A criterion for existence of Néron models

Given an abelian variety A over the fraction field K of a Dedekind domain R, there is a canonical model for A over R, called Néron model. It enjoys a number of useful properties. However, if K is the fraction field of a higher dimensional ring R, there may be no Néron model for A. In the first 30 minutes, I will introduce Néron models, motivate their definition, and show an example of jacobian not admitting a Néron model over a 2-dimension base. In the following 45 minutes, I will introduce a criterion, called toric additivity, for an abelian family with semistable reduction to admit a Néron model over a higher dimensional base scheme. If time allows, I will discuss work in progress about how logarithmic jacobians constitute a good replacement for Néron models.