# Birational geometry of Deligne-Mumford stacks

Andrew Kresch  Institut für Mathematik, Universität Zürich,
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
<EMAIL_ADDRESS>and  Yuri Tschinkel  Courant Institute, 251 Mercer
Street, New York, NY 10012, USA<EMAIL_ADDRESS>Simons Foundation
160 Fifth Avenue
New York, NY 10010
USA

(Date: December 21, 2023)

###### Abstract.

We investigate the birational geometry of Deligne-Mumford stacks and define
new birational invariants in this context.

##  1\. Introduction

Birational geometry of algebraic varieties is a classical subject in algebraic
geometry. Closely related to this is $G$-equivariant birational geometry for a
finite group $G$, also a well-studied area. There are meaningful parallels
between these theories: the study of birationality over an algebraically
nonclosed field $k$ is analogous, in many aspects, to equivariant
birationality over an algebraically closed field, where the role of $G$ is
played by the absolute Galois group of $k$. This point of view has been
brought into focus by Manin [37].

In this paper, we explore the birational geometry of Deligne-Mumford stacks
over fields of characteristic zero. In the setting of a projective $G$-variety
$X$ the associated Deligne-Mumford stack $[X/G]$ carries features such as
stabilizer groups, but does not carry the full information of $X$ with
$G$-action. The passage

$X\righttoleftarrow G\quad\rightsquigarrow\quad[X/G]\quad\rightsquigarrow\quad
k(X)^{G}=k(X/G)$

from equivariant geometry to geometry of Deligne-Mumford stacks and
subsequently to classical birational geometry involves a loss information at
each step. There is extensive literature on rationality of fields of
invariants and relevant obstructions, such as unramified cohomology (see,
e.g., [45], [13], [17], [14]), as well as a growing literature on equivariant
birationality (see, e.g., [41] and references therein, [31], [25]).

The Burnside ring, introduced in [27], encodes classical birational types:

$\operatorname{Burn}=\bigoplus_{n=0}^{\infty}\operatorname{Burn}_{n};$

the grading is by dimension. Refinements to Deligne-Mumford stacks and
$G$-varieties, respectively, appear in [29] and [31], with corresponding
Burnside groups and compatibility homomorphisms:

$\operatorname{Burn}_{n}(G)\quad\longrightarrow\quad{\overline{\operatorname{Burn}}}_{n}\quad\longrightarrow\quad\operatorname{Burn}_{n}.$

Important to the birational geometry of Deligne-Mumford stacks is the
appropriate notion of birational equivalence. In [29, Thm. 4.1], the class in
${\overline{\operatorname{Burn}}}_{n}$ of an $n$-dimensional Deligne-Mumford
stack (when this is defined) is shown to satisfy
$[{\mathcal{X}}]=[{\mathcal{Y}}]$ in ${\overline{\operatorname{Burn}}}_{n}$
whenever ${\mathcal{X}}$ and ${\mathcal{Y}}$ are _birationally equivalent_ ,
in the sense of fitting into a diagram

$\textstyle{{\mathcal{X}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}}$$\textstyle{\mathcal{Y}}$

where the morphisms are representable, proper, and birational.

A subtle phenomenon is the existence of ${\mathcal{X}}$ and ${\mathcal{Y}}$,
not birationally equivalent, with $[{\mathcal{X}}]=[{\mathcal{Y}}]$ in
${\overline{\operatorname{Burn}}}_{n}$. For instance, over an algebraically
closed field we consider the unique (nontrivial) action of
$G={\mathbb{Z}}/2{\mathbb{Z}}$ on elliptic curves. If $E$ and $F$ are non-
isomorphic elliptic curves, then the quotient stacks ${\mathcal{X}}=[E/G]$ and
${\mathcal{Y}}=[F/G]$ constitute such a pair ($n=1$). By way of contrast, the
class in $\operatorname{Burn}_{n}$, respectively in
$\operatorname{Burn}_{n}(G)$, fully captures the birational type of a
projective variety, respectively the equivariant birational type of a
projective $G$-variety of dimension $n$.

In this paper, we introduce a refinement of
${\overline{\operatorname{Burn}}}_{n}$ which fully captures the relevant
birational type. This is an abelian group

$\operatorname{\mathcal{B}\mathit{urn}}=\bigoplus_{n=0}^{\infty}\operatorname{\mathcal{B}\mathit{urn}}_{n}$

defined by explicit generators and relations. This is a refinement, in that a
quotient by further explicit relations

${\overline{\operatorname{\mathcal{B}\mathit{urn}}}}=\bigoplus_{n=0}^{\infty}{\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n},$

can be identified with the group from [29]:

${\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}\cong{\overline{\operatorname{Burn}}}_{n}.$

Our principal constructions and results are:

  * •

The groups $\operatorname{\mathcal{B}\mathit{urn}}$ and
${\overline{\operatorname{\mathcal{B}\mathit{urn}}}}$, in Section 7;

  * •

Associated birational invariants of Deligne-Mumford stacks, in Section 8;

  * •

Specialization homomorphisms, in Section 9;

  * •

Comparisons among various Burnside and Grothendieck groups, in Section 10.

