Cubic surface explained
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
. 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 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
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.
[2] 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.
[3] More strongly, Clebsch showed that every smooth cubic surface in
over an algebraically closed field is isomorphic to the
blow-up of
at 6 points.
[4] As a result, every smooth cubic surface over the complex numbers is
diffeomorphic to the
connected sum
, 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. 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
is isomorphic to
.) More precisely,
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 in
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
.
; 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
, 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 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.
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.
[7] One explanation for this connection is that the
lattice arises as the orthogonal complement to the
anticanonical class
in the
Picard group \operatorname{Pic}(X)\congZ7
, 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
.
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.[8]
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.
[9] 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 classifications in mathematics. 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.
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.
[11] The next most symmetric smooth cubic surface is the Clebsch surface, whichcan 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 is the unique surface with the maximal number of nodes, 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 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).[12] 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
. 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.[13]
The average number of real lines on X is
[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
.
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.
[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
: 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 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.
[16] 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
) 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
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
\operatorname{Pic}(X\overline{k
})\cong \mathbf^7.) 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.
[19] 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.[20] 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
in
with local coordinates
is said to be in
normal form if it is given by
F=x3f2(x0,x1,x2)-f3(x0,x1,x2)=0
. Depending on the type of singularity
contains, it is
isomorphic to the projective surface in
given by
F=x3f2(x0,x1,x2)-f3(x0,x1,x2)=0
where
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:
are three distinct elements of
, the parameters
are in
and
is an element of
. Notice that there are two different singular cubic surfaces with singularity
.
[21] Classification of singular cubic surfaces by singularity type !Singularity!
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| (x0-ax1)(-x0+(b+1)x1-bx2)(x1-cx2)
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| (x0-2x1+x2)(x0-ax1)(x1-bx2)
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| (x0-x1)(-x1+x2)(x0-(a+1)x1+ax2)
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| (x0-x1)(-x1+x2)(x0-2x1+x2)
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In normal form, whenever a cubic surface
contains at least one
singularity, it will have an
singularity at
.
Lines on singular cubic surfaces
According to the classification of singular cubic surfaces, the following table shows the number of lines each surface contains.
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No. of lines | 21 | 16 | 11 | 12 | 7 | 8 | 9 | 4 | 5 | 5 | 2 | 15 | 7 | 3 | 10 | 6 | 3 | 6 | 3 | 1 |
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Automorphism groups of singular cubic surfaces with no parameters
An automorphism of a normal singular cubic surface
is the
restriction of an automorphism of the
projective space
to
. 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.
Automorphism groups of singular cubic surfaces with no parameters!Singularity!Automorphism group of
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, the symmetric group of order
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| (C\rtimesZ/3Z)\rtimesZ/2Z
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See also
References
- Book: Robin Hartshorne . 1977 . 1997 . Algebraic geometry . Springer-Verlag . 978-0-387-90244-9 . 0463157 . Algebraic Geometry (book). Robin Hartshorne .
- Book: Reid, Miles. Miles Reid. Undergraduate algebraic geometry. 1988. Cambridge University Press. 978-0-521-35662-6. 0982494. registration.
External links
Notes and References
- Reid (1988), Corollary 7.4.
- Kollár, Smith, Corti (2004), Example 1.28.
- Kollár, Smith, Corti (2004), Exercise 1.59.
- Dolgachev (2012), Chapter 9, Historical notes.
- Reid (1988), section 7.6.
- Hartshorne (1997), Exercise V.4.11.
- Bruce & Wall (1979), section 4; Dolgachev (2012), Table 9.1.
- Dolgachev (2012), section 9.1.4.
- Hartshorne (1997), Theorem V.4.9.
- Serganova & Skorobogatov (2007).
- Dolgachev (2012), Table 9.6.
- Degtyarev and Kharlamov (2000), section 3.5.2. The various types of real cubic surfaces, and the lines on them, are pictured in Holzer & Labs (2006).
- Silhol (1989), section VI.5.
- Basu. S.. Lerario. A.. Lundberg. E.. Peterson. C.. 2019. Random fields and the enumerative geometry of lines on real and complex hypersurfaces. Mathematische Annalen. 374. 3–4 . 1773–1810. 10.1007/s00208-019-01837-0. 1610.01205. 253717173 .
- Dolgachev (2012), equation (9.57).
- Hartshorne (1997), Theorem V.4.11.
- Kollár, Smith, Corti (2004), Exercise 1.29.
- Kollár, Smith, Corti (2004), Theorems 1.37 and 1.38.
- Kollár, Smith, Corti (2004), Theorems 2.1 and 2.2.
- Bruce. J. W.. Wall. C. T. C.. 1979. On the Classification of Cubic Surfaces. Journal of the London Mathematical Society. en. s2-19. 2. 245–256. 10.1112/jlms/s2-19.2.245. 1469-7750.
- SAKAMAKI. YOSHIYUKI. Automorphism Groups on Normal Singular Cubic Surfaces with No Parameters. 2010. Transactions of the American Mathematical Society. 362. 5. 2641–2666. 10.1090/S0002-9947-09-05023-5. 25677798. 0002-9947. free.