  We investigate N\'eron 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 N\'eron 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 N\'eron models. Previously, only
split semiabelian varieties were known to have this property.
