Compactifying the relative Jacobian over families of reduced curves

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.