Hyperelliptic surface
Encyclopedia
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.
Hodge diamond:
on F by translations). There are seven families of hyperelliptic curves as in the following table.
Here ω is a primitive cube root of 1 and i is a primitive 4th root of 1.
. 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).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a hyperelliptic surface, or bi-elliptic surface, is a surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
with an elliptic fibration over an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
. Any such surface can be written as the quotient
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
of a product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
of two elliptic curves by a finite abelian group.
Hyperelliptic surfaces form one of the classes of surfaces of Kodaira dimension
Kodaira dimension
In algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...
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/Γ and F are elliptic curves, and G is a subgroup of F (actingGroup action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
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 cuspCusp (singularity)
In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
. They only exist in characteristics
Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
2 or 3. Their second Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
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
Group scheme
In mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not...
of F (acting on F by translations).