The trivial $\mathcal{G}$-gerbe over $\mathcal{Y}$ is the classifying stack $\mathrm{B}\mathcal{G} \times \mathcal{Y}$, i.e., the stack of $\mathcal{G}$-torsors over $\mathcal{Y}$. Its class in $H^2(\mathcal{Y}, \mathcal{G})$ is zero. Every $\mathcal{G}$-gerbe that has a global section (a global object) is isomorphic to the trivial gerbe.

A $\mu_n$-gerbe over $\operatorname{Spec} k$ is classified by $H^2(\operatorname{Spec} k, \mu_n)$. By the Kummer sequence $1 \to \mu_n \to \mathbb{G}_m \xrightarrow{(\cdot)^n} \mathbb{G}_m \to 1$, we get $H^2_{\mathrm{et}}(\operatorname{Spec} k, \mu_n) \cong \operatorname{Br}(k)[n]$ (the $n$-torsion in the Brauer group). So $\mu_n$-gerbes over a point are classified by $n$-torsion Brauer classes.

Over $k = \mathbb{R}$: $\operatorname{Br}(\mathbb{R}) = \mathbb{Z}/2\mathbb{Z}$ (generated by the quaternion algebra $\mathbb{H}$). So there is a nontrivial $\mu_2$-gerbe over $\operatorname{Spec} \mathbb{R}$, corresponding to $\mathbb{H}$.

$\mathbb{G}_m$-gerbes over a scheme $X$ are classified by $H^2_{\mathrm{et}}(X, \mathbb{G}_m)$. The cohomological Brauer group is $\operatorname{Br}'(X) = H^2_{\mathrm{et}}(X, \mathbb{G}_m)_{\mathrm{tors}}$. By a theorem of Gabber (and de Jong), this equals the Azumaya Brauer group $\operatorname{Br}(X)$ when $X$ has an ample line bundle.

A Brauer class $\alpha \in \operatorname{Br}(X)$ defines a $\mathbb{G}_m$-gerbe $\mathcal{X}_\alpha \to X$. $\alpha$-twisted sheaves on $X$ are quasi-coherent sheaves on the gerbe $\mathcal{X}_\alpha$. This is the foundation of Caldararu's theory of twisted sheaves.

Let $C$ be a smooth projective curve over an algebraically closed field $k$. Then $H^2_{\mathrm{et}}(C, \mu_n) \cong \mathbb{Z}/n\mathbb{Z}$ (by the cycle class map to $H^2(C, \mathbb{Z}/n) \cong \mathbb{Z}/n$). So there are $n$ isomorphism classes of $\mu_n$-gerbes over $C$.

The nontrivial $\mu_n$-gerbe can be constructed as follows: take a degree-1 line bundle $\mathcal{L}$ on $C$, and consider the $\mu_n$-gerbe of $n$-th roots of $\mathcal{L}$ (locally, an $n$-th root exists, but globally there is an obstruction in $\operatorname{Pic}(C)/n\operatorname{Pic}(C) \cong (\mathbb{Z}/n)^{2g} \times \mathbb{Z}/n$).

Let $\mathcal{X}$ be an algebraic stack and $x$ a point of $\mathcal{X}$ (i.e., a morphism $\operatorname{Spec} k \to \mathcal{X}$ from a field). The residual gerbe at $x$ is the reduced closed substack $\mathcal{G}_x \hookrightarrow \mathcal{X}$ supported at $x$. It is a gerbe over $\operatorname{Spec} k(x)$ banded by $\operatorname{Aut}(x)^{\circ}$ (the identity component of the automorphism group).

For a DM stack, the residual gerbe at a closed point $x$ with automorphism group $G$ is the classifying stack $\mathrm{B}G$, since $G$ is finite.

