Fine compactified Jacobians

We study Esteves's fine compactified Jacobians for nodal curves. We give a proof of the fact that, for a one-parameter regular local smoothing of a nodal curve $X$, the relative smooth locus of a relative fine compactified Jacobian is isomorphic to the N\'eron model of the Jacobian of the general fiber, and thus it provides a modular compactification of it. We show that each fine compactified Jacobian of $X$ admits a stratification in terms of certain fine compactified Jacobians of partial normalizations of $X$ and, moreover, that it can be realized as a quotient of the smooth locus of a suitable fine compactified Jacobian of the total blowup of $X$. Finally, we determine when a fine compactified Jacobian is isomorphic to the corresponding Oda-Seshadri's coarse compactified Jacobian.