For $\mathcal{M}_g$ ($g \geq 2$), the inertia is finite (automorphism groups are finite by the Hurwitz bound). Keel--Mori gives the coarse moduli space $M_g$, a quasi-projective variety of dimension $3g - 3$. The map $\pi : \mathcal{M}_g \to M_g$ is proper and quasi-finite, and an isomorphism over the open locus of curves without extra automorphisms.

For $g = 2$: every genus-2 curve has the hyperelliptic involution, so $\pi$ is nowhere an isomorphism. The coarse space $M_2 \cong \mathbb{A}^3/S_6$ (identified via the six branch points of the hyperelliptic map, modulo the $S_6$-action by relabeling).

For $\mathrm{B}G = [\operatorname{Spec} k/G]$ with $G$ finite, the coarse space is $\operatorname{Spec} k$. The map $\pi : \mathrm{B}G \to \operatorname{Spec} k$ sends every $G$-torsor to the unique $k$-point. The pushforward $\pi_* : \operatorname{Rep}(G) \to \operatorname{Vect}_k$ is the functor $V \mapsto V^G$ (taking $G$-invariants).

In characteristic zero, this is exact (Maschke's theorem: $V^G$ is a direct summand of $V$). In characteristic $p$ dividing $|G|$, the functor $V \mapsto V^G$ is only left-exact, and $R^i\pi_*$ can be nonzero.

For $G = \mu_n$ acting on $\mathbb{A}^2$ (type $A_{n-1}$), Keel--Mori gives $[\mathbb{A}^2/\mu_n] \to \mathbb{A}^2/\mu_n$. The coarse space has the $A_{n-1}$ singularity at the origin, while the stack is smooth. The map $\pi$ is a bijection on geometric points (both have the same underlying set of orbits), but $\pi$ is not an isomorphism at the origin.

More generally, for $G \subset \mathrm{SL}(2, k)$ a finite subgroup (giving ADE singularities), the coarse space $\mathbb{A}^2/G$ has the corresponding surface singularity, and $[\mathbb{A}^2/G]$ is the "canonical stacky resolution."

For $\mathcal{P}(a_0, \ldots, a_n)$ with $\gcd(a_0, \ldots, a_n) = 1$, the stabilizers are cyclic (hence finite), and Keel--Mori gives the coarse space $\mathbb{P}(a_0, \ldots, a_n)$ (the classical weighted projective variety).

For $\mathcal{P}(1, 1, 2)$: the coarse space is $\mathbb{P}(1,1,2) \cong V(xy - z^2) \subset \mathbb{P}^3$ (the quadric cone). The Keel--Mori map $\pi$ is an isomorphism away from $[0:0:1]$ (where the $\mu_2$-stabilizer lives) and contracts the $\mu_2$-gerbe at $[0:0:1]$ to the singular point of the cone.

For $\mathcal{M}_{1,1}$, the inertia is finite (stabilizers are $\mu_2$, $\mu_4$, or $\mu_6$). Keel--Mori gives $M_{1,1} = \mathbb{A}^1_j$ (the $j$-line). The map $\pi$ satisfies:

• $\pi^{-1}(j) \cong \mathrm{B}\operatorname{Aut}(E_j)$ for each $j$-value (the fiber is the residual gerbe).
• $\pi_*\mathcal{O}_{\mathcal{M}_{1,1}} = \mathcal{O}_{\mathbb{A}^1_j}$.
• The universal elliptic curve exists on the stack $\mathcal{M}_{1,1}$ but not on $\mathbb{A}^1_j$.

For $n \geq 3$, $\mathcal{M}_{0,n}$ has trivial automorphism groups (a rational curve with $\geq 3$ marked points is rigid), so the coarse moduli map is an isomorphism: $\mathcal{M}_{0,n} \cong M_{0,n}$. The compactification $\overline{\mathcal{M}}_{0,n} \cong \overline{M}_{0,n}$ is a smooth projective variety, explicitly described as an iterated blowup of $(\mathbb{P}^1)^{n-3}$.

