Taking $X = \operatorname{Spec} k$ with the trivial $\mathrm{GL}_n$-action gives the classifying stack $\mathrm{B}\mathrm{GL}_n = [\operatorname{Spec} k / \mathrm{GL}_n].$ An object over $T$ is a $\mathrm{GL}_n$-torsor $P \to T$ (i.e., a frame bundle), which is equivalent to a rank-$n$ vector bundle $E \to T$. So $\mathrm{B}\mathrm{GL}_n$ is the moduli stack of rank-$n$ vector bundles.

The single $k$-point of $\mathrm{B}\mathrm{GL}_n$ (the trivial bundle $k^n$) has $\operatorname{Aut} = \mathrm{GL}_n$, which is $n^2$-dimensional. Hence $\mathrm{B}\mathrm{GL}_n$ is an Artin stack of dimension $-n^2$ (in the stacky sense: $\dim \mathrm{B}G = -\dim G$).

Let $\mathrm{GL}_n$ act on $\mathrm{Mat}_{n \times n} = \mathbb{A}^{n^2}$ by conjugation: $g \cdot A = gAg^{-1}$. The quotient stack $[\mathrm{Mat}_{n \times n}/\mathrm{GL}_n]$ parametrizes pairs (vector bundle $E$, endomorphism $\phi : E \to E$).

The GIT quotient is $\mathrm{Mat}_{n \times n} /\!/ \mathrm{GL}_n = \operatorname{Spec} k[\sigma_1, \ldots, \sigma_n] \cong \mathbb{A}^n$, where $\sigma_i$ are the coefficients of the characteristic polynomial. Two matrices have the same image if and only if they have the same characteristic polynomial -- but the stack $[\mathrm{Mat}_{n \times n}/\mathrm{GL}_n]$ distinguishes non-conjugate matrices with the same characteristic polynomial (e.g., the nilpotent matrix with a single Jordan block vs. the zero matrix in the $n = 2$ case).

Let $C$ be a smooth projective curve and fix a line bundle $\mathcal{L}$ of sufficiently large degree. Consider the Quot scheme $\mathcal{Q}$ parametrizing quotients $\mathcal{L}^{\oplus N} \twoheadrightarrow E$ where $E$ is a rank-$n$ vector bundle. Then $\mathrm{GL}_N$ acts on $\mathcal{Q}$ and $\mathrm{Bun}_n(C) \cong [\mathcal{Q}^{\mathrm{ss}} / \mathrm{GL}_N]$ (after restricting to the semistable locus). This gives an explicit presentation of the moduli stack of vector bundles as a quotient stack.

$\mathbb{P}^n$ is the geometric quotient $(\mathbb{A}^{n+1} \setminus \{0\})/\mathbb{G}_m$, and since $\mathbb{G}_m$ acts freely on $\mathbb{A}^{n+1} \setminus \{0\}$, the quotient stack agrees with the scheme: $[(\mathbb{A}^{n+1} \setminus \{0\}) / \mathbb{G}_m] \cong \mathbb{P}^n.$ More precisely, the $\mathbb{G}_m$-torsors over $T$ with equivariant map to $\mathbb{A}^{n+1} \setminus \{0\}$ are exactly the line bundles $\mathcal{L}$ on $T$ together with $n+1$ sections $(s_0, \ldots, s_n)$ generating $\mathcal{L}$. This recovers the functor of points of $\mathbb{P}^n$.

Let $\mathbb{G}_m$ act on $\mathbb{A}^1$ by scaling: $t \cdot x = tx$. The quotient stack $[\mathbb{A}^1/\mathbb{G}_m]$ has:

• Over $x \neq 0$: the orbit is all of $\mathbb{A}^1 \setminus \{0\}$, with trivial stabilizer, giving a single point (isomorphic to $\operatorname{Spec} k$).
• At $x = 0$: the stabilizer is $\mathbb{G}_m$, giving a point with automorphism group $\mathbb{G}_m$.

So $[\mathbb{A}^1/\mathbb{G}_m]$ consists of "a point glued to $\mathrm{B}\mathbb{G}_m$". The GIT quotient is $\operatorname{Spec} k[x]^{\mathbb{G}_m} = \operatorname{Spec} k$, which collapses everything.

This stack is sometimes called the "quotient point" or the "stacky point" -- it serves as a fundamental building block in the theory.

With $\mathbb{G}_m$ acting on $\mathbb{A}^{n+1} \setminus \{0\}$ by $t \cdot (x_0, \ldots, x_n) = (t^{a_0} x_0, \ldots, t^{a_n} x_n)$ for positive integers $a_i$, the quotient stack $\mathcal{P}(a_0, \ldots, a_n) = [(\mathbb{A}^{n+1} \setminus \{0\}) / \mathbb{G}_m]$ is the weighted projective stack. For $\gcd(a_0, \ldots, a_n) = 1$, the generic stabilizer is trivial, but at coordinate points $[0:\cdots:0:1:0:\cdots:0]$ the stabilizer is $\mu_{a_i}$.

