Seminar

Let $\mathcal Pic_{d,g,n}$ be the stack parametrizing degree $d$ line bundles over smooth curves of genus $g$ with $n$ marked points. In this talk I will give a construction of smooth and irreducible algebraic stacks yeldying a modular compactification of $\mathcal Pic_{d,g,n}$ over the moduli stack of $n$-pointed stable curves, $\overline{\mathcal M}_{g,n}$. By this we mean an algebraic stack with a proper (or at least universally closed) map onto $\o{\mathcal M}_{g,n}$, containing $\mathcal Pic_{d,g,n}$ as a dense open substack. These stacks parametrize what we will call balanced line bundles over $n$-pointed quasistable curves, generalizing L. Caporaso's compactification of the universal degree $d$ Picard variety over $\overline{\mathcal M}_{g}$. In fact, for $n=0$, we just give a stack theoretical description of Caporaso's compactification and then, following the lines of Knudsen's construction of $\overline{\mathcal M}_{g,n}$, we go on by induction on the number of points.