Models of Jacobians of curves

We show that the Jacobians of prestable curves over toroidal varieties always admit N\'eron models. These models are rarely quasi-compact or separated, but we also give a complete classification of quasi-compact separated group-models of such Jacobians. In particular we show the existence of a maximal quasi-compact separated group model, which we call the saturated model, which has the extension property for all torsion sections. The N\'eron model and the saturated model coincide over a Dedekind base, so the saturated model gives an alternative generalisation of the classical notion of N\'eron models to higher-dimensional bases; in the general case we give necessary and sufficient conditions for the N\'eron model and saturated model to coincide. The key result, from which most others descend, is that the logarithmic Jacobian of \cite{Molcho2018The-logarithmic} is a log Neron model of the Jacobian.