An arithmetic variant of Raynaud's theorem

TL;DR

This paper generalizes Raynaud's theorem relating dual graphs of special fibers to Néron models of Jacobians, incorporating arithmetic data without assuming algebraically closed residue fields.

Contribution

It introduces a combinatorially enriched dual graph framework to extend Raynaud's theorem to more general arithmetic settings.

Findings

Established a bijection between enriched dual graphs and Néron model components

Generalized Raynaud's theorem to non-algebraically closed residue fields

Provided a new combinatorial approach to arithmetic geometry problems

Abstract

It is well known that for a regular semistable curve $\mathfrak X$ over a DVR with algebraically closed residue field, the spanning trees of the dual graph of the special fiber of $\mathfrak X$ are in bijection with components of the special fiber of the N\'eron model of the Jacobian of $\mathfrak X$. We prove a generalization of this fact that does not require the residue field to be algebraically closed, using a combinatorially enriched version of the dual graph to encode arithmetic information about divisors on $\mathfrak X$.