Hyperelliptic surface

In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a surface with an elliptic fibration over an elliptic curve. Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group. Hyperelliptic surfaces form one of the classes of surfaces of Kodaira dimension 0 in the Enriques–Kodaira classification.

Contents

Invariants

The Kodaira dimension is 0.

Hodge diamond:

1
1 1
0 2 0
1 1
1

Classification

Any hyperelliptic surface is a quotient (E×F)/G, where E = C/Γ and F are elliptic curves, and G is a subgroup of F (acting on F by translations). There are seven families of hyperelliptic curves as in the following table.

order of K Γ G Action of G on E
2 Any Z/2Z e → −e
2 Any Z/2Z ⊕ Z/2Z e → −e, e → e+c, −c=c
3 Z ⊕ Zω Z/3Z e → ωe
3 Z ⊕ Zω Z/3Z ⊕ Z/3Z e → ωe, e → e+c, ωc=c
4 Z ⊕ Zi; Z/4Z e → ie
4 Z ⊕ Zi Z/4Z ⊕ Z/2Z e → ie, e → e+c, ic=c
6 Z ⊕ Zω Z/6Z e → −ωe

Here ω is a primitive cube root of 1 and i is a primitive 4th root of 1.

Quasi hyperelliptic surfaces

A quasi-hyperelliptic surface is a surface whose canonical divisor is numerically equivalent to zero, the Albanese mapping maps to an elliptic curve, and all its fibers are rational with a cusp. They only exist in characteristics 2 or 3. Their second Betti number is 2, the second Chern number vanishes, and the holomorphic Euler characteristic vanishes. They were classified by (Bombieri & Mumford 1976), who found six cases in characteristic 3 (in which case 6K= 0) and eight in characteristic 2 (in which case 6K or 4K vanishes). Any quasi-hyperelliptic surface is a quotient (E×F)/G, where E is a rational curve with one cusp, F is an elliptic curve, and G is a finite subgroup scheme of F (acting on F by translations).

References