We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields, and their
behaviour under tame base change. For a semiabelian variety, this behaviour is governed by a finite sequence of (a
priori) real numbers between 0 and 1, called
jumps.
The jumps are conjectured to be rational, which is known in some
cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of
a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori
which are induced along finite separable extensions, and generalize Raynaud’s description of the identity component of
the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model)
to our situation. The main technical result of this paper is that the exact sequence which decomposes the Jacobian of
one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously,
only split semiabelian varieties were known to have this property.