Resolving plane curves using stack-theoretic blow-ups

TL;DR

This paper advances the use of stack-theoretic blow-ups for resolving plane curve singularities in positive characteristic, addressing challenges related to tangent space dimension increases with a novel multi-weighted blow-up approach.

Contribution

It introduces a canonical multi-weighted blow-up method to resolve curve singularities in positive characteristic, overcoming inductive obstacles present in previous techniques.

Findings

Successfully resolves certain curve singularities in positive characteristic.

Provides a canonical construction for multi-weighted blow-ups.

Addresses tangent space dimension issues in stack-theoretic resolutions.

Abstract

Stack-theoretic blow-ups have proven to be efficient in resolving singularities over fields of characteristic zero. In this article, we move forward towards positive characteristic where new challenges arise. In particular, the dimension of the tangent space of the Artin stack created after a weighted blow-up may increase, which makes it hard to apply inductive arguments -- even if the maximal order decreases. We focus on the case of curve singularities embedded into a smooth surface defined over a perfect field. For this special situation we propose a solution to overcome the inductive challenge through canonically constructed multi-weighted blow-ups.