I'm not very skilled in sheaf theory, but my question is the following:
Consider a family of sheaves $\{\mathcal{O}(D)\}$ where $[D]\in\operatorname{Pic}(X).$ Is it possible to define some scheme (I think it should be just an algebraic variety) such that $\{H^0(X,\mathcal{O}(D))\}$ becomes a sheaf on it?
My naive thoughts go like this: we can take a product $\operatorname{Pic}(X)\times X$ then we should be able (or not) to make this family into a sheaf. For the discrete Picard group, it should work for trivial reasons, I'm of course most interested in the case where the Picard group is continuous (e.g. $X$ is an abelian variety).
I have this idea from complex geometry (also without rigorous proof), where the Picard group is often continuous, and it made me think that something similar should work algebraically.