The moduli stack $\mathcal{M}_g$ ($g \geq 2$) of smooth projective curves of genus $g$ is a smooth, separated DM stack of dimension $3g - 3$ over $\operatorname{Spec} \mathbb{Z}$.

Why DM: For $g \geq 2$, $\operatorname{Aut}(C)$ is a finite group for every smooth curve $C$ of genus $g$. By the Hurwitz bound, $|\operatorname{Aut}(C)| \leq 84(g-1)$.

Atlas construction: Choose $n$ large enough so that $\omega_C^{\otimes n}$ is very ample. This embeds $C \hookrightarrow \mathbb{P}^N$ (where $N = n(2g-2) - g$). The Hilbert scheme $\mathcal{H}$ parametrizing such embedded curves with marking provides a scheme with a natural map $\mathcal{H} \to \mathcal{M}_g$. The group $\mathrm{PGL}_{N+1}$ acts on $\mathcal{H}$ and $\mathcal{M}_g \simeq [\mathcal{H} / \mathrm{PGL}_{N+1}]$ -- but this presentation is as an Artin stack. The DM property ensures we can find an etale atlas.

$\overline{\mathcal{M}}_g$ ($g \geq 2$) is the moduli stack of stable curves: connected, projective, at-worst-nodal curves of arithmetic genus $g$ with finite automorphism group. It is a proper, smooth DM stack.

Deligne and Mumford proved that the coarse moduli space $\overline{M}_g$ is a projective variety, using the fact that $\overline{\mathcal{M}}_g$ is proper and the Keel--Mori theorem (existence of coarse moduli spaces for separated DM stacks with finite inertia).

The boundary $\overline{\mathcal{M}}_g \setminus \mathcal{M}_g$ is a divisor with $\lfloor g/2 \rfloor$ irreducible components $\Delta_0, \Delta_1, \ldots, \Delta_{\lfloor g/2 \rfloor}$, where $\Delta_i$ parametrizes curves with a node separating components of genera $i$ and $g - i$ (with $\Delta_0$ for non-separating nodes).

The stack $\mathcal{M}_{g,n}$ of smooth $n$-pointed genus-$g$ curves (where $2g - 2 + n > 0$) is a smooth DM stack of dimension $3g - 3 + n$. The stability condition ensures finite automorphism groups.

Key cases:

• $\mathcal{M}_{0,3} \cong \operatorname{Spec} k$ (three points on $\mathbb{P}^1$ are rigid).
• $\mathcal{M}_{0,4}$ is an algebraic space isomorphic to $\mathbb{P}^1 \setminus \{0, 1, \infty\}$ via the cross-ratio.
• $\overline{\mathcal{M}}_{0,n}$ is actually a smooth projective variety (not just a DM stack) for $n \geq 3$, since pointed rational curves have no nontrivial automorphisms.
• $\mathcal{M}_{1,1}$ is a DM stack but not a scheme -- the curve $y^2 = x^3 - x$ (with $j = 1728$) has $\operatorname{Aut} \cong \mathbb{Z}/4\mathbb{Z}$ and $y^2 = x^3 - 1$ (with $j = 0$) has $\operatorname{Aut} \cong \mathbb{Z}/6\mathbb{Z}$.

The stack $\mathcal{A}_g$ of principally polarized abelian varieties of dimension $g$ is a separated DM stack of dimension $g(g+1)/2$. The automorphism group of a general principally polarized abelian variety is $\{\pm 1\}$ (finite), confirming the DM property.

Over $\mathbb{C}$, the coarse moduli space is $A_g = \Gamma_g \backslash \mathfrak{H}_g$, where $\mathfrak{H}_g$ is the Siegel upper half-space and $\Gamma_g = \mathrm{Sp}(2g, \mathbb{Z})$. The Torelli map $\mathcal{M}_g \to \mathcal{A}_g$ sends a curve $C$ to its Jacobian $(\mathrm{Jac}(C), \Theta)$.

The stack $\mathcal{M}_{1,1}$ of elliptic curves (genus-1 curves with a marked point) is a smooth DM stack of dimension 1 over $\operatorname{Spec} \mathbb{Z}[1/6]$. The $j$-invariant gives a map $j : \mathcal{M}_{1,1} \to \mathbb{A}^1$, which is the coarse moduli map.

The stacky structure is concentrated at two points:

• $j = 1728$: $E : y^2 = x^3 - x$, $\operatorname{Aut}(E) \cong \mu_4$ (generated by $(x,y) \mapsto (-x, iy)$).
• $j = 0$: $E : y^2 = x^3 - 1$, $\operatorname{Aut}(E) \cong \mu_6$ (generated by $(x, y) \mapsto (\zeta_3 x, -y)$).
• All other $j$: $\operatorname{Aut}(E) \cong \mu_2 = \{\pm 1\}$ (the elliptic involution).

