Stable reduction of curves and tame ramification

TL;DR

This paper provides a geometric criterion and explicit minimal extension for achieving stable reduction of curves with tame ramification, avoiding vanishing cycles, and clarifies the conditions under which tame extensions suffice.

Contribution

It offers a geometric proof of Saito's criterion and constructs an explicit minimal extension for stable reduction in tame ramification cases.

Findings

Established a geometric criterion for tame stable reduction

Constructed an explicit minimal extension for stable reduction

Provided a geometric proof avoiding vanishing cycles

Abstract

We study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring, there is a criterion, due to T. Saito, that describes precisely, in terms of the geometry of the minimal model with strict normal crossings of X, when a tamely ramified extension suffices in order for X to obtain stable reduction. For such curves we construct an explicit extension that realizes the stable reduction, and we furthermore show that this extension is minimal. We also obtain purely geometric proof of Saito's criterion, avoiding the use of vanishing cycles.