Compactifying the relative Jacobian over families of reduced curves

We construct natural relative compactifications for the relative Jacobian o ver a family X/S of reduced curves. In contrast with all the available comp actifications so far, ours admit a Poincare sheaf after an etale base chang e. Our method consists of studying the etale sheaf F of simple, torsion-fre e, rank-1 sheaves on X/S, and showing that certain open subsheaves of F hav e the completeness property. Strictly speaking, the functor F is only repre sentable by an algebraic space, but we show that F is representable by a sc heme after an etale base change. Finally, we use theta functions originatin g from vector bundles to compare our new compactifications with the availab le ones.