Over $\operatorname{Spec} \mathbb{Z}[1/6]$, we have $\mathcal{M}_{1,1} \cong [\operatorname{Spec} \mathbb{Z}[1/6][a_4, a_6, \Delta^{-1}] / \mathbb{G}_m]$, where the Weierstrass equation is $y^2 = x^3 + a_4 x + a_6$ with $\Delta = -16(4a_4^3 + 27a_6^2)$, and $\mathbb{G}_m$ acts by $\lambda \cdot (a_4, a_6) = (\lambda^4 a_4, \lambda^6 a_6)$. This is actually an Artin stack presentation; the DM structure comes from the fact that stabilizers are always finite subgroups of $\mathbb{G}_m$.

Given a scheme $X$, a line bundle $\mathcal{L}$, a section $s \in \Gamma(X, \mathcal{L})$, and an integer $r \geq 1$, the $r$-th root stack $\sqrt[r]{(X, \mathcal{L}, s)}$ is a DM stack. Over the locus where $s \neq 0$, the root stack is isomorphic to $X$. Over the zero locus $D = V(s)$, points acquire $\mu_r$-automorphisms.

For example, $\sqrt[r]{\mathcal{O}_{\mathbb{P}^1}(p)/\mathbb{P}^1}$ (the $r$-th root at the point $p$) is a DM stack that looks like $\mathbb{P}^1$ away from $p$ and has a $\mu_r$-gerbe at $p$.

A toric DM stack is a DM stack $\mathcal{X}$ with a dense open torus $T \subset \mathcal{X}$ and a $T$-action extending the multiplication on $T$. These are classified by stacky fans $(N, \Sigma, \beta)$, where $N$ is a finitely generated abelian group, $\Sigma$ is a fan in $N_{\mathbb{R}}$, and $\beta : \mathbb{Z}^n \to N$ encodes the stacky structure.

For instance, the weighted projective stack $\mathcal{P}(1,1,2)$ is a toric DM stack. Its stacky fan is obtained from the fan of $\mathbb{P}^2$ by modifying the ray corresponding to the weight-2 coordinate.

Given a DM stack $\mathcal{X}$ with a central subgroup $G \subset \operatorname{Aut}(\xi)$ for every object $\xi$ (a "common automorphism"), the rigidification $\mathcal{X} \!\sslash\! G$ is the DM stack obtained by removing these automorphisms.

For example, $\mathcal{M}_{1,1} \!\sslash\! \mu_2$ is an algebraic space (the $j$-line $\mathbb{A}^1$) -- rigidifying by the elliptic involution removes the universal $\mu_2$-automorphism, converting the stack into its coarse space.

The Hurwitz stack $\mathcal{H}_{d,g}$ parametrizes degree-$d$ covers $f : C \to \mathbb{P}^1$ where $C$ has genus $g$, with simple branching. This is a smooth DM stack (the automorphism group of a general cover is trivial).

The Hurwitz stack connects to $\mathcal{M}_g$ via the source map $\mathcal{H}_{d,g} \to \mathcal{M}_g$ (sending $f$ to $C$). Hurwitz used the monodromy representation and the connectivity of $\mathcal{H}_{d,g}$ (proved using the transitivity of the braid group action on factorizations in $S_d$) to show that $\mathcal{M}_g$ is connected.

The moduli stack of K3 surfaces is not DM: a general K3 surface $X$ has $\operatorname{Aut}(X) \supseteq \{\pm \mathrm{id}\}$ on $H^{2,0}(X)$, but this is finite (so one might expect DM). However, the period map shows that the moduli problem is better understood through the coarse moduli space $\mathcal{F}_d = \Gamma_d \backslash \Omega_d$ (an arithmetic quotient of a type IV domain). The full moduli stack has subtle issues with non-reduced automorphism schemes in positive characteristic and is typically studied via marked or polarized variants that are DM.

Let $X$ be a scheme and $G$ a finite group acting on $X$. Then $[X/G]$ is a DM stack. The etale atlas is $X \to [X/G]$, and the groupoid is $(X, G \times X)$ with source $s(g, x) = x$ and target $t(g, x) = g \cdot x$. Both $s$ and $t$ are etale since $G$ is etale.

For $X = \mathbb{A}^2$ with $G = \mathbb{Z}/n\mathbb{Z}$ acting by $(x, y) \mapsto (\zeta x, \zeta^{-1} y)$ (where $\zeta$ is a primitive $n$-th root of unity), $[X/G]$ is a smooth DM stack. The coarse moduli space is $\operatorname{Spec} k[x,y]^G = \operatorname{Spec} k[u, v, w]/(uv - w^n)$, the $A_{n-1}$ surface singularity. The stack $[X/G]$ is the "stacky resolution" of this singularity.

The moduli stack $\overline{\mathcal{M}}_{g,n}(X, \beta)$ of stable maps to a projective variety $X$ is a proper DM stack (assuming $2g - 2 + n > 0$ or $\beta \neq 0$). The key point is that stability (no infinitesimal automorphisms) ensures the DM property. The automorphism group of a stable map $(C, p_1, \ldots, p_n, f)$ is the finite group of automorphisms of $C$ commuting with $f$ and preserving the markings.

When $X = \operatorname{Spec} k$ and $\beta = 0$, this reduces to $\overline{\mathcal{M}}_{g,n}$.