Compactifying the relative Jacobian over families of reduced curves
Abstract
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.