Explicit resolution of weak wild quotient singularities on arithmetic surfaces

TL;DR

This paper provides explicit methods to resolve weak wild quotient singularities on arithmetic surfaces, using deformation and valuation theories, and addresses questions posed by Lorenzini.

Contribution

It offers a detailed, local resolution technique for these singularities, expanding understanding of their structure and regular models.

Findings

Explicit resolution procedures for weak wild quotient singularities

Valuation-theoretic criterion for regular snc-models of P^1

Answers to several questions posed by Lorenzini

Abstract

A weak wild arithmetic quotient singularity arises from the quotient of a smooth arithmetic surface by a finite group action, where the inertia group of a point on a closed characteristic p fiber is a p-group acting with smallest possible ramification jump. In this paper, we give complete explicit resolutions of these singularities using deformation theory and valuation theory, taking a more local perspective than previous work has taken. Our descriptions answer several questions of Lorenzini. Along the way, we give a valuation-theoretic criterion for a normal snc-model of P^1 over a discretely valued field to be regular.