  We construct natural relative compactifications for the relative Jacobian
over a family $X/S$ of reduced curves. In contrast with all the available
compactifications so far, ours admit a universal sheaf, after an etale base
change. Our method consists of considering the functor $F$ of relatively
simple, torsion-free, rank 1 sheaves on $X/S$, and showing that certain open
subsheaves of $F$ have good properties. Strictly speaking, the functor $F$ is
only representable by an algebraic space, but we show that $F$ is representable
by a scheme after an etale base change. Finally, we use theta functions
originating from vector bundles to compare our new compactifications with the
available ones.
