Abelian surface

In algebraic geometry an Abelian surface over a field $K$ is a two dimensional Abelian variety. Every abelian surface is a finite quotient of a Jacobian variety of a smooth hyperelliptic curve of genus two or a product of two elliptic curves. Abelian surfaces are one of the two types of algebraic surfaces with trivial canonical class, the other type being algebraic K3 surfaces.

Polarization

Abelian surfaces have a trivial canonical class. Therefore they are usually considered together with a choice of some non-trivial effective divisor on them. This divisor is called the polarization on the Abelian surface; A pair $(A,C)$ of an Abelian surface and a polarization is call a polarized Abelian surface. Given a polarized Abelian variety $(A,D)$ we define the polarization map $A\to Pic^{0}(A)$ by sending a point $a$ to the divisor class $[\tau _{-a}C-C]$. This map is a group morphism. The kernel of the map is a finite Abelian group with at most four generators. The ismorphism type of the kernel is called the type of the polarization. If the kerenl is trivial then polarization is called principal; in this case the arithmetic genus of $C$ is $2$. Most of the classical theory of Abelian surfaces deal with the case where $C$ is a smooth curve of genus 2.