Torus Quotients as Global Quotients by Finite Groups

Details Torus Quotients as Global Quotients by Finite Groups Torus Quotients as Global Quotients by Finite Groups

2012-01-23

arxiv.org/abs/1201.4807v1

Mathematics

Algebraic Geometry

Scientific paper

This article concerns 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. We end the paper by giving a characterization of varieties which are expressible as the quotient of a smooth variety by a split torus. As an application, we show that the following variant of the above local-to-global question is false: is every variety with tame abelian quotient singularities globally the 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.

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