Minimal log regular models of hyperbolic curves over discrete valuation fields

TL;DR

This paper develops a theory of minimal log regular models for hyperbolic curves over discrete valuation fields, extending classical results on stable reduction without residue field restrictions.

Contribution

It introduces a new framework for minimal log regular models and provides a criterion for log regularity, broadening the understanding of stable reduction in algebraic geometry.

Findings

Established a criterion for log regularity via minimal desingularization.

Proved the equivalence of stable reduction of curves and their Jacobians without residue field assumptions.

Extended classical stable reduction results to more general settings.

Abstract

In the famous paper of Deligne and Mumford, they proved that a proper hyperbolic curve over a discrete valuation field has stable reduction if and only if the Jacobian variety of the curve has stable reduction in the case where the residue field of its valuation ring is algebraically closed. In the proof, the theory of minimal regular models played an important role. In this paper, we establish a theory of minimal log regular models of curves. As a key tool for this theory, we give a criterion for $2$-dimensional local schemes to be log regular in terms of their minimal desingularization. Moreover, as an application of this theory, we prove the above equivalence without the assumption on the residue field.