A Jacobian variety is a fundamental concept in algebraic geometry and is associated with algebraic curves. Specifically, it is the complex torus formed by the points of a smooth projective algebraic curve and is used to study the algebraic properties of the curve.
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The Jacobian variety of a smooth projective algebraic curve parametrizes degree-zero line bundles. Its dimension is the genus of the algebraic curve. An effective degree- Cartier divisor defines a point in a degree- torsor through the Abel map of an algebraic curve. The theta divisor induces an isomorphism with the dual abelian variety.