The fibers that are not elliptic curves are called the singular fibers and were classified by Kunihiko Kodaira. In the context of string theory, both elliptic and singular fibers are crucial in the descriptions using F-theory.
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-manifolds. They are similar to (have analogies with, that is), elliptic curves over number fields.
- The product of any elliptic curve with any curve is an elliptic surface (with no singular fibers).
- All surfaces of Kodaira dimension 1 are elliptic surfaces.
- Every complex Enriques surface is elliptic, and has an elliptic fibration over the projective line.
- Kodaira surfaces.
- Dolgachev surfaces
- 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 Néron independently classified the possible fibers, and Tate's algorithm can be used to find the type of the fibers of an elliptic curve over a number field.
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 Néron'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.
- The multiplicities of each fiber are indicated in the Dynkin diagram.
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:
- 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)
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.
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- Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius. Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 4 (2nd enlarged ed.). Berlin: Springer-Verlag. ISBN 3-540-00832-2. Zbl 1036.14016..
- Kodaira, Kunihiko (1964). "On the structure of compact complex analytic surfaces. I". Am. J. Math. 86: 751–798. Zbl 0137.17501.
- Kodaira, Kunihiko (1966). "On the structure of compact complex analytic surfaces. II". Am. J. Math. 88: 682–721. Zbl 0193.37701.
- Néron, André (1964). "Modeles minimaux des variétés abeliennes sur les corps locaux et globaux". Publications Mathématiques de l'IHÉS (in French) 21: 5–128. MR 0179172. Zbl 0132.41403.