The Jacobian variety of a smooth algebraic curve C is the variety of degree 0 divisors of C, up to rational 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 principally polarized Abelian variety of dimension g.
Principal polarization: The principal polarization of the Jacobian variety is given by the theta divisor: some shift from Picg-1 to Jacobian of the image of Symg-1C in Picg-1.
- A genus 1 curve is naturally isomorphic to the variety of degree 1 divisors, and therefore to is isomorphic to it's Jacobian.
Related theorems and problems: