Jacobian of a curve (awaiting review)

The Jacobian of a (smooth, projective, geometrically integral) curve of genus $g$ is an abelian variety of dimension $g$ that is canonically associated to the curve.

Points on the Jacobian correspond to certain formal sums of points on the curve, modulo an equivalence relation. By choosing a rational point on the curve, one can embed the curve in its Jacobian (this embedding is known as the Abel-Jacobi map); for elliptic curves this embedding is an isomorphism, but otherwise not.

Note that it is possible for an abelian variety to be isogenous to the Jacobian of a curve without being isomorphic to one.

Any Jacobian is a principally polarized abelian variety, and the Torelli theorem states that if $J(C)$ and $J(C')$ are isomorphic as principally polarized abelian varieties then $C$ and $C'$ are isomorphic curves. It is possible, however, for non-isomorphic curves to have Jacobians that are isomorphic as unpolarized abelian varieties.