A substantial part of the paper, Sections 2 to 6, is devoted to foundations of
birational geometry of stacks. Throughout, we provide examples illustrating
various aspects of the theory.

Acknowledgments: The second author was partially supported by NSF grant
2301983.

##  2\. Orbifolds and stabilizer groups

We work over a field $k$ of characteristic zero. We are interested in the
birational geometry of _orbifolds_ over $k$.

Algebraic orbifolds are smooth separated irreducible Deligne-Mumford (DM)
stacks of finite type over $k$, with trivial generic stabilizer. All algebraic
stacks in this section are of finite type over $k$, unless indicated
otherwise. All morphisms are morphisms over $k$.

######  Example 2.1.

A standard example of a DM stack is

$\mathcal{X}=[X/G],$

where $X$ is an algebraic variety over $k$ with regular action of a finite
group $G$. This is the category, where an object is a $G$-torsor $E\to T$
(over an arbitrary $k$-scheme $T$) with a $G$-equivariant morphism $E\to X$,
and a morphism is a $G$-equivariant isomorphism of torsors (over a morphisms
of $k$-schemes), compatible with the morphisms to $X$. Informally, the stack
$\mathcal{X}$ encodes the $G$-equivariant geometry of $X$, e.g., when $k$ is
algebraically closed, the category $\mathcal{X}(k)$ of $k$-points of
$\mathcal{X}$ has isomorphism classes of objects in bijective correspondence
with the $G$-orbits of $X(k)$. The case $X=\operatorname{Spec}(k)$ gives

$BG,$

the category of $G$-torsors (over $k$-schemes) and $G$-equivariant morphisms
(over morphisms of $k$-schemes).

A separated DM stack, or more generally an algebraic stack satisfying the
weaker hypothesis of having finite inertia, has a coarse moduli space. For
instance, $\mathcal{X}=[X/G]$ has coarse moduli space $X/G$, the quotient
variety when this exists (e.g., when $X$ is quasi-projective), generally the
algebraic space quotient. When $X$ is nonsingular and the $G$-action is
faithful, $\mathcal{X}=[X/G]$ is an example of an orbifold.

For an orbifold, when the coarse moduli space is a projective scheme, we speak
of a _projective orbifold_ , and when quasi-projective, a _quasi-projective
orbifold_. The same definitions apply to smooth separated DM stacks; cf. [28].
If $\mathcal{X}$ is quasi-projective, then $\mathcal{X}\cong[X/G]$ for some
linear algebraic group $G$ acting on a smooth quasi-projective scheme $X$; in
fact, we may take $G$ to be $GL_{N}$ for some $N$. But it is a subtle
question, whether this can be achieved with a _finite_ group $G$. Even in
dimension one, there exist orbifolds not of the form $[X/G]$, for any variety
$X$ and finite group $G$. E.g., for any positive integers $a$ and $b$, not
both equal to $1$, the stack

${\mathbb{P}}(a,b)=[{\mathbb{A}}^{2}\setminus\\{0\\}/{\mathbb{G}}_{m}],$

with action $t\cdot(x,y)=(t^{a}x,t^{b}y)$, cannot be expressed as stack
quotient of a scheme by a finite group (cf. [8]). This is an orbifold when
$\gcd(a,b)=1$.

Algebraic stacks form a $2$-category, where besides morphisms (over $k$) there
are $2$-morphisms given by natural transformations of functors (over identity
morphisms in the base category of $k$-schemes). The $2$-morphisms are all
invertible.

We often perform local constructions, such as the projective fibration
$\operatorname{Proj}\operatorname{Sym}{\mathcal{E}}\to{\mathcal{X}}$
associated with a coherent sheaf ${\mathcal{E}}$ on a given algebraic stack
${\mathcal{X}}$. This, along with a more general _projective_ morphism to
${\mathcal{X}}$, which by definition is one that admits a factorization up to
$2$-isomorphism as a closed immersion to a projective fibration, provides an
example of _representable_ morphism. A morphism
${\mathcal{Y}}\to{\mathcal{X}}$ is representable if, for any scheme $T$ with
morphism to ${\mathcal{X}}$, the fiber product
$T\times_{{\mathcal{X}}}{\mathcal{Y}}$ is an algebraic space.

