The Jacobian of a (smooth, projective, geometrically integral) curve $X$ of genus $g$ over a field $k$ is a $g$-dimensional principally polarized abelian variety $J$ that is canonically associated to $X$.
If $X$ has a $k$-rational point, then $J(k)$ is isomorphic to the group of degree zero divisors on $X$ modulo linear equivalence. A choice of rational point on $X$ determines a morphism $X \to J$ called an Abel-Jacobi map; it is an embedding if and only if $g \ge 1$, and an isomorphism if and only if $g=1$.
The Torelli theorem states that if $X$ and $Y$ are curves whose Jacobians are isomorphic as principally polarized abelian varieties, then $X$ and $Y$ are isomorphic. It is possible, however, for non-isomorphic curves to have Jacobians that are isomorphic as unpolarized abelian varieties.
- Review status: beta
- Last edited by Bjorn Poonen on 2022-03-26 16:16:15