N\'eron models of jacobians over base schemes of dimension greater than 1

Abstract

We investigate to what extent the theory of N\'eron models of jacobians and of abel-jacobi maps extends to relative curves over base schemes of dimension greater than 1. We give a necessary and sufficient criterion for the existence of a N\'eron model. We use this to show that, in general, N\'eron models do not exist even after making a modification or even alteration of the base. On the other hand, we show that N\'eron models do exist outside some codimension-2 locus.