This is a wild guess, but I can not find anything in the literature. (Probably I do not know the right terms to search for.) Please bear with me.
I am looking for some way to assign a smooth projective curve (or an abelian variety, e.g. the Jacobian of the curve) to $\mathbb{P}^{1} - \{n \text{ points}\}$.
Motivation
Over $\mathbb{C}$, if one takes $\mathbb{P}^{1}$ and pinches together $g$ pairs of points, this looks pretty much like a surface with $g$ handles, except that all the handles are contracted to 1 point, resulting in singularities. If we remove the singularities, we obtain $\mathbb{P}^{1} - \{2g \text{ points}\}$.
Besides that (or maybe: on the other hand), the fundamental groups of $\mathbb{P}^{1} - \{2g + 1 \text{ points}\}$ and a surface with $g$ handles seem related.
I realise that this is all very topological, and ignores all algebro-geometric data. Nevertheless, maybe something like this is possible.
Question
So I guess my question boils down to:
Let $k$ be an algebraically closed field. Let $S \subset \mathbb{P}^{1}(k)$ be a finite set of cardinality $n$. Is there an algebro-geometric method to assign an abelian variety $A/k$ to $\mathbb{P}^{1} - S$?
If yes, please give some indication of how canonical the method is.
Notes:
- The reason I ask for an abelian variety, is that if it is possible to assign a curve, we can take the Jacobian. However, maybe it is not possible to assign a curve in a canonical way, but there might still be a method to reach an abelian variety (in some way I currently do not see…).
- I would already be interested in the situation for $k = \mathbb{C}$, however ultimately I am interested in the situation for $k$ of characteristic $p > 0$.