######  Lemma 2.2.

Let $f\colon{\mathcal{Y}}\to{\mathcal{X}}$ be a projective morphism of
separated DM stacks. Then the induced morphism of coarse moduli spaces $e:Y\to
X$ is projective.

###### Proof.

The morphism $e$ is proper, since the composite ${\mathcal{Y}}\to X$ is proper
and ${\mathcal{Y}}\to Y$ is proper surjective. We will exhibit a line bundle
$M$ on $Y$, whose pullback to a finite scheme cover of $Y$ is ample, relative
to the composite morphism to $X$. When $X$ and $Y$ are schemes, [20, (2.6.2)]
tells us that $e$ is projective, with relatively ample line bundle $M$. The
proof given in [20, (2.6.3)] uses Chevalley’s theorem. This is available for
algebraic spaces, and we reach the same conclusion, supposing just $X$ to be a
scheme. The general case reduces to this one, by base change to an étale atlas
of $X$, since the condition of obtaining a closed immersion
$Y\to\operatorname{Proj}\operatorname{Sym}e_{*}(M^{n})$, for $n\gg 0$, may be
checked étale locally.

By [35, (16.6)], there exists a scheme $Z$ with a finite surjective morphism
$Z\to{\mathcal{X}}$. We let $W={\mathcal{Y}}\times_{{\mathcal{X}}}Z$, with
projection morphisms $p$ to ${\mathcal{Y}}$ and $g$ to $Z$, so $p$ is finite
and surjective, and $g$ is projective. The composite morphisms $r\colon Z\to
X$ and $s\colon W\to Y$ are proper and quasi-finite, hence finite. As well,
they are surjective.

Factoring $f$ up to $2$-isomorphism as
${\mathcal{Y}}\to\operatorname{Proj}\operatorname{Sym}{\mathcal{E}}\to{\mathcal{X}}$
we let $L$ denote the restriction to ${\mathcal{Y}}$ of the line bundle
${\mathcal{O}}_{\operatorname{Proj}\operatorname{Sym}{\mathcal{E}}}(1)$. By
[33, Lemma 2], after replacing $L$ by a tensor power, we may suppose that $L$
is isomorphic to the pullback of a line bundle $M$ on $Y$. So $s^{*}M\cong
p^{*}L$ is ample relative to $g$, hence also relative to $r\circ g$, and $e$
is projective. ∎

Stacks with representable morphism to a given stack ${\mathcal{Y}}$ form, in a
natural way, an _ordinary_ category. An object is a stack $\mathsf{X}$ with
representable morphism $\mathsf{f}\colon\mathsf{X}\to{\mathcal{Y}}$. A
morphism to another object
$\mathsf{f}^{\prime}\colon\mathsf{X}^{\prime}\to{\mathcal{Y}}$ is an
equivalence class of pairs $(g,\alpha)$, consisting of a morphism
$g\colon\mathsf{X}\to\mathsf{X}^{\prime}$ and $2$-isomorphism
$\alpha\colon\mathsf{f}\Rightarrow\mathsf{f}^{\prime}\circ g$. The equivalence
relation is defined by $(g,\alpha)\sim(g^{\prime},\alpha^{\prime})$ when there
exists a $2$-isomorphism $\beta\colon g\Rightarrow g^{\prime}$, such that

$\alpha^{\prime}(\mathsf{x})=\mathsf{f}^{\prime}(\beta(\mathsf{x}))\circ\alpha(\mathsf{x})$

for all objects $\mathsf{x}$ of $\mathsf{X}$.

A useful variant, given a stack ${\mathcal{X}}$ with morphism
$f\colon{\mathcal{X}}\to{\mathcal{Y}}$, is the category of factorizations of
$f$ through a stack with representable morphism to ${\mathcal{Y}}$. An object
is a stack $\mathsf{X}$ with representable morphism
$\mathsf{f}\colon\mathsf{X}\to{\mathcal{Y}}$, morphism
$\pi\colon{\mathcal{X}}\to\mathsf{X}$, and $2$-isomorphism $\iota\colon
f\Rightarrow\mathsf{f}\circ\pi$. A morphism to
$\mathsf{f}^{\prime}\colon\mathsf{X}^{\prime}\to{\mathcal{Y}}$, with
$\pi^{\prime}\colon{\mathcal{X}}\to\mathsf{X}^{\prime}$ and
$\iota^{\prime}\colon f\Rightarrow\mathsf{f}^{\prime}\circ\pi^{\prime}$, is an
equivalence class of pairs $(g,\alpha)$ as above, admitting a compatible
$2$-isomorphism $\pi^{\prime}\Rightarrow g\circ\pi$.

