Is there a non-trivial class $S$ of smooth Deligne-Mumford stacks over a base $B$ with the property that if $\mathcal{X}, \mathcal{Y} \in S$ have isomorphic coarse moduli spaces (assumed to exist) then $\mathcal{X} \cong \mathcal{Y}$? If $B = Spec(k)$, $k$ a field (of characteristic $0$ if necessary), can one take $S$ to be the class of all irreducible, smooth, separated DM stacks with trivial inertia in codimension $\leq 1$?
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$\begingroup$ See the paper "Functorial reconstruction theorems for stacks" by Lieblich and Osserman (on arxiv, etc.). It's somewhat contrary to the whole point of introducing stacks to pose this kind of question, but anyway that paper does give affirmative result in some classes of examples. $\endgroup$– BCnrdCommented Apr 12, 2010 at 15:35
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$\begingroup$ @Brian I was aware of the paper of Lieblich and Osserman but it seemed to me that they address a somewhat different question i.e. they consider the functor associated to the stack rather than the coarse moduli space, and the former retains more information about the stack. $\endgroup$– nafCommented Apr 12, 2010 at 15:52
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$\begingroup$ @unknown: sorry, I misread the question, Mea culpa. Feel free to delete both my comment and yours if you wish. $\endgroup$– BCnrdCommented Apr 12, 2010 at 16:57
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2 Answers
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Yes, the class of all smooth, separated DM stacks over a field of characteristic $0$, with trivial inertia in codimension at most $1$, over a field of characteristic $0$, has the propery you want. The point is that every moduli space of such a stack has quotient singularities; and every variety with quotient singularities is the moduli space of a unique such stack. I believe that this was first proved in Angelo Vistoli: Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math. 97 (1989), no. 3, 613-670, Proposition 2.8 (uniqueness is not stated there, but it follows from the proof).
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The answer to the second question is No. For example the weighted projective stack $\mathbb{P}(1,2)$ and $\mathbb{P}^1$ have isomorphic coarse moduli spaces but are not isomorphic stacks. More generally, you can always ``add stack structure" in codimension one by using the root stack construction first developed by Cadman.
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2$\begingroup$ @mdeland In the second question I specified that there is no inertia in codimension 1 precisely in order to rule out such examples. $\endgroup$– nafCommented Apr 12, 2010 at 15:55
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$\begingroup$ Ah sorry, I also misread the question! $\endgroup$– mdelandCommented Apr 12, 2010 at 18:33