A "bottom up" characterization of smooth Deligne-Mumford stacks

TL;DR

This paper provides a bottom-up approach to characterize smooth Deligne-Mumford stacks using their coarse space and ramification divisor, showing they can be reconstructed via canonical and root stack procedures.

Contribution

It establishes that smooth tame Deligne-Mumford stacks with trivial generic stabilizer are uniquely determined by their coarse space and ramification divisor, and can be constructed through iterative canonical and root stack operations.

Findings

Stacks are characterized by coarse space and ramification divisor.

Stacks can be reconstructed using canonical and root stack procedures.

The stabilizer groups are determined by codimension 1 data.

Abstract

In casual discussion, a stack is often described as a variety (the coarse space) together with stabilizer groups attached to some of its subvarieties. However, this description does not uniquely specify the stack. Our main result shows that for a large class of stacks one typically encounters, this description does indeed characterize them. Moreover, we prove that each such stack can be described in terms of two simple procedures applied iteratively to its coarse space: canonical stack constructions and root stack constructions. More precisely, if $\mathcal X$ is a smooth separated tame Deligne-Mumford stack of finite type over a field $k$ with trivial generic stabilizer, it is completely determined by its coarse space $X$ and the ramification divisor (on $X$) of the coarse space morphism $\pi\colon \mathcal X \to X$. Therefore, to specify such a stack, it is enough to specify a variety and the orders of the stabilizers of codimension 1 points. The group structures, as well as the stabilizer groups of higher codimension points, are then determined.