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Let $\mathcal{X}$ be a Deligne-Mumford stack over a field $k$ which admits a coarse scheme $c : \mathcal{X}\rightarrow X$. This will be the case if $\mathcal{X}$ is separated and locally of finite presentation, and I am happy to assume this.

What do the fibers of $c$ look like?

For example, if $k$ contains a primitive $n$th root of unity $\zeta$, $D := \text{Spec }k[[t]]$, $G$ is a cyclic group of order $n$ acting by $t\mapsto \zeta t$, and $\mathcal{X} := [D/G]$ (stack quotient), then $X = D/G\cong \text{Spec }k[[t^n]]$ (the scheme quotient). In this case, if $x :\text{Spec }k\rightarrow D/G$ is the closed point, then it is easy to read off from the categorial definition of the fiber product that $c^{-1}(x) = [D/G]\times_{D/G} x$ is a Deligne-Mumford stack with a single point with automorphism group $G$. The first expectation is that $c^{-1}(x) \cong [x/G]$, where $G$ acts on $x$ trivially, but I don't think this is correct since $$\text{Hom}(\text{Spec }k[\epsilon],D) = \text{Hom}(\text{Spec }k[\epsilon],[D/G]) = \text{Hom}(\text{Spec }k[\epsilon],c^{-1}(x))$$ is 1-dimensional. This would lead me to guess that $c^{-1}(x) \cong [(\text{Spec }k[t]/t^n) / G]$, but I don't have a proof. Is this second guess correct?

In general is there a good way to understand the fibers of the coarse map $c$? Maybe there is a good structure theory when $\mathcal{X}$ is 1-dimensional?

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You're guess is correct. Here's an approach: For any stack and any global section of the structure sheaf, we can define the vanishing set in the obvious way: It's $R$-points are the $R$-points of the stack where that function is sent to $0\in R$.

This definition is so natural that it's easy to see that the inverse image of the vanishing set of $f$ is the vanishing set of the inverse image of $f$, and the vanishing set of an invariant function $f$ in the quotient is the quotient of the vanishing set of $f$.

Clearly the closed point of $X$ is the vanishing set of the uniformizer $t^n$, so the inverse image of the closed point is the vanishing set of $t^n$ in $\mathcal X$, which is the quotient of the vanishing set of $t^n$ in $D$ by $G$, which is exactly the formula you give.

I don't know about a formula for general 1-dimensional stacks, but certainly this idea can be generalized to arbitrary quotients of schemes by finite groups. The inverse image of a closed point will be the quotient of the vanishing set of pullbacks of the generators of the ideal of that point.

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