Hyperelliptic surface
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.
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/Λ and F are elliptic curves, and G is a subgroup of F (acting on F 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 |
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
- Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225 - the standard reference book for compact complex surfaces
- Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts, vol. 34 (2nd ed.), Cambridge University Press, ISBN 978-0-521-49510-3, MR 1406314, ISBN 978-0-521-49842-5
- Bombieri, Enrico; Mumford, David (1976), "Enriques' classification of surfaces in char. p. III.", Inventiones Mathematicae, 35: 197–232, doi:10.1007/BF01390138, ISSN 0020-9910, MR 0491720
- Bombieri, Enrico; Mumford, David (1977), "Enriques' classification of surfaces in char. p. II", Complex analysis and algebraic geometry, Tokyo: Iwanami Shoten, pp. 23–42, MR 0491719