An arithmetic variant of Raynaud's theorem

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 $\mathfrak X$.