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

P3

. 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

x3+y3+z3+w3=0

in

P3

. 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

P2

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

P3

, which are never rational. In characteristic zero, smooth surfaces of degree at least 4 in

P3

are not even uniruled.[3]

More strongly, Clebsch showed that every smooth cubic surface in

P3

over an algebraically closed field is isomorphic to the blow-up of

P2

at 6 points.[4] As a result, every smooth cubic surface over the complex numbers is diffeomorphic to the connected sum

CP2\#6(-CP2)

, where the minus sign refers to a change of orientation. Conversely, the blow-up of

P2

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

P3

is isomorphic to

P1

.) 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

P3

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

P3

.

S27

; 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

E6

, a group generated by reflections on a 6-dimensional real vector space, related to the Lie group

E6

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

E6

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

E6

root system.[7] One explanation for this connection is that the

E6

lattice arises as the orthogonal complement to the anticanonical class

-KX

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

H2(X,Z)

.

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

P2

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

P2

, 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

P2

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

E6

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

E6

.[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

P3

with the largest automorphism group is the Fermat cubic surface, defined by

x3+y3+z3+w3=0.

Its automorphism group is an extension
3:S
3
4
, of order 648.[11]

The next most symmetric smooth cubic surface is the Clebsch surface, whichcan be defined in

P4

by the two equations

x0+x1+x2+x3+x4=x

3=0.
4
Its automorphism group is the symmetric group

S5

, of order 120. After a complex linear change of coordinates, the Clebsch surface can also be defined by the equation

x2y+y2z+z2w+w2x=0

in

P3

.

Among singular complex cubic surfaces, Cayley's nodal cubic surface is the unique surface with the maximal number of nodes, 4:

wxy+xyz+yzw+zwx=0.

Its automorphism group is

S4

, 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

P3

, distinguished by the topology of the space of real points

X(R)

. The space of real points is diffeomorphic to either

W7,W5,W3,W1

, or the disjoint union of

W1

and the 2-sphere, where

Wr

denotes the connected sum of r copies of the real projective plane

RP2

. 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

6\sqrt{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

P3

. 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

P3

: they are exactly the (−1)-curves on X, meaning the curves isomorphic to

P1

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

u2=-1

and

-KXu=1

. (This uses that the restriction of the hyperplane line bundle O(1) on

P3

to X is the anticanonical line bundle

-KX

, 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

N1(X)

of 1-cycles modulo numerical equivalence, or in the homology group

H2(X,R)

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

N1(X)\congR7

with a large symmetry group, the Weyl group of

E6

. There is a similar description of the cone of curves for any del Pezzo surface.

Cubic surfaces over a field

Qp

) 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

\overline{k}

of k (through some subgroup of the Weyl group of

E6

). 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

X

in
3
bf{P}
C
with local coordinates

[x0:x1:x2:x3]

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

X

contains, it is isomorphic to the projective surface in

bf{P}3

given by

F=x3f2(x0,x1,x2)-f3(x0,x1,x2)=0

where

f2,f3

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:

a,b,c

are three distinct elements of

C\setminus\{0,1\}

, the parameters

d,e

are in

C\setminus\{0,-1\}

and

u

is an element of

C\setminus\{0\}

. Notice that there are two different singular cubic surfaces with singularity

D4

.[21]
Classification of singular cubic surfaces by singularity type !Singularity!

f2(x0,x1,x2)

!

f3(x0,x1,x2)

A1

x0x2-x

2
1

(x0-ax1)(-x0+(b+1)x1-bx2)(x1-cx2)

2A1

x0x2-x

2
1

(x0-2x1+x2)(x0-ax1)(x1-bx2)

A1A2

x0x2-x

2
1

(x0-x1)(-x1+x2)(x0-(a+1)x1+ax2)

3A1

x0x2-x

2
1

x0x2(x0-(a+1)x1+ax2)

A1A3

x0x2-x

2
1

(x0-x1)(-x1+x2)(x0-2x1+x2)

2A1A2

x0x2-x

2
1
2(x
x
0-x

1)

4A1

x0x2-x

2
1

(x0-x1)(x1-x2)x1

A1A4

x0x2-x

2
1
2x
x
1

2A1A3

x0x2-x

2
1

x0x

2
1

A12A2

x0x2-x

2
1
3
x
1

A1A5

x0x2-x

2
1
3
x
0

A2

x0x1

x2(x0+x1+x2)(dx0+ex1+dex2)

2A2

x0x1

x2(x1+x2)(-x1+dx2)

3A2

x0x1

3
x
2

A3

x0x1

x2(x0+x1+x2)(x0-ux1)

A4

x0x1

2x
x
2

+

3
x
1

-x1x

2
2

A5

x0x1

3
x
0

+

3
x
1

-x1x

2
2

D4(1)

2
x
0
3
x
2

D4(2)

2
x
0
3+x
x
0x

1x2

D5

2
x
0

x0x

2
2

+

2x
x
2

E6

2
x
0

x0x

2
2

+

3
x
1

\tilde{E}6

0

2x
x
2-x

0(x0-x2)(x0-ax2)

In normal form, whenever a cubic surface

X

contains at least one

A1

singularity, it will have an

A1

singularity at

[0:0:0:1]

.

Lines on singular cubic surfaces

According to the classification of singular cubic surfaces, the following table shows the number of lines each surface contains.

A1

2A1

A1A2

3A1

A1A3

2A1A2

4A1

A1A4

2A1A3

A12A2

A1A5

A2

2A2

3A2

A3

A4

A5

D4

D5

E6

\tilde{E}6

No. of lines21161112789455215731063631

infty

Automorphism groups of singular cubic surfaces with no parameters

An automorphism of a normal singular cubic surface

X

is the restriction of an automorphism of the projective space

bf{P}3

to

X

. 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

X

A1A3

Z/2Z

2A1A2

Z/2Z

4A1

\Sigma4

, the symmetric group of order

4!

A1A4

C x =C\setminus\{0\}

2A1A3

C x \rtimesZ/2Z

A12A2

C x \rtimesZ/2Z

A1A5

C\rtimesC x

3A2

(C)2\rtimes\Sigma3

A4

Z/4Z

A5

(C\rtimesZ/3Z)\rtimesZ/2Z

D4(1)

C x \rtimes\Sigma3

D4(2)

\Sigma3

D5

C x

E6

C\rtimesC x

See also

References

External links

Notes and References

  1. Reid (1988), Corollary 7.4.
  2. Kollár, Smith, Corti (2004), Example 1.28.
  3. Kollár, Smith, Corti (2004), Exercise 1.59.
  4. Dolgachev (2012), Chapter 9, Historical notes.
  5. Reid (1988), section 7.6.
  6. Hartshorne (1997), Exercise V.4.11.
  7. Bruce & Wall (1979), section 4; Dolgachev (2012), Table 9.1.
  8. Dolgachev (2012), section 9.1.4.
  9. Hartshorne (1997), Theorem V.4.9.
  10. Serganova & Skorobogatov (2007).
  11. Dolgachev (2012), Table 9.6.
  12. 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).
  13. Silhol (1989), section VI.5.
  14. 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 .
  15. Dolgachev (2012), equation (9.57).
  16. Hartshorne (1997), Theorem V.4.11.
  17. Kollár, Smith, Corti (2004), Exercise 1.29.
  18. Kollár, Smith, Corti (2004), Theorems 1.37 and 1.38.
  19. Kollár, Smith, Corti (2004), Theorems 2.1 and 2.2.
  20. 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.
  21. 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.