# Fibers of the coarse moduli space map

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?

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.