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 curves.
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-manifolds. They are similar to (have analogies with, that is), elliptic curves over number fields.
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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 | 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.
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.
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.