For $n = 4$: $\overline{M}_{0,4} \cong \mathbb{P}^1$ (the cross-ratio). For $n = 5$: $\overline{M}_{0,5} \cong \operatorname{Bl}_4 \mathbb{P}^2$ (the del Pezzo surface of degree 5).

For the root stack $\sqrt[r]{D/X}$ with $D$ a smooth divisor in a smooth variety $X$, the stabilizers along $D$ are $\mu_r$ (finite). Keel--Mori gives the coarse space $X$ back: $\sqrt[r]{D/X} \xrightarrow{\pi} X.$ The map $\pi$ is an isomorphism over $X \setminus D$ and a $\mu_r$-gerbe along $D$.

The stack $\mathcal{A}_g$ has finite stabilizers (the automorphism group of a ppav is finite). Keel--Mori gives the coarse space $A_g$, which over $\mathbb{C}$ is the Siegel modular variety $\mathrm{Sp}(2g, \mathbb{Z}) \backslash \mathfrak{H}_g$.

For $g = 1$: $A_1 \cong \mathbb{A}^1_j$. For $g = 2$: $A_2$ is a normal quasi-projective variety of dimension 3, with the Igusa compactification $A_2^* = A_2 \cup \{0\text{-dim cusp}\}$ being projective.

Consider $\mathcal{X} = [(\mathbb{A}^1 \sqcup \mathbb{A}^1) / \mathbb{Z}_2]$, where $\mathbb{Z}_2$ swaps the two copies of $\mathbb{A}^1 \setminus \{0\}$ but fixes both origins. The coarse space $M$ is the "line with doubled origin" -- a non-separated algebraic space. This happens because $\mathcal{X}$ itself is not separated.

The stack $\mathrm{B}\mathbb{G}_m$ has stabilizer $\mathbb{G}_m$ at its unique point -- this is not finite, so Keel--Mori does not apply. Indeed, there is no coarse moduli space: the universal property would require that every line bundle on every scheme $T$ is pulled back from a map $T \to \operatorname{Spec} k$. But $\operatorname{Pic}(T)$ can be nontrivial (e.g., $T = \mathbb{P}^1$), contradicting the pullback requirement.

The "good moduli space" theory of Alper extends the Keel--Mori theorem to certain Artin stacks with non-finite stabilizers, replacing "coarse moduli space" with "good moduli space."

A DM stack $\mathcal{X}$ is tame if the stabilizer groups at all geometric points have order invertible in the base. The Keel--Mori theorem for tame stacks has the additional property that $\pi_* : \operatorname{Coh}(\mathcal{X}) \to \operatorname{Coh}(M)$ is exact.

For example, over $\operatorname{Spec} \mathbb{Z}[1/n]$, any DM stack with stabilizers of order dividing $n$ is tame. The stack $\mathcal{M}_{1,1}$ over $\operatorname{Spec} \mathbb{Z}[1/6]$ is tame (stabilizers have order dividing 6).

In the wild case (characteristic divides the order of stabilizers), the Keel--Mori theorem still gives a coarse space, but the pushforward $\pi_*$ may fail to be exact.

If $\mathcal{X}$ is a DM stack with all stabilizers isomorphic to a fixed finite group $G$, then $\pi : \mathcal{X} \to M$ makes $\mathcal{X}$ into a $G$-gerbe over $M$ (in the etale topology). The gerbe class $[\mathcal{X}] \in H^2_{\mathrm{et}}(M, G)$ is the obstruction to $\mathcal{X}$ having a section (i.e., to $\mathcal{X}$ being a "trivial" gerbe).

For $\mathcal{M}_{1,1} \to \mathbb{A}^1_j$: generically this is a $\mu_2$-gerbe (the generic stabilizer is $\mu_2$). The gerbe class in $H^2_{\mathrm{et}}(\mathbb{A}^1_j \setminus \{0, 1728\}, \mu_2) \cong \operatorname{Br}(\mathbb{A}^1_j \setminus \{0, 1728\})[2]$ is nontrivial -- there is no universal elliptic curve over the $j$-line.