A criterion for existence of N\'eron models of jacobians

Abstract

N\'eron models of abelian varieties do not necessarily exist if the base $S$ has dimension higher than 1. We introduce a new condition, called toric additivity, on a family of smooth curves having nodal reduction over a normal crossing divisor $D\subset S$. The condition is necessary and sufficient for existence of a N\'eron model of the jacobian of the family; it depends only on the Betti numbers of the dual graphs of the fibres of the family, or on the toric ranks of the fibres of the jacobian.