Geometric description of the Deligne-Mumford stacks It is well known that a one-dimensional smooth Deligne-Mumford stack (over $\mathbb{C}$) could be described as a collection of its "stacky" points (finitely many) on its coarse moduli space with the orders of their stabilizers. As wccanard noted in the comments, here we need to assume that all but finitely many points of the stack have the trivial automorphisms group.


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*Is there an analogous description for two-dimensional DM stacks? For $\dim > 2$ DM stacks? Maybe for stacks with some additional restrictions (for example, assume that its coarse moduli is a smooth scheme)? By analogous description I mean a collection of closed subschemes on its coarse moduli space with the orders of their stabilizers. Is there an example of two non-isomorphic DM-stacks for which such descriptions coincide?

*Is there some geometric description for one-dimensional DM 2-stacks (n-stacks)?

 A: For question 2, I've never seen a definition of DM 2-stack, so I don't know where to start.  For question 1, I have a tentative counterexample even for the case where both the stack and the coarse moduli space are smooth.
The étale local picture near a stacky point is that we take an affine space, quotient by a finite group $G$, and take the coarse moduli space to get an affine space.  By the Chevalley-Shepard-Todd theorem, if you want your coarse moduli space to be smooth, it is necessary and sufficient that $G$ be a complex reflection group, acting by complex reflections.  In order to find a pair of suitable non-isomorphic stacks for which labelings of the coarse space by orders of groups does not distinguish them, we need to find two complex reflection groups satisfying the following conditions:


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*They have the same order.

*The orders of stabilizers of affine subspaces of a given dimension form identical sets of positive integers.

*There is an isomorphism of the coarse moduli spaces taking the images of suitably labeled subspaces to each other.
The relevant information is in the big table at the Wikipedia page on complex reflection groups.
I will confine myself to rank 2 groups, because then the nontrivial subspace stabilizers are precisely the reflections.  For some reason, I'm unable to think clearly right now (this may have something to do with the awkward wording of condition 3), so I'm going to make an assumption about transitivity that might not be true.
Assumption: For each reflection order in an irreducible complex reflection group $G$, all reflection hyperplanes of that order in the source affine space are mapped to a single hyperplane in the target.  Equivalently, irreducible complex reflection groups act transitively by conjugation on the quotient of the set of reflections of any fixed order by the corresponding hyperplane stabilizers.
If the assumption is true, we only need to find two rank 2 complex reflection groups of the same order, whose reflections have the same order.  For example, the exceptional groups of order 144 listed as numbers 7 and 14 in the Wikipedia table work, as well as several other exceptional groups matched with the semidirect products $G(m,p,2)$.
