hyperelliptic surface
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.{{Clarification needed|date=August 2024}}
Hodge diamond:
| style="font-weight:bold" | ||||
| 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), which acts on E not only by translations. There are seven families of hyperelliptic surfaces as in the following table.
| class="wikitable"
!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 {{harv|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
- {{Citation | last1=Barth | first1=Wolf P. | last2=Hulek | first2=Klaus | last3=Peters | first3=Chris A.M. | last4=Van de Ven | first4=Antonius | title=Compact Complex Surfaces | publisher= Springer-Verlag, Berlin | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-00832-3 | mr=2030225 | year=2004 | volume=4}} - the standard reference book for compact complex surfaces
- {{Citation | last1=Beauville | first1=Arnaud | title=Complex algebraic surfaces | publisher=Cambridge University Press | edition=2nd | series=London Mathematical Society Student Texts | isbn=978-0-521-49510-3 |id={{ISBN|978-0-521-49842-5}} | mr=1406314 | year=1996 | volume=34}}
- {{Citation | last1=Bombieri | first1=Enrico | author1-link=Enrico Bombieri | last2=Mumford | first2=David | author2-link=David Mumford | title=Enriques' classification of surfaces in char. p. III. | doi=10.1007/BF01390138 | mr=0491720 | year=1976 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=35 | pages=197–232| bibcode=1976InMat..35..197B | url=https://dash.harvard.edu/bitstream/handle/1/3612776/Mumford_EnriquesClassIII.pdf?sequence=3 }}
- {{Citation | last1=Bombieri | first1=Enrico | author1-link=Enrico Bombieri | last2=Mumford | first2=David | author2-link=David Mumford | title=Complex analysis and algebraic geometry | publisher=Iwanami Shoten | location=Tokyo | mr=0491719 | year=1977 | chapter=Enriques' classification of surfaces in char. p. II | pages=23–42}}