Let $k$ be an algebraically closed field and $X$ a smooth, projective curve over $k$. Consider a finite set $S$ of closed points of $X$ and let $U = X \setminus S$. Consider the divisor $\mathfrak{m} = \sum_{P \in S} (P)$ on $X$.
Following the book Groupes algébriques et corps de classes by Serre, to $\mathfrak{m}$ there is associated a generalized Jacobian variety $J_{\mathfrak{m}}$ of $X$ with a universal morphism $f_{\mathfrak{m}}: U = X \setminus S \to J_{\mathfrak{m}}$ that satisfies a certain universal property.
It is said in various places, that $f_\mathfrak{m}$ is injective, but I can't find a reference on this. In particular, if $X = \mathbb{P}^1_k$ and $S = \emptyset$, then $J_{\mathfrak{m}}$ is the Jacobian of $\mathbb{P}^1_k$ which is zero. Hence there must be some condition on $\# S$ for this to hold.