Let $G$ be a smooth algebraic group over $S$. The classifying stack $\mathrm{B}G = [\operatorname{Spec} k / G]$ is the stack whose objects over a scheme $T$ are $G$-torsors (principal $G$-bundles) $P \to T$. A morphism is an isomorphism of $G$-torsors.

The smooth atlas is $\operatorname{Spec} k \to \mathrm{B}G$ (the trivial torsor), which is smooth and surjective. The fiber product $\operatorname{Spec} k \times_{\mathrm{B}G} \operatorname{Spec} k \cong G$, so the groupoid is $({\operatorname{Spec} k}, G)$.

For $G = \mathbb{G}_m$, $\mathrm{B}\mathbb{G}_m$ classifies line bundles. For $G = \mathrm{GL}_n$, $\mathrm{B}\mathrm{GL}_n$ classifies rank-$n$ vector bundles.

Since $\operatorname{Aut}(P) = G$ for any $G$-torsor $P$, the automorphism groups are $G$ itself -- this is positive-dimensional for nontrivial $G$, so $\mathrm{B}G$ is an Artin stack but not DM (unless $G$ is etale).

The stack $\mathcal{M}_g$ of smooth projective curves of genus $g \geq 2$ is a Deligne--Mumford stack. A smooth atlas is provided by a sufficiently fine Hilbert scheme parametrizing embedded curves. The automorphism group of a genus $g \geq 2$ curve is finite, which is why $\mathcal{M}_g$ is DM rather than merely Artin.

For $g = 1$, the stack $\mathcal{M}_1$ of elliptic curves (without marked point) has automorphism groups containing $\{\pm 1\}$ (and larger groups for $j = 0$ and $j = 1728$), but these are still finite, so $\mathcal{M}_1$ remains DM. However, $\mathcal{M}_{1,1}$ (with a marked point) is a cleaner DM stack.

Let $G$ be a smooth algebraic group acting on a scheme $X$. The quotient stack $[X/G]$ is an algebraic (Artin) stack. The atlas is $X \to [X/G]$, which is a $G$-torsor (hence smooth). The groupoid is $(X, G \times X)$ with $s(g,x) = x$ and $t(g,x) = g \cdot x$.

If the action has finite stabilizers, $[X/G]$ is DM. If stabilizers are positive-dimensional (e.g., $\mathbb{G}_m$ acting on $\mathbb{A}^1$ with the origin as a fixed point), $[X/G]$ is Artin but not DM.

Let $C$ be a smooth projective curve of genus $g \geq 2$. The stack $\operatorname{Bun}_n(C)$ of rank-$n$ vector bundles on $C$ is an algebraic (Artin) stack, smooth of dimension $n^2(g-1)$. Every vector bundle $E$ has $\operatorname{Aut}(E) \supseteq \mathbb{G}_m$ (scalar automorphisms), so this cannot be DM.

For $n = 1$, $\operatorname{Bun}_1(C) \cong \operatorname{Pic}(C) \times \mathrm{B}\mathbb{G}_m$ decomposes as the Picard scheme times the classifying stack. For higher rank, the structure is much richer: the Harder--Narasimhan stratification provides a filtration by locally closed substacks.

The weighted projective stack $\mathcal{P}(a_0, \ldots, a_n)$ is defined as the quotient stack $\mathcal{P}(a_0, \ldots, a_n) = [(\mathbb{A}^{n+1} \setminus \{0\}) / \mathbb{G}_m]$ where $\mathbb{G}_m$ acts by $t \cdot (x_0, \ldots, x_n) = (t^{a_0} x_0, \ldots, t^{a_n} x_n)$ with positive integer weights $a_i$.

If all $a_i = 1$, this is $\mathbb{P}^n$ (a scheme). If the $a_i$ are not all equal, points with nontrivial stabilizers appear, and $\mathcal{P}(a_0, \ldots, a_n)$ is a DM stack (the stabilizers are cyclic, hence finite). The coarse moduli space is the classical weighted projective variety $\mathbb{P}(a_0, \ldots, a_n)$.

For example, $\mathcal{P}(1,2)$ has the atlas $\mathbb{A}^2 \setminus \{0\} \to \mathcal{P}(1,2)$. The point $[0:1]$ has stabilizer $\mu_2 = \{\pm 1\}$.

For a finite group $G$ (viewed as a constant group scheme), $\mathrm{B}G = [\operatorname{Spec} k / G]$ classifies $G$-torsors, i.e., Galois $G$-covers. Over an algebraically closed field, $\mathrm{B}G$ has a single $k$-point (the trivial torsor $\operatorname{Spec} k$ with its unique $G$-structure up to isomorphism), but $\operatorname{Aut}$ of this point is $G$.

Since $G$ is finite (hence etale), $\mathrm{B}G$ is a DM stack. Its coarse moduli space is $\operatorname{Spec} k$, but $\mathrm{B}G \neq \operatorname{Spec} k$ as stacks -- they differ in their stacky structure.

