mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a hyperelliptic surface, or bi-elliptic surface, is a minimal

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

whose Albanese morphism is an elliptic fibration without singular fibres. Any such surface can be written as the

quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...

of a product of two elliptic curves by a finite abelian group. Hyperelliptic surfaces form one of the classes of surfaces of

Kodaira dimension In algebraic geometry, the Kodaira dimension measures the size of the canonical model of a projective variety . Soviet mathematician Igor Shafarevich in a seminar introduced an important numerical invariant of surfaces with the notation . ...

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 who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode. Acting involves a broad range of sk ...

on ''F'' by translations), which acts on ''E'' not only 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 The adjective canonical is applied in many contexts to mean 'according to the canon' the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, ''canonical examp ...

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

fiber Fiber (spelled fibre in British English; from ) 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 inco ...

s are

rational Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...

with a

cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifu ...

. They only exist in characteristics 2 or 3. Their second

Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...

is 2, the second

Chern number In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches o ...

vanishes, and the holomorphic Euler characteristic 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