Elliptic surface
Encyclopedia
In mathematics
, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism
to an algebraic curve
, almost all of whose fibers are elliptic curve
s.
The fibers that are not elliptic curves are called the singular fibers and were classified by Kunihiko Kodaira.
Elliptic surfaces form a large class of surfaces that contains many of the interesting examples of surfaces, and are relatively well-understood from the viewpoint of complex manifold
theory and the theory of smooth
4-manifold
s. They are similar to (have analogies with, that is), elliptic curve
s over number fields.
The following table lists the possible fibers of a minimal elliptic fibration. ("Minimal" means roughly one that cannot be factored through a "smaller" one; for surfaces this means that the singular fibers should contain no minimal curves.) It gives:
This table can be found as follows. Geometric arguments show that the intersection matrix of the components of the fiber must be negative semidefinite, connected, symmetric, and have no diagonal entries equal to − 1 (by minimality). Such a matrix must be 0 or a multiple of the Cartan matrix of an affine Dynkin diagram of type ADE
.
The intersection matrix determines the fiber type with three exceptions:
This gives all the possible non-multiple fibers.
Multiple fibers can only exist for non-simply connected fibers, which are the fibers of type Iv.
Logarithmic transformations can be quite violent: they can change the Kodaira dimension, and can turn algebraic surfaces into non-algebraic surfaces.
Example:
Let L be the lattice Z+iZ of C, and let E
be the elliptic curve C/L. Then the projection map from E×C to C is an elliptic fibration. We will show how to replace the fiber over 0 with a fiber of multiplicity 2.
There is an automorphism of E×C of order 2 that maps
(c,s) to (c+1/2, −s). We let X be the quotient of
E×C by this group action. We make X into a fiber space over C by mapping (c,s) to s2.
We construct an isomorphism from X minus the fiber over 0 to
E×C minus the fiber over 0 by mapping (c,s) to (c-log(s)/2πi,s2). (The two fibers over 0 are non-isomorphic elliptic curves, so the fibration X is certainly not isomorphic to the fibration E×C over all of C.)
Then the fibration X has a fiber of multiplicity 2 over 0, and otherwise looks like
E×C. We say that X is obtained by applying a logarithmic transformation of order 2 to E×C with center 0.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
to an algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
, almost all of whose fibers are elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s.
The fibers that are not elliptic curves are called the singular fibers and were classified by Kunihiko Kodaira.
Elliptic surfaces form a large class of surfaces that contains many of the interesting examples of surfaces, and are relatively well-understood from the viewpoint of complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
theory and the theory of smooth
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
4-manifold
4-manifold
In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...
s. They are similar to (have analogies with, that is), elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s over number fields.
Examples
- The product of any elliptic curve with any curve is an elliptic surface (with no singular fibers).
- All surfaces of Kodaira dimensionKodaira dimensionIn algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...
1 are elliptic surfaces. - Every complex Enriques surfaceEnriques surfaceIn mathematics, Enriques surfaces, discovered by , are complex algebraic surfacessuch that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square...
is elliptic, and has an elliptic fibration over the projective line. - Kodaira surfaceKodaira surfaceIn mathematics, a Kodaira surface is a compact complex surface of Kodaira dimension 0 and odd first Betti number.These are never algebraic, though they have non-constant meromorphic functions...
s. - Dolgachev surfaceDolgachev surfaceIn mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by . They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds no two of which are diffeomorphic.-Properties:...
s - Shioda modular surfaces
Kodaira's table of singular fibers
Most of the fibers of an elliptic fibration are (non-singular) elliptic curves. The remaining fibers are called singular fibers: there are a finite number of them, and they consist of unions of rational curves, possibly with singularities or non-zero multiplicities (so the fibers may be non-reduced schemes). Kodaira and Neron independently classified the possible fibers, and Tate's algorithm can be used to find the type of a fiber.The following table lists the possible fibers of a minimal elliptic fibration. ("Minimal" means roughly one that cannot be factored through a "smaller" one; for surfaces this means that the singular fibers should contain no minimal curves.) It gives:
- Kodaira's symbol for the fiber,
- André NéronAndré NéronAndré Néron was a French mathematician at the Université de Poitiers who worked on elliptic curves and Abelian varieties...
