Curves and semi-abelian varieties

Question

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 $C \rightarrow J_D$?

Answer 1

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.