A note on toric Deligne-Mumford stacks

TL;DR

This paper provides a new framework for describing morphisms to toric Deligne-Mumford stacks using line bundles and sections, and explores their structure, actions, and morphisms in detail.

Contribution

It introduces a novel description of morphisms to toric Deligne-Mumford stacks via line bundles and sections, and compares different definitions of toric stacks.

Findings

Characterizes toric Deligne-Mumford stacks as products of roots of line bundles.

Describes the torus action explicitly.

Provides a new characterization of morphisms between such stacks.

Abstract

We give a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack. The description is given in terms of a collection of line bundles and sections which satisfy certain conditions. As applications, we characterize any toric Deligne-Mumford stack as a product of roots of line bundles over the rigidified stack, describe the torus action, describe morphisms between toric Deligne-Mumford stacks with complete coarse moduli spaces in terms of homogeneous polynomials, and compare two different definitions of toric stacks.