Jacobians

The Jacobian variety of a smooth algebraic curve C is the variety of degree 0 divisors of C, up to ratinal equivalence; i.e. it is the kernel of the degree map from Pic(C) to the integers; sometimes also denoted as Pic0. It is an Abelian variety of dimension g.

Examples:
 * A genus 1 curve is naturally ismorphic to the variety of degree 1 divisors, and therefor to is isomorphic to it's Jacobian.

Related theorems and problems:
 * Abels theorem states that the map $$\mathcal{M}_g\to\mathcal{A}_g$$, which takes a curve to it's jacobian is an injection.
 * The Shottcky problem calss for the classification of the map above.