Torus Quotients as Global Quotients by Finite Groups

TL;DR

This paper investigates when varieties with tame quotient singularities can be globally expressed as quotients of smooth varieties by finite groups, providing explicit constructions for certain cases and counterexamples for others.

Contribution

It establishes that varieties as quotients of smooth varieties by split tori are globally quotients by finite groups and offers explicit toric constructions, also identifying limitations with abelian quotient singularities.

Findings

Quasi-projective varieties as quotients of smooth varieties by split tori are globally quotients by finite groups.

Explicit toric techniques are provided for constructing quotient structures.

Some varieties with abelian quotient singularities are not quotients of smooth varieties by finite abelian groups.

Abstract

This article is motivated by the following local-to-global question: is every variety with tame quotient singularities globally the quotient of a smooth variety by a finite group? We show that this question has a positive answer for all quasi-projective varieties which are expressible as a quotient of a smooth variety by a split torus (e.g. simplicial toric varieties). Although simplicial toric varieties are rarely toric quotients of smooth varieties by finite groups, we give an explicit procedure for constructing the quotient structure using toric techniques. This result follow from a characterization of varieties which are expressible as the quotient of a smooth variety by a split torus. As an additional application of this characterization, we show that a variety with abelian quotient singularities may fail to be a quotient of a smooth variety by a finite abelian group. Concretely, we show that $\mathbb{P}^2/A_5$ is not expressible as a quotient of a smooth variety by a finite abelian group.