On the Complexity of Stacks

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

We prove that if is a smooth separated tame deligne-mumford stack of finite type over a field with trivial generic stabilizer, it is completely determined by its coarse space and the ramification divisor (on of the coarse space morphism).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.