In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a surface whose Albanese morphism is an elliptic fibration. 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.
The Kodaira dimension is 0.
Hodge diamond:
| 1 | ||||
| 1 | 1 | |||
| 0 | 2 | 0 | ||
| 1 | 1 | |||
| 1 |
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.
| 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 |
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).