# Curves and semi-abelian varieties

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

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.

• 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? Jun 29 at 12:26
• A reference: A construction of generalized Jacobian varieties by group extensions by F. Oort, Math. Ann. 147 (1962), 277-286.
– abx
Jun 29 at 14:07
• 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).
– abx
Jun 29 at 14:11