Universal property of the Jacobian variety
= Universal property of the Jacobian variety
For a smooth connected projective curve $C$ with chosen basepoint $p$, the <Abel map of an algebraic curve> $i_p:C\to J(C)$ sends $q$ to $[q-p]$. Every morphism $g:C\to A$ to an <abelian variety> with $g(p)=0$ factors uniquely through a homomorphism $J(C)\to A$. This universal property converts a curve embedded in an abelian variety into a homomorphism from its Jacobian.