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.

  1. 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?
  2. Is there some geometric description for one-dimensional DM 2-stacks (n-stacks)?
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    $\begingroup$ This doesn't seem to me to be true for one-dimensional stacks. Can't I take any 1-dimensional space, take any finite group, let the finite group act trivially and take the stack quotient? And then your assertion becomes "a finite group is determined by its order". What am I missing? Can you give a reference for this well-known fact? $\endgroup$ – user30035 Mar 25 '13 at 20:17
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    $\begingroup$ Yes, certainly I've missed some conditions. I think this fact about 1-dimensional DM-stacks will be true if we shall assume that general points have trivial stabilizer. Unfortunately, I can't provide a reference for this fact. $\endgroup$ – user32511 Mar 25 '13 at 21:17
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    $\begingroup$ You've edited the question since my previous comment but let me try to cause more trouble. Consider two lines meeting transversally at a point, and let the group of order 2 switch the lines. The coarse moduli space of the stack quotient is the line, with one point having a group of order 2 as automorphisms. Now consider the group of order 2 acting on the affine line with the non-trivial element acting as -1. The coarse moduli space is again the line with one point having automorphism group of order 2. But those stacks are not isomorphic, if my understanding is correct (one is smooth andoneisnt $\endgroup$ – user30035 Mar 25 '13 at 22:28
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    $\begingroup$ Assuming that previous counterexample is OK, here's a suggestion. Instead of just asserting that something that sort-of looks true in your world view is "well-known", why not actually try and find, or write down, a proof yourself? Then you'll see what actually is true, and then you'll probably be able to ask a much better question. $\endgroup$ – user30035 Mar 25 '13 at 22:31
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    $\begingroup$ wccanard: Thanks. I forgot about some conditions, because my question was not about the one-dimensional case and also rather general. I asked if there is something in this style... So, certainly, we need to add a smoothness condition. And now, I think, that's it! I think the proof is straightforward. Use the fact that smooth DM stacks locally (in the etale topology) are quotients of a smooth scheme by a finite group. $\endgroup$ – user32511 Mar 25 '13 at 23:53

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:

  1. They have the same order.

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

  3. 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 $G(m,p,2)$.

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    $\begingroup$ The assumption is not true. For example, the exceptional group of order 144 listed as number 7 gives three different lines with non-trivial stabilizers on the target, and the exceptional group of order 144 listed as number 14 gives only two different lines with non-trivial stabilizers on the target. Moreover, I'm absolutely sure that there is no counterexamples in dimension 2 constructed in such way. $\endgroup$ – user32511 Apr 7 '13 at 15:39
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    $\begingroup$ By the way, I don't understand, why the étale local picture near a stacky point need to be an affine space, quotient by a finite group G. Could you explain it? Moreover, for the using of Chevalley-Shepard-Todd theorem we need to assume that the action is linearizable that is not true in general. So, I think there still could be some ways to construct a counterexample in the dimention 2. $\endgroup$ – user32511 Apr 7 '13 at 15:57
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    $\begingroup$ Okay, thanks for letting me know about the two groups. Can you also compare them with the quotient by $G(24,8,2)$? Regarding the étale local picture, if you choose an étale neighborhood of your stacky point that is a smooth scheme, then by EGA IV 17.11.4 it is étale locally isomorphic to affine space. I haven't thought through the reduction to Chevalley-Shepard-Todd in full detail, but I figured that passing to the induced action on the tangent space of a point would not lose substantial information in characteristic zero. $\endgroup$ – S. Carnahan Apr 8 '13 at 0:29

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