Functorial resolution by torus actions

TL;DR

This paper introduces a new functorial resolution method for varieties using torus actions and cobordant blow-ups, achieving canonical resolutions with simple normal crossings divisors in characteristic zero and applications to singularities.

Contribution

It presents a novel, efficient resolution technique employing torus actions and cobordant blow-ups, extending to positive and mixed characteristic cases.

Findings

Achieves canonical functorial resolution in characteristic zero.

Produces a smooth variety with a torus action and SNC divisor.

Establishes resolution results for certain singularities in positive and mixed characteristic.

Abstract

We present a simple and fast embedded resolution of varieties and principalization of ideals using torus actions on ambient smooth varieties with simple normal crossings (SNC) divisors. The canonical functorial resolution in characteristic zero is achieved via the newly introduced cobordant blow-ups along smooth weighted centers. These centers are defined by a geometric invariant measuring the singularities on smooth schemes with SNC divisors. The output is a smooth variety with a torus action and an SNC exceptional divisor. Its geometric quotient is birational to the resolved variety, has only abelian quotient singularities, and can be desingularized by purely combinatorial methods. The method is rooted in ideas from the joint work with Abramovich and Temkin and is closely related to McQuillan's resolution via stack-theoretic weighted blow-ups. As an application, we establish resolution results for certain classes of singularities in positive and mixed characteristic. This paper is a shortened and revised version of an earlier preprint.