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 $\leq 1$?
[Math] Stacks determined by their coarse moduli spaces
ag.algebraic-geometry
Related Solutions
An example is Deligne's theorem on the existence of good notion of quotient $X/G$ of a separated algebraic space $X$ under the action of a finite group $G$, or relativizations or generalizations (with non-constant $G$) due to D. Rydh. See Theorem 3.1.13 of my paper with Lieblich and Olsson on Nagata compactification for algebraic spaces for the statement and proof of Deligne's result in a relative situation, and Theorem 5.4 of Rydh's paper "Existence of quotients..." on arxiv or his webpage for his generalization.
Note that in the above, there is no mention of DM stacks, but they come up in the proof! The mechanism to construct $X/G$ (say in the Deligne situation or its relative form) is to prove existence of a coarse space for the DM stack $[X/G]$ via Keel-Mori and show it has many good properties to make it a reasonable notion of quotient. Such quotients $X/G$ are very useful when $X$ is a scheme (but $X/G$ is "only" an algebraic space), such as for reducing some problems for normal noetherian algebraic spaces to the scheme case; cf. section 2.3 of the C-L-O paper. I'm sure there are numerous places where coarse spaces are convenient to do some other kinds of reduction steps in proofs of general theorems, such as reducing a problem for certain DM stacks to the case of algebraic spaces.
Also, Mazur used a deep study of the coarse moduli scheme associated to the DM stack $X_0(p)$ in his pioneering study of torsion in and rational isogenies between elliptic curves over $\mathbf{Q}$ (and these modular curves show up in numerous other places). But those specific coarse spaces are schemes and can be constructed and studied in more concrete terms without needing the fact that they are coarse spaces in the strong sense of the Keel-Mori theorem, so I think the example of Deligne's theorem above is a "better" example.
In the smooth case, I think that the answer is positive over an arbitrary regular excellent base. The argument was in my PhD thesis; it was done over a field, but I think it adapts to this case.
Cover your $X$ in the étale topology with schemes of the form $U/G$, where $G$ is prime to all the residue characteristics of $U$, and $U$ is smooth over $S$. Choose a geometric point $p$ of $U$; we can restrict to the stabilizer of $G$, and assume that $G$ leaves $p$ fixed. Call $H$ the subgroup generated by the elements of $G$ that are pseudoreflections, or the identity, when restricted to the fiber along $p$; this is normal in $G$. The scheme $U/H$ is flat over $S$, because of the tameness hypothesis. Furthermore taking quotients by $H$ commutes with base change on $S$, again because of tameness; hence the geometric fiber of $U/H \to S$ along the image of $p$ is the quotient of the geometric fiber, which is smooth, because of Cartan’s and Chevalley’s theorems. In this way we can assume that $G$ stabilizes $p$, and the restriction of $G$ to the fiber through $p$ contains no pseudoreflexions.
Now look at the locus in which $U \to X$ is étale. Notice that if the restriction of $U \to X$ to the fiber is étale at $p$, then $U \to X$ is étale at $p$, by the local criterion of flatness. The locus on which $U \to X$ is étale is open in X; its complement must have codimension larger than $1$, because otherwise it would intersect the fiber in codimension $1$.
Now take two of these charts $U \to X$ and $V \to X$; the normalization of the part of the fibered product $U \times_X V$ that dominates $X$ is étale over $U$ and $V$, by purity of the branch locus. These data give a Q-variety, in the sense of Mumford; from them you get an étale groupoid that defines the stack that you are looking for.
When S is not regular, I am really not sure, I suspect it might be false.
Best Answer
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).