Compactifying the relative Jacobian over families of reduced curves

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