Functorial destackification of tame stacks with abelian stabilisers

TL;DR

This paper presents a functorial algorithm for removing stackiness from tame Artin stacks with abelian stabilizers, enabling destackification and desingularization in a broad algebraic geometry context.

Contribution

It introduces a new functorial destackification algorithm applicable over general bases, extending to Deligne-Mumford stacks and varieties with toric quotient singularities.

Findings

Algorithm effectively removes stackiness via stacky blow-ups.

Applicable over arbitrary fields without toroidal assumptions.

Enables weak factorization for certain algebraic stacks.

Abstract

We give an algorithm for removing stackiness from smooth, tame Artin stacks with abelian stabilisers by repeatedly applying stacky blow-ups. The construction works over a general base and is functorial with respect to base change and compositions with gerbes and smooth, stabiliser preserving maps. As applications, we indicate how the result can be used for destackifying general Deligne-Mumford stacks in characteristic zero, and to obtain a weak factorisation theorem for such stacks. Over an arbitrary field, the method can be used to obtain a functorial algorithm for desingularising varieties with simplicial toric quotient singularities, without assuming the presence of a toroidal structure.