Chai's conjecture for semiabelian Jacobians

TL;DR

This paper proves Chai's conjecture for Jacobians of proper curves, extending Néron model constructions, and confirms related conjectures, providing new examples of wildly ramified semiabelian varieties.

Contribution

It establishes Chai's conjecture for Jacobians, extends Raynaud's Néron model construction to arbitrary seminormal curves, and proves conjectures on Néron models over fields with imperfection degree at most one.

Findings

Proved additivity of the base change conductor for Jacobians.

Constructed Néron models for a broader class of curves.

Confirmed conjectures on the existence of Néron models in specific cases.

Abstract

We prove Chai's conjecture on the additivity of the base change conductor of semiabelian varieties in the case of Jacobians of proper curves. This includes the first infinite family of non-trivial wildly ramified examples. Along the way, we extend Raynaud's construction of the N\'eron lft-model of a Jacobian in terms of the Picard functor to arbitrary seminormal curves (beyond which Jacobians admit no N\'eron lft-models, as shown by our more general structural results). Finally, we investigate the structure of Jacobians of (not necessarily geometrically reduced) proper curves over fields of degree of imperfection at most one and prove two conjectures about the existence of N\'eron models and N\'eron lft-models due to Bosch-L\"utkebohmert-Raynaud for Jacobians of general proper curves in the case of perfect residue fields, thus strengthening the author's previous results in this situation.