Suppose that ${\mathcal{X}}$ has a coarse moduli space and ${\mathcal{Y}}$ has
separated diagonal. The construction of the coarse moduli space of
${\mathcal{X}}$, carried out (smooth) locally over ${\mathcal{Y}}$, yields the
_relative coarse moduli space_ [3, §3]. This is characterized as the initial
object in the category of factorizations of $f$ through a stack with
representable morphism to ${\mathcal{Y}}$. When this is $1_{{\mathcal{Y}}}$,
with $f$ and $1_{f}$, we call $f$ a _coarsening morphism_ [6, §2.3]. Given a
morphism ${\mathcal{Y}}\to{\mathcal{Z}}$, the morphism
$\pi\colon{\mathcal{X}}\to\mathsf{X}$ induces an isomorphism of relative
coarse moduli spaces over ${\mathcal{Z}}$. In particular, $\pi$ induces an
isomorphism of coarse moduli spaces [6, Lemma 2.3.4(iii)].

######  Example 2.3.

The diagonal

$\Delta\colon{\mathcal{X}}\to{\mathcal{X}}\times{\mathcal{X}}$

is representable. For a point of an algebraic stack ${\mathcal{X}}$,
represented by a morphism $x\colon\operatorname{Spec}(\ell)\to{\mathcal{X}}$
for a field $\ell/k$, we obtain a stabilizer group scheme $G_{x}$, locally of
finite type over $\ell$, fitting into a fiber diagram

$\textstyle{G_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{\operatorname{Spec}(\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta\circ
x}$$\textstyle{\mathcal{X}\times\mathcal{X}}$

with the diagonal $\Delta$ of $\mathcal{X}$. For the stacks that we typically
consider, DM stacks with finite inertia, $G_{x}$ will be reduced and finite.
Passing to an algebraic closure of $\ell$ leads to a finite group, the
_geometric stabilizer group_ at $x$. Fixing $\ell$, an isomorphism in
$\mathcal{X}(\ell)$ is a $2$-isomorphism

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(2.1)

and this determines an isomorphism

$G_{x}\to G_{x^{\prime}}.$ (2.2)

A different choice of $2$-isomorphism $\Downarrow$ in (2.1) determines an
isomorphism that differs from (2.2) by an inner automorphism of
$G_{x^{\prime}}$. In the sequel, DM stacks with _abelian_ geometric stabilizer
groups, at least over a dense open substack $\mathcal{U}$ of ${\mathcal{X}}$
will play the biggest role. Then the geometric stabilizer groups of points of
$\mathcal{U}$ are defined up to _canonical_ isomorphism.

######  Remark 2.4.

All of the stabilizers are packaged together in the _inertia stack_ , which is
the stack ${\mathcal{I}}_{{\mathcal{X}}}$ in the fiber diagram

$\textstyle{{\mathcal{I}}_{{\mathcal{X}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{\mathcal{X}\times\mathcal{X}}$

If ${\mathcal{X}}$ is a smooth separated DM stack, then so is
${\mathcal{I}}_{{\mathcal{X}}}$, and so are the locally closed substacks of
${\mathcal{X}}$, appearing in the stratification by isomorphism type of
geometric stabilizer.

##  3\. Normal crossing divisor and root construction

We continue to work with stacks of finite type over a field $k$ of
characteristic $0$.

A _normal crossing divisor_ on a smooth separated DM stack ${\mathcal{X}}$ is
a divisor ${\mathcal{D}}\subset{\mathcal{X}}$, whose pre-image in one or
equivalently every étale altas $U$ of ${\mathcal{X}}$ is a normal crossing
divisor on $U$. If, furthermore, we can write
${\mathcal{D}}={\mathcal{D}}_{1}\cup\dots\cup{\mathcal{D}}_{\ell}$, where each
${\mathcal{D}}_{i}$ is smooth, then we call ${\mathcal{D}}$ a _simple normal
crossing (snc) divisor_. Equivalently, an snc divisor on ${\mathcal{X}}$ is a
normal crossing divisor whose irreducible components are smooth.

######  Remark 3.1.

Desingularization of a reduced Noetherian quasi-excellent algebraic stack in
characteristic $0$ is established in [47]. This is applicable to reduced
stacks, of finite type over a field of characteristic zero. The case of
reduced stacks of finite type over a DVR with residue field of characteristic
zero, needed in Section 9, is also covered. Embedded resolution of
singularities is also available