Abelian surface

In algebraic geometry an Abelian surface over a field $K$ is a two dimensional Abelian variety. Every abelian surface is either 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.

Jacobians of curves of genus 2

Below we consider the case where an Abelian variety is the Jacobian of a curve $H$ of genus 2.