's symbol for the fiber, - The number of irreducible components of the fiber (all rational except for type I0)
- The intersection matrix of the components. This is either a 1×1 zero matrix, or an affine Cartan matrix, whose Dynkin diagram is given.
| Kodaira | Neron | Components | intersection matrix | |
|---|---|---|---|---|
| I0 | A | 1 (elliptic) | 0 | |
| I1 | B1 | 1 (with double point) | 0 | |
| Iv (v≥2) | Bv | v (v distinct intersection points) | affine Av-1 | |
| mIv (v≥0, m≥2) | Iv with multiplicity m | |||
| II | C1 | 1 (with cusp) | 0 | |
| III | C2 | 2 (meet at one point of order 2) | affine A1 | |
| IV | C3 | 3 (all meet in 1 point) | affine A2 | |
| I0* | C4 | 5 | affine D4 | |
| Iv* (v>0) | C5,v | 5+v | affine D4+v | |
| IV* | C6 | 7 | affine E6 | |
| III* | C7 | 8 | affine E7 | |
| II* | C8 | 9 | affine E8 |
This table can be found as follows. Geometric arguments show that the intersection matrix of the components of the fiber must be negative semidefinite, connected, symmetric, and have no diagonal entries equal to − 1 (by minimality). Such a matrix must be 0 or a multiple of the Cartan matrix of an affine Dynkin diagram of type ADE
ADE classification
In mathematics, the ADE classification is the complete list of simply laced Dynkin diagrams or other mathematical objects satisfying analogous axioms; "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of \pi/2 = 90^\circ ...
.
The intersection matrix determines the fiber type with three exceptions:
- If the intersection matrix is 0 the fiber can be either an elliptic curve (type I0), or have a double point (type I1), or a cusp (type II).
- If the intersection matrix is affine A1, there are 2 components with intersection multiplicity 2. They can meet either in 2 points with order 1 (type I2), or at one point with order 2 (type III).
- If the intersection matrix is affine A2, there are 3 components each meeting the other two. They can meet either in pairs at 3 distinct points (type I3), or all meet at the same point (type IV)
This gives all the possible non-multiple fibers.
Multiple fibers can only exist for non-simply connected fibers, which are the fibers of type Iv.
Logarithmic transformations
A logarithmic transformation (of order m with center p) of an elliptic surface or fibration turns a fiber of multiplicity 1 over a point p of the base space into a fiber of multiplicity m. It can be reversed, so fibers of high multiplicity can all be turned into fibers of multiplicity 1, and this can be used to eliminate all multiple fibers.Logarithmic transformations can be quite violent: they can change the Kodaira dimension, and can turn algebraic surfaces into non-algebraic surfaces.
Example:
Let L be the lattice Z+iZ of C, and let E
be the elliptic curve C/L. Then the projection map from E×C to C is an elliptic fibration. We will show how to replace the fiber over 0 with a fiber of multiplicity 2.
There is an automorphism of E×C of order 2 that maps
(c,s) to (c+1/2, −s). We let X be the quotient of
E×C by this group action. We make X into a fiber space over C by mapping (c,s) to s2.
We construct an isomorphism from X minus the fiber over 0 to
E×C minus the fiber over 0 by mapping (c,s) to (c-log(s)/2πi,s2). (The two fibers over 0 are non-isomorphic elliptic curves, so the fibration X is certainly not isomorphic to the fibration E×C over all of C.)
Then the fibration X has a fiber of multiplicity 2 over 0, and otherwise looks like
E×C. We say that X is obtained by applying a logarithmic transformation of order 2 to E×C with center 0.