  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.
