Jumps and motivic invariants of semiabelian Jacobians

TL;DR

This paper proves the rationality of jumps in the Néron models of Jacobians of certain singular curves, extending known results and describing the structure of their Néron models in new cases.

Contribution

It establishes the rationality of jumps for Jacobians of curves with push-out singularities and generalizes Raynaud's description of Néron models to these cases.

Findings

Proves the conjecture for Jacobians of curves with push-out singularities.

Extends the exact sequence of Néron models to non-split semiabelian varieties.

Provides new insights into the structure of Néron models for singular curves.

Abstract

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.