2.png)
Among singular complex cubic surfaces,
Cayley's nodal cubic surface
In algebraic geometry, the Cayley surface, named after Arthur Cayley, is a cubic nodal surface in 3-dimensional projective space with four conical points. It can be given by the equation
: wxy+ xyz+ yzw+zwx =0\
when the four singular point ...
is the unique surface with the maximal number of
node
In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex).
Node may refer to:
In mathematics
* Vertex (graph theory), a vertex in a mathematical graph
*Vertex (geometry), a point where two or more curves, lines ...
s, 4:
:
Its automorphism group is
, 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
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
in the classical
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
(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
Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional space ...
(1863),
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
(1865), and
H. G. Zeuthen (1875). Namely, there are 5 isotopy classes of smooth real cubic surfaces ''X'' in
, distinguished by the topology of the space of
real points . The space of real points is diffeomorphic to either
, or the disjoint union of
and the 2-sphere, where
denotes the connected sum of ''r'' copies of the
real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
. 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
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
.
Geometric invariant theory
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in clas ...
gives a
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
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 In algebraic geometry, a weighted projective space P(''a''0,...,''a'n'') is the projective variety Proj(''k'' 'x''0,...,''x'n'' associated to the graded ring ''k'' 'x''0,...,''x'n''where the variable ''x'k'' has degree ''a'k''.
Prop ...
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
: they are exactly the (−1)-curves on ''X'', meaning the curves isomorphic to
that have self-intersection −1. Also, the classes of lines in the Picard lattice of ''X'' (or equivalently the
divisor class group
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumfo ...
) are exactly the elements ''u'' of Pic(''X'') such that
and
. (This uses that the restriction of the
hyperplane line bundle O(1) on
to ''X'' is the anticanonical line bundle
, by the
adjunction formula In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedde ...
.)
For any projective variety ''X'', the
cone of curves
In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety X is a combinatorial invariant of importance to the birational geometry of X.
Definition
Let X be a proper variety. By definition, a (real) ''1-cycle'' ...
means the
convex cone spanned by all curves in ''X'' (in the real vector space
of 1-cycles modulo numerical equivalence, or in the
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
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
with a large symmetry group, the Weyl group of
. 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
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
Q (or the
p-adic numbers
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ...
) with no
rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fiel ...
s, in which case ''X'' is certainly not rational. If ''X''(''k'') is nonempty, then ''X'' is at least
unirational In mathematics, a rational variety is an algebraic variety, over a given field ''K'', which is birationally equivalent to a projective space of some dimension over ''K''. This means that its function field is isomorphic to
:K(U_1, \dots , U_d),
t ...
over ''k'', by
Beniamino Segre
Beniamino Segre (16 February 1903 – 2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of finite geometry.
Life and career
He was born and studied in Turin ...
and
János Kollár
János Kollár (born 7 June 1956) is a Hungarian mathematician, specializing in algebraic geometry.
Professional career
Kollár began his studies at the Eötvös University in Budapest and later received his PhD at Brandeis University in 1984 ...
. For ''k'' infinite, unirationality implies that the set of ''k''-rational points is
Zariski dense in ''X''.
The
absolute Galois group
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' t ...
of ''k'' permutes the 27 lines of ''X'' over the algebraic closure
of ''k'' (through some subgroup of the Weyl group of
). 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
.) In the latter case, Segre showed that ''X'' is never rational. More strongly,
Yuri Manin
Yuri Ivanovich Manin (russian: Ю́рий Ива́нович Ма́нин; born 16 February 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical log ...
proved a birational rigidity statement: two smooth cubic surfaces with Picard number 1 over a
perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds:
* Every irreducible polynomial over ''k'' has distinct roots.
* Every irreducible polynomial over ''k'' is separable.
* Every finite extension of ''k' ...
''k'' are
birational
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational f ...
if and only if they are isomorphic.
[Kollár, Smith, Corti (2004), Theorems 2.1 and 2.2.] For example, these results give many cubic surfaces over Q that are unirational but not rational.
Singular cubic surfaces
In contrast to
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
cubic surfaces which contain 27 lines,
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, ...
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
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras ...
.
Classification
A normal singular cubic surface
in
with local coordinates