Let $X$ be a projective curve of genus $g$ and $\operatorname{Pic}^d(X)$ its degree $d$ Picard variety, parameterizing isomorphism classes of invertible sheaves of degree $d$ on $X$. $\operatorname{Pic}^d(X)$ is a very important invariant of the curve and it is well known that it is complete if and only if $X$ is a curve of compact type. The problem of compactifying $\operatorname{Pic}^d(X)$ has been widely studied in the last decades and in nowadays there exist several solutions, differing from one another in various aspects such as the geometrical interpretation or the functorial properties.

By the other hand, the introduction of (algebraic) stacks in the study of moduli problems in algebraic geometry allowed a deeper understanding of the objects one wants to parametrize exactly because of the good functorial properties of moduli stacks.

In the first part of my talk I will explain how to construct geometrically meaningful algebraic (Artin) stacks $\overline{\mathcal P}_{d,g}$ over the moduli stack of stable curves, $\overline{\mathcal M}_g$, giving a functorial way of compactifying the relative degree $d$ Picard variety for families of stable curves.

In the secont part, I will explain how to generalize this construction to curves with marked points using an inductive argument that consists, at each step, on giving a geometrical description of the universal family over the previous stack.