mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a hyperelliptic surface, or bi-elliptic surface, is a surface whose Albanese morphism is an

elliptic fibration In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed fi ...

. Any such surface can be written as the quotient of a product of two elliptic curves by a

finite abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

. 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:

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 Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode. Acting involves a broad r ...

on ''F'' by translations). There are seven families of hyperelliptic surfaces as in the following table. 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 In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...

is numerically equivalent to zero, the Albanese mapping maps to an elliptic curve, and all its

fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...

s 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 In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the ex ...

vanishes. They were classified by , who found six cases in characteristic 3 (in which case 6''K''= 0) and eight in characteristic 2 (in which case 6''K'' or 4''K'' 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

* - the standard reference book for compact complex surfaces * * *{{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 Complex surfaces Algebraic surfaces