Hyperelliptic surface
title: "Hyperelliptic surface" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["complex-surfaces", "algebraic-surfaces"] topic_path: "general/complex-surfaces" source: "https://en.wikipedia.org/wiki/Hyperelliptic_surface" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a minimal surface whose Albanese morphism is an elliptic fibration without singular fibres. 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.
Invariants
The Kodaira dimension is 0.
Hodge diamond: ::data[format=table]
| 1 | ||
|---|---|---|
| :: |
Classification
Any hyperelliptic surface is a quotient (E×F)/G, where E = C/Λ and F are elliptic curves, and G is a subgroup of F (acting on F by translations), which acts on E not only by translations. There are seven families of hyperelliptic surfaces as in the following table. ::data[format=table]
| order of K | Λ | 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 , 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
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- the standard reference book for compact complex surfaces
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