For example, $\mathcal{P}(1, 2, 3)$ is a DM stack with:

• At $[1:0:0]$: stabilizer is trivial.
• At $[0:1:0]$: stabilizer is $\mu_2$.
• At $[0:0:1]$: stabilizer is $\mu_3$.

The coarse moduli space $\mathbb{P}(1,2,3)$ is a rational surface with cyclic quotient singularities of types $\frac{1}{2}(1,1)$ and $\frac{1}{3}(1,1)$.

Let $\mathcal{L}$ be a line bundle on a scheme $X$. The total space $\operatorname{Tot}(\mathcal{L})$ carries a natural $\mathbb{G}_m$-action (scaling in the fibers). Then $[\operatorname{Tot}(\mathcal{L}) / \mathbb{G}_m] \cong X \times \mathrm{B}\mathbb{G}_m$ via the zero section. More generally, removing the zero section gives $[(\operatorname{Tot}(\mathcal{L}) \setminus X) / \mathbb{G}_m] \cong X$ since $\mathbb{G}_m$ acts freely on the complement of the zero section.

Let $G = S_n$ (the symmetric group) act on $X^n$ by permuting factors. The quotient stack $[X^n / S_n]$ parametrizes "unordered $n$-tuples of points on $X$ with multiplicity data." The coarse moduli space is the symmetric product $\operatorname{Sym}^n(X) = X^n / S_n$.

For $X = \mathbb{A}^1$ and $n = 2$: $S_2$ acts on $\mathbb{A}^2$ by $(x, y) \mapsto (y, x)$. The quotient $\mathbb{A}^2/S_2 = \operatorname{Spec} k[x+y, xy] \cong \mathbb{A}^2$ is smooth (an accident of small $n$). But the stack $[\mathbb{A}^2/S_2]$ has a $\mathbb{Z}/2\mathbb{Z}$-gerbe along the diagonal $\{x = y\}$.

Let $\mu_n = \mathbb{Z}/n\mathbb{Z}$ act on $\mathbb{A}^2$ by $\zeta \cdot (x, y) = (\zeta x, \zeta^{-1} y)$. The quotient stack $[\mathbb{A}^2/\mu_n]$ is a smooth DM stack. Its coarse space is $\mathbb{A}^2/\mu_n = \operatorname{Spec} k[u, v, w]/(uv - w^n)$ which is the $A_{n-1}$ singularity. The stack "remembers" the smooth structure that the coarse space has lost.

The minimal resolution of the $A_{n-1}$ singularity has $n-1$ exceptional $(-2)$-curves arranged in a chain, corresponding to the Dynkin diagram $A_{n-1}$. The McKay correspondence relates this resolution to the representation theory of $\mu_n$.

The binary dihedral group $BD_{4n}$ (of order $4n$) acts on $\mathbb{A}^2$ via its natural 2-dimensional representation. The quotient stack $[\mathbb{A}^2/BD_{4n}]$ is a smooth DM stack, while the coarse space $\mathbb{A}^2/BD_{4n}$ is a $D_{n+2}$ surface singularity: $k[x,y,z]/(x^2 y + y^{n+1} + z^2).$ Similarly, the exceptional binary groups $\tilde{E}_6, \tilde{E}_7, \tilde{E}_8$ (binary tetrahedral, octahedral, icosahedral) give the $E_6, E_7, E_8$ surface singularities. This is part of the McKay correspondence.

Let $C$ be a hyperelliptic curve of genus $g$ with involution $\iota : C \to C$. The quotient stack $[C/\langle\iota\rangle]$ is a smooth DM stack with $\mu_2$-stabilizers at the $2g + 2$ Weierstrass points. The coarse space is $C/\langle\iota\rangle \cong \mathbb{P}^1$, and the map $C \to \mathbb{P}^1$ is a degree-2 cover branched at $2g + 2$ points.

The Hurwitz formula reads $2g - 2 = 2(0 - 2) + (2g + 2)$, which is $2g - 2 = 2g - 2$, consistent.

For $\mathrm{B}\mathbb{G}_m = [\operatorname{Spec} k / \mathbb{G}_m]$, a quasi-coherent sheaf is a $\mathbb{G}_m$-equivariant $k$-module, i.e., a $\mathbb{Z}$-graded $k$-vector space $V = \bigoplus_{n \in \mathbb{Z}} V_n$.

Line bundles on $\mathrm{B}\mathbb{G}_m$ correspond to one-dimensional representations of $\mathbb{G}_m$, which are indexed by $\mathbb{Z}$ (via the weight). So $\operatorname{Pic}(\mathrm{B}\mathbb{G}_m) \cong \mathbb{Z}$. The line bundle of weight $n$ is denoted $\mathcal{O}(n)$.