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
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 geometrica ...
. The theory is simplified by working in
projective space 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 numbers rather than the
real numbers; note that a complex surface has real dimension 4. A simple example is the
Fermat cubic surface
:
in
. Many properties of cubic surfaces hold more generally for
del Pezzo surfaces.
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 algebraic ...
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 abili ...
, as shown by
Alfred Clebsch in 1866. That is, there is a one-to-one correspondence defined by
rational functions 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
A projective cone (or just cone) in projective geometry is the union of all lines that intersect a projective subspace ''R'' (the apex of the cone) and an arbitrary subset ''A'' (the basis) of some other subspace ''S'', disjoint from ''R''.
In ...
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 Hemmings ...
of
at 6 points.
[Dolgachev (2012), Chapter 9, Historical notes.] As a result, every smooth cubic surface over the complex numbers is
diffeomorphic 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 classifica ...
, where the minus sign refers to a change of
orientation. 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 speci ...
. As a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a c ...
(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. ...
), 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
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.
As a child, Cayley enjoyed solving complex maths problems ...
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
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 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
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...
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 of the family of cubic surfaces. A remarkable 19th-century discovery was that the monodromy group is neither trivial nor the whole
symmetric group ; 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
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 afte ...
(1936)) as the
Weyl group 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 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
root system. 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 , 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
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
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals ...
-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, ...
s in mathematics. Pursuing these analogies,
Vera Serganova and
Alexei Skorobogatov 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 (6 momenta; 15
membranes; 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 combining S-duality and T-duality transformations. The term is most often met in the context of t ...
group. This map between
del Pezzo surfaces 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's ...
.
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
In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 exceptional lines
can be defined over the real numbers. The term Klein's icosahedral sur ...
, 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
.
Among singular complex cubic surfaces,
Cayley's nodal cubic surface
In algebraic geometry, the Cayley surface, named after Arthur Cayley, is a cubic
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a three-dimensional solid object bounded by six square faces, facets or ...
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 (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
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 group ...
(1865), and
H. G. Zeuthen
Hieronymus Georg Zeuthen (15 February 1839 – 6 January 1920) was a Danish mathematician.
He is known for work on the enumerative geometry of conic sections, algebraic surfaces, and history of mathematics.
Biography
Zeuthen was born in Grims ...
(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 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 s ...
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''.
Prope ...
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) 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.)
For any projective variety ''X'', the
cone of curves means the
convex cone
In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every .
...
spanned by all curves in ''X'' (in the real vector space
of 1-cycles modulo numerical equivalence, or in the
homology group 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 extens ...
) with no
rational points, 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),
th ...
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'' tha ...
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 proved a birational rigidity statement: two smooth cubic surfaces with Picard number 1 over a
perfect field ''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 ...
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 algebraic ...
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
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