  We introduce the abstract notion of a \emph{smoothable fine compactified
Jacobian} of a nodal curve, and of a family of nodal curves whose general
element is smooth. Then we introduce the notion of a combinatorial stability
condition for line bundles and their degenerations.
  We prove that smoothable fine compactified Jacobians are in bijection with
these stability conditions.
  We then turn our attention to \emph{fine compactified universal Jacobians},
that is, fine compactified Jacobians for the moduli space
$\overline{\mathcal{M}}_g$ of stable curves (without marked points). We prove
that every fine compactified universal Jacobian is isomorphic to the one first
constructed by Caporaso, Pandharipande and Simpson in the nineties. In
particular, without marked points, there exists no fine compactified universal
Jacobian unless $\gcd(d+1-g, 2g-2)=1$.
