Elliptic surface


In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. This is equivalent to the generic fiber being a smooth curve of genus one. This follows from proper base change.
The surface and the base curve are assumed to be non-singular. The fibers that are not elliptic curves are called the singular fibers and were classified by Kunihiko Kodaira. Both elliptic and singular fibers are important in string theory, especially in F-theory.
Elliptic surfaces form a large class of surfaces that contains many of the interesting examples of surfaces, and are relatively well understood in the theories of complex manifolds and smooth 4-manifolds. They are similar to, elliptic curves over number fields.

Examples

Most of the fibers of an elliptic fibration are 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. 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. It gives:
KodairaNéronComponentsIntersection matrixDynkin diagramFiber
I0A1 0
I1B11 0
I2B22 affine A1
Iv Bvv affine Av-1
mIv Iv with multiplicity m--
IIC11 0
IIIC22 affine A1
IVC33 affine A2
I0*C45affine D4
Iv* C5,v5+vaffine D4+v
IV*C67affine E6
III*C78affine E7
II*C89affine 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. 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:
The monodromy around each singular fiber is a well-defined conjugacy class in the group SL of 2 × 2 integer matrices with determinant 1. The monodromy describes the way the first homology group of a smooth fiber changes as we go around a singular fiber. Representatives for these conjugacy classes associated to singular fibers are given by:
FiberIntersection matrixMonodromyj-invariantGroup structure on smooth locus
Iνaffine Aν-1
II00
IIIaffine A11728
IVaffine A20
Iν*affine D4+νif ν is even, if ν is odd
IV*affine E60
III*affine E71728
II*affine E80

For singular fibers of type II, III, IV, IV*, III*, or II*, the monodromy has finite order in SL. This reflects the fact that an elliptic fibration has potential good reduction at such a fiber. That is, after a ramified finite covering of the base curve, the singular fiber can be replaced by a smooth elliptic curve. Which smooth curve appears is described by the j-invariant in the table. Over the complex numbers, the curve with j-invariant 0 is the unique elliptic curve with automorphism group of order 6, and the curve with j-invariant 1728 is the unique elliptic curve with automorphism group of order 4.
For an elliptic fibration with a section, called a Jacobian elliptic fibration, the smooth locus of each fiber has a group structure. For singular fibers, this group structure on the smooth locus is described in the table, assuming for convenience that the base field is the complex numbers. Knowing the group structure of the singular fibers is useful for computing the Mordell-Weil group of an elliptic fibration, in particular its torsion subgroup.

Logarithmic transformations

A logarithmic transformation 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 to. We let X be the quotient of E×C by this group action. We make X into a fiber space over C by mapping to s2. We construct an isomorphism from X minus the fiber over 0 to E×C minus the fiber over 0 by mapping to.
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.