Smooth toric DM stacks

TL;DR

This paper introduces a new geometric definition of smooth toric Deligne-Mumford stacks, demonstrating its equivalence to existing definitions and providing a classification framework based on combinatorial and geometric data.

Contribution

It offers a novel geometric perspective on smooth toric DM stacks and establishes their equivalence with prior combinatorial definitions using stacky fans.

Findings

New geometric definition of smooth toric DM stacks

Equivalence with Borisov, Chen, and Smith's stacky fan approach

Classification via simplicial toric varieties and root stacks

Abstract

We give a new definition of smooth toric DM stacks in the same spirit of toric varieties. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.