Consider $\mathbb{G}_m$ acting on $\operatorname{Spec} k$ trivially. Then $[\operatorname{Spec} k / \mathbb{G}_m] = \mathrm{B}\mathbb{G}_m$. Now consider $\mathbb{G}_m$ acting on $\mathbb{A}^1$ by scaling: $t \cdot x = tx$. Then $[\mathbb{A}^1/\mathbb{G}_m]$ is an Artin stack with two kinds of points:

• The origin $0$: stabilizer is $\mathbb{G}_m$ (the full group).
• Any $x \neq 0$: stabilizer is trivial (the orbit is $\mathbb{A}^1 \setminus \{0\}$, which is free).

So $[\mathbb{A}^1/\mathbb{G}_m]$ has a "stacky point" at the origin (with automorphism group $\mathbb{G}_m$) and is a scheme (in fact $\operatorname{Spec} k$) away from the origin. The coarse moduli space is $\operatorname{Spec} k[x]^{\mathbb{G}_m} = \operatorname{Spec} k$ -- a point.

Let $C$ be a smooth projective curve. A Higgs bundle on $C$ is a pair $(E, \phi)$ where $E$ is a vector bundle on $C$ and $\phi : E \to E \otimes \omega_C$ is a morphism (the Higgs field). The moduli stack $\mathcal{H}iggs_n(C)$ of rank-$n$ Higgs bundles is an algebraic (Artin) stack. It plays a central role in the geometric Langlands program and in non-abelian Hodge theory.

The Hitchin map $h : \mathcal{H}iggs_n(C) \to \bigoplus_{i=1}^n H^0(C, \omega_C^{\otimes i})$ sends $(E, \phi)$ to its characteristic polynomial. The generic fibers are abelian varieties (Hitchin fibers), giving $\mathcal{H}iggs_n(C)$ the structure of an integrable system.

For a projective variety $X$, the stack $\operatorname{Coh}(X)$ of coherent sheaves on $X$ is an algebraic (Artin) stack, locally of finite type. Unlike the moduli of vector bundles, this stack is not of finite type -- there is no bound on the "complexity" of a coherent sheaf.

The substack of torsion sheaves supported at a point $p \in X$ is equivalent to $\coprod_n [\operatorname{Spec} k / \mathrm{GL}_n]$ (one component for each length $n$), which captures the combinatorics of partitions via the Jordan normal form.

The stack $\overline{\mathcal{M}}_{g,n}$ of $n$-pointed stable curves of genus $g$ (with $2g - 2 + n > 0$) is a proper, smooth DM stack of dimension $3g - 3 + n$. It compactifies $\mathcal{M}_{g,n}$ by allowing nodal curves with finite automorphism groups.

The boundary $\overline{\mathcal{M}}_{g,n} \setminus \mathcal{M}_{g,n}$ is a normal crossing divisor. Its components are indexed by the combinatorial data of dual graphs: a stable graph $\Gamma$ with vertices (irreducible components), edges (nodes), and markings (labeled points). For $\overline{\mathcal{M}}_{0,4}$, the coarse space is $\mathbb{P}^1$ with three boundary points corresponding to the three ways to partition four points into two pairs.

Let $X = \mathbb{A}^2$ with the $\mathbb{G}_m$-action $t \cdot (x, y) = (tx, t^{-1}y)$. The quotient stack $[\mathbb{A}^2 / \mathbb{G}_m]$ is Artin but not DM: the origin $(0,0)$ has stabilizer $\mathbb{G}_m$. The ring of invariants is $k[xy] \cong k[z]$, so the GIT quotient (coarse space) is $\mathbb{A}^1$. But the stack remembers the full orbit structure: the fiber over $z = 0$ consists of the three orbits $\{(x,0) : x \neq 0\}$, $\{(0,y) : y \neq 0\}$, and $\{(0,0)\}$, all collapsed to one point in the GIT quotient.

Fix a smooth projective variety $X$ and a class $\beta \in H_2(X, \mathbb{Z})$. The Kontsevich moduli stack $\overline{\mathcal{M}}_{g,n}(X, \beta)$ of $n$-pointed genus-$g$ stable maps $f : C \to X$ with $f_*[C] = \beta$ is a proper DM stack (when $X$ is convex, e.g., $X = \mathbb{P}^n$). It is the foundation of Gromov--Witten theory.

For $g = 0$, $X = \mathbb{P}^2$, $\beta = d[\ell]$ (degree $d$ curves), $\overline{\mathcal{M}}_{0,0}(\mathbb{P}^2, d)$ has dimension $3d - 1$. The virtual fundamental class $[\overline{\mathcal{M}}_{0,0}(\mathbb{P}^2, d)]^{\mathrm{vir}}$ computes the number $N_d$ of rational curves of degree $d$ through $3d - 1$ general points: $N_1 = 1$, $N_2 = 1$, $N_3 = 12$, $N_4 = 620$, ...

The classifying stack $\mathrm{B}G$ is separated if and only if $G$ is proper. So $\mathrm{B}(\mathbb{G}_m)$ is not separated (the diagonal $\mathrm{B}\mathbb{G}_m \to \mathrm{B}\mathbb{G}_m \times \mathrm{B}\mathbb{G}_m$ has fibers isomorphic to $\mathbb{G}_m$, which is not proper). On the other hand, $\mathrm{B}(\mu_n)$ is separated since $\mu_n$ is finite (hence proper).