  To every singular reduced projective curve X one can associate many fine
compactified Jacobians, depending on the choice of a polarization on X, each of
which yields a modular compactification of a disjoint union of the generalized
Jacobian of X. We investigate the geometric properties of fine compactified
Jacobians focusing on curves having locally planar singularities. We give
examples of nodal curves admitting non isomorphic (and even non homeomorphic
over the field of complex numbers) fine compactified Jacobians. We study
universal fine compactified Jacobians, which are relative fine compactified
Jacobians over the semiuniversal deformation space of the curve X. Finally, we
investigate the existence of twisted Abel maps with values in suitable fine
compactified Jacobians.
