As far as I know, the main representability result for the relative Picard functor $Pic_{X/k}$, for a noeth. sep. scheme of finite type over a field $k$ is:

If $X$ is proper then $Pic_{X/k}$ is representable by a $k$-scheme loc. of finite type. (This is attributed to Murre and Oort in Bosch-Lüttkebohmert-Raynaud)

I am interested in what can be said once the requirement of properness is dropped, e.g. what can be said for quasi-projective varieties?

Representability is probably to much to ask for (even as an algebraic space), but do you have references or know of examples where the Picard functor of a non-projective quasi-projective variety is representable?

Is there a weaker sense of representability in which sense the "open" Picard functor is representable?

Is the group somehow controlled by (the group of $k$-points of) representable objects. (I have the naive impression that if $X$ is my quasi-projective variety, then a proper hypercovering of $X$ should be able to compute $H^1(X,\mathcal{O}_X^*)$, and that then one might be able to use representability theorems for proper/projective maps, but I know nearly nothing about the involved technical requirements.)

**Edit**: I should have added that I do not want to assume resolution of singularities.