Néron models over bases of higher dimension

Néron models for 1-parameter families of abelian varieties were defined and constructed by Néron in the 1960’s, and provide a `best possible’ model for the degenerating family. For a degenerating family of abelian varieties over a base scheme of dimension greater than 1, it is much less clear what the `best possible' model for the family would be. If one naively extends Néron’s original definition to this setting then these objects fail to exist, even if we allow blowups or alterations of the base space of the family - more precisely, we give a combinatorial characterisation of exactly when such Néron models of jacobians exist. In the case of the jacobian of the universal curve we will describe the minimal base-change required in order that a Néron model exists, giving a possible answer to the shape of the `best possible degeneration’. If time allows, we will also relate this to questions concerning height jumping and rational points on abelian varieties.