Compactifications of the universal Jacobian over curves with marked points

Abstract: We construct modular compactifications of the universal Jacobian stack over the moduli stack of reduced curves with marked points depending on stability parameters obtained out of fixing a vector bundle on the universal curve. When restricted to the locus of stable marked curves, our compactifications are Deligne-Mumford irreducible smooth stacks endowed with projective moduli spaces and, following Esteves approach to the construction of fine compactifications of Jacobians, they parametrize torsion-free rank-1 simple sheaves satisfying a stability condition with respect to the fixed vector bundle. We also study a number of properties of our compactifications as the existence of forgetful and clutching morphisms and as well of sections from the moduli stack of stable curves with marked points. We conclude by indicating a number of different possible applications for our constructions.