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 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 t ...

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 In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...

of a

product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...

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 In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...

0 in the

Enriques–Kodaira classification In mathematics, the Enriques–Kodaira classification is a classification of compact complex surfaces into ten classes. For each of these classes, the surfaces in the class can be parametrized by a moduli space. For most of the classes the modu ...

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 Rationality is the quality of being guided by or based on reasons. 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 abili ...

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 bifurca ...

. 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 found applications in physics, Calabi–Yau m ...

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 In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...

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