Fix $k$ a number field, and let $C$ be a smooth geometrically integral affine curve over $k$. We can "associate" to $C$ a semi-abelian variety in the following way:


One knows that $C$ is a finite number of (closed) points away from its smooth compactification, which is a projective curve $X$. Let these points be denoted by $p\_1,...,p\_n$, they lie in $X(\bar{k})$. Let $D$ be the divisor in $\mathrm{Div}\_\bar{k}(X)$ given by $$D = p\_1+...+p\_n.$$
In the book *Algebraic groups and class fields* by Serre, one can talk about the *generalized Jacobian* $J\_D$ of $X$ which fits in the exact sequence $$0 \rightarrow T \rightarrow J\_D \rightarrow J(X) \rightarrow 0,$$
where $T$ is an $(n-1)$-dimensional torus and $J(X)$ is the Jacobian of $X$, it is an abelian variety.


Thus $J\_D$ is an extension of $J(X)$ by $T$ and by definition, it is a semi-abelian variety. I have the following questions:


1. How is this extension defined? I know it is almost never true that this is the direct product $J\_D = T \times J(X)$.
2. Does one have a (canonical) map $C \rightarrow J\_D$?



Curves and semi-abelian varieties
Let me treat in some details the case $n=2$ — the general case is similar. Consider the nodal curve $Y$ obtained from $X$ by identifying $p\_1$ and $p\_2$. Then $J\_D$ is the Jacobian $JY$ of $Y$. Pulling back to $X$ gives an exact sequence
$$0\rightarrow \mathbb{G}\_m\rightarrow JY\rightarrow JX\rightarrow 0\,.$$
Such an extension is parameterized by a class in $\operatorname{Ext}^1(JX,\mathbb{G}\_m) $, say in the category of abelian group schemes. This group is canonically isomorphic to the dual abelian variety of $JX$, which is canonically isomorphic to $JX$ through the principal polarization; the extension class corresponds to the class of the divisor $p\_1-p\_2$.


To answer your question 2), if $s$ is the node of $Y$ and $p\in Y\smallsetminus s$, there is an Abel-Jacobi map $\alpha \_p:Y\smallsetminus s\rightarrow JY$ mapping a point $x\in X$ to the class of the divisor $x-p$. Note that this map is canonical only up to translation, as is the case already when $Y$ is smooth. And it is definitely not defined at $s$ — in fact there is a natural compactification of $JY$, and $\alpha \_p$ maps $s$ into the divisor at infinity.


Finally this works for any $n$ — you must consider the curve obtained by identifying the $n$ points.


