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 $C \rightarrow J_D$?
  • 2
    $\begingroup$ This is discussed on page 2 of Serre's book that you mention $\endgroup$
    – George R
    Jun 29 at 8:12

1 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.

  • $\begingroup$ What you've said is totally new to me, so if you could provide some reference it'll be great, otherwise pardon me for asking silly questions. 1) When you construct $Y$ by identifying the two points, can the fact that $J_D = JY$ be checked by definitions of generalized Jacobian? 2) In your exact sequence you wrote $\mathbb{G}_m$, are you considering it over $\bar{k}$, where $T \cong \mathbb{G}_m$? 3) I suppose when we consider the general case, it would be $\mathbb{G}_m^{n-1}$ instead if what I've written in the question was correct? $\endgroup$ Jun 29 at 12:26
  • 1
    $\begingroup$ A reference: A construction of generalized Jacobian varieties by group extensions by F. Oort, Math. Ann. 147 (1962), 277-286. $\endgroup$
    – abx
    Jun 29 at 14:07
  • 1
    $\begingroup$ Some answers: 1) Yes. 2) I think that everything is defined over $k$, provided $p_1$ and $p_2$ are in $X(k)$. 3) Yes, in the general case you get an extension by $\Bbb{G}_m^{n-1}$ (this is explained in Oort's paper). $\endgroup$
    – abx
    Jun 29 at 14:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.