A Severi--Brauer variety $V$ over a field $k$ is a variety that becomes isomorphic to $\mathbb{P}^n$ after base change to $\bar{k}$. These are classified by $H^1(k, \mathrm{PGL}_{n+1})$. The associated $\mathbb{G}_m$-gerbe is obtained from the exact sequence $1 \to \mathbb{G}_m \to \mathrm{GL}_{n+1} \to \mathrm{PGL}_{n+1} \to 1$ via the connecting map $\delta : H^1(k, \mathrm{PGL}_{n+1}) \to H^2(k, \mathbb{G}_m) = \operatorname{Br}(k)$.

For example, the conic $x^2 + y^2 + z^2 = 0$ over $\mathbb{R}$ is a nontrivial Severi--Brauer variety, giving the nontrivial class in $\operatorname{Br}(\mathbb{R}) = \mathbb{Z}/2$.

Let $X = \mathbb{P}^1$, $D = \{p\}$ a point, $\mathcal{L} = \mathcal{O}(p)$, and $s$ the canonical section vanishing at $p$. The $r$-th root stack $\sqrt[r]{D/\mathbb{P}^1}$ is a smooth DM stack that:

• Is isomorphic to $\mathbb{P}^1$ over $\mathbb{P}^1 \setminus \{p\}$.
• Has a $\mu_r$-stabilizer at $p$.

Locally near $p$, with coordinate $z$ vanishing at $p$, the root stack is $[\operatorname{Spec} k[u]/(u^r - z) / \mu_r] = [\mathbb{A}^1 / \mu_r]$, where $\mu_r$ acts by $\zeta \cdot u = \zeta u$. The map $\sqrt[r]{D/\mathbb{P}^1} \to \mathbb{P}^1$ is the coarse moduli map.

An orbifold Riemann surface (or orbicurve) with underlying curve $C$ and orbifold points $p_1, \ldots, p_k$ of orders $r_1, \ldots, r_k$ is the iterated root stack $\mathcal{C} = \sqrt[r_1]{p_1} \times_C \sqrt[r_2]{p_2} \times_C \cdots \times_C \sqrt[r_k]{p_k}.$ This is a smooth DM stack with $\mu_{r_i}$-stabilizer at each $p_i$.

The orbifold Euler characteristic is $\chi^{\mathrm{orb}}(\mathcal{C}) = 2 - 2g - \sum_{i=1}^k \left(1 - \frac{1}{r_i}\right)$ where $g$ is the genus of $C$. For example, $\mathbb{P}^1$ with orbifold points of orders $(2, 3, 7)$ has $\chi^{\mathrm{orb}} = 2 - 1/2 - 2/3 - 6/7 = -1/42 < 0$, so the corresponding orbifold is hyperbolic.

Let $D = D_1 + \cdots + D_k$ be a simple normal crossing divisor on a smooth variety $X$, with line bundles $\mathcal{L}_i = \mathcal{O}_X(D_i)$ and canonical sections $s_i$. The root stack $\sqrt[r]{D/X} = \sqrt[r]{(\mathcal{L}_1, s_1)} \times_X \cdots \times_X \sqrt[r]{(\mathcal{L}_k, s_k)}$ is a smooth DM stack. Along $D_i \setminus \bigcup_{j \neq i} D_j$, the stabilizer is $\mu_r$; along $D_i \cap D_j$ (for $i \neq j$), the stabilizer is $\mu_r \times \mu_r$; and so on.

This construction is central to logarithmic geometry and orbifold birational geometry: root stacks provide a stacky way to encode the log structure of a pair $(X, D)$.

Let $X = \mathbb{A}^2$ with $D = V(xy)$ (the coordinate axes). The root stack $\mathcal{X} = \sqrt[2]{D/\mathbb{A}^2}$ is locally $[\operatorname{Spec} k[u,v]/(u^2 - x, v^2 - y) / (\mu_2 \times \mu_2)]$. The coarse space is $\mathbb{A}^2$, but the stack has $\mu_2$-stabilizers along each axis and $\mu_2 \times \mu_2$ at the origin.

The iterated root stack $\sqrt[r]{D/X}$ for large $r$ provides a "stacky resolution" of the pair $(X, D)$ that is often better behaved for cohomological computations than a classical log resolution.

There is a remarkable equivalence (due to Borne, Biswas, and others) between:

• Parabolic bundles on $(X, D)$ with rational weights of denominator dividing $r$, and
• Vector bundles on the root stack $\sqrt[r]{D/X}$.

More precisely, $\operatorname{Vect}(\sqrt[r]{D/X}) \cong \operatorname{ParVect}_r(X, D)$. This provides a geometric interpretation of Mehta--Seshadri's parabolic structures: a parabolic bundle is simply a vector bundle on the root stack.

For a curve $C$ with a single marked point $p$ and weights $0 \leq \alpha_1 < \cdots < \alpha_n < 1$ with denominators dividing $r$, a parabolic bundle $(E, E_{p,\bullet})$ on $(C, p)$ corresponds to a vector bundle $\mathcal{E}$ on $\sqrt[r]{p/C}$.

The $A_{n-1}$ singularity $\operatorname{Spec} k[u,v]/(uv)$ can be given a "canonical stack structure" as a root stack. Starting with $\mathbb{A}^1$ and the point $0$ (with its reduced structure), the $n$-th root stack $\sqrt[n]{0/\mathbb{A}^1}$ is the DM stack $[\mathbb{A}^1/\mu_n]$, which is the canonical stacky resolution.

More generally, for a normal surface singularity $p \in X$ that is a cyclic quotient of type $\frac{1}{n}(1, a)$, the canonical stack $\mathcal{X} \to X$ is a smooth DM stack with a single $\mu_n$-stacky point at $p$. This canonical stack is a root stack for the appropriate divisor.

Let $X$ be a smooth projective surface over an algebraically closed field. The Brauer group $\operatorname{Br}(X) = H^2_{\mathrm{et}}(X, \mathbb{G}_m)_{\mathrm{tors}}$ classifies $\mathbb{G}_m$-gerbes on $X$. For a K3 surface, $\operatorname{Br}(X) \cong (\mathbb{Q}/\mathbb{Z})^{22-\rho}$ where $\rho$ is the Picard rank. Each element $\alpha \in \operatorname{Br}(X)$ defines a gerbe $\mathcal{X}_\alpha \to X$, and the derived category $D^b(\mathcal{X}_\alpha)$ of the gerbe (i.e., $\alpha$-twisted sheaves on $X$) is a central object of study.

Mukai showed that certain Fourier--Mukai equivalences $D^b(X) \cong D^b(X')$ between K3 surfaces can be generalized to $D^b(\mathcal{X}_\alpha) \cong D^b(\mathcal{X}'_{\alpha'})$ for appropriate Brauer twists.

Consider the moduli problem of rank-$n$ vector bundles on $X$ with fixed determinant $\mathcal{L}$. The moduli stack $\mathcal{M}^{\mathcal{L}}_n$ has automorphism groups $\mu_n$ (the $n$-th roots of unity acting by scalar multiplication preserving the determinant). The natural map $\mathcal{M}^{\mathcal{L}}_n \to M^{\mathcal{L}}_n$ (to the coarse moduli space, when it exists) makes $\mathcal{M}^{\mathcal{L}}_n$ into a $\mu_n$-gerbe over $M^{\mathcal{L}}_n$.

The gerbe is trivial if and only if there exists a universal bundle on $M^{\mathcal{L}}_n \times X$. The obstruction to triviality lies in $H^2(M^{\mathcal{L}}_n, \mu_n) \cong \operatorname{Br}(M^{\mathcal{L}}_n)[n]$. For the moduli of stable bundles on a curve $C$ with coprime rank and degree, this Brauer class is nontrivial for $n \geq 2$.