Jacobians

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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 Pic^g-1 to Jacobian of the image of Sym^g-1C in Pic^g-1.

Examples:

• 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:

• Abels theorem states that the map ${\displaystyle {\mathcal {M}}_{g}\to {\mathcal {A}}_{g}}$, which takes a curve to it's Jacobian is an injection.
• The Schottky problem calls for the classification of the map above.