Elliptic surface

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.

Examples

• 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 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 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.

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:

• 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 I_v.

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.

See also

• Enriques-Kodaira classification
• Néron minimal model

References

• Compact Complex Surfaces by Wolf P. Barth, Klaus Hulek, Chris A.M. Peters, Antonius Van de Ven ISBN 3-540-00832-2 This is the standard reference book for compact complex surfaces.