Models of Jacobians of Curves

We show that the Jacobians of prestable curves over toroidal varieties always admit Néron 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, and has the extension property for all torsion sections. The Néron model and the saturated model coincide over a Dedekind base, so the saturated model gives an alternative generalization of the classical notion of Néron models to higher-dimensional bases; in the general case we give necessary and sufficient conditions for the Néron model and saturated model to coincide. The key result, from which most others descend, is that the logarithmic Jacobian of [S. Molcho and J. Wise, The logarithmic Picard group and its tropicalization, preprint 2018] is a log Néron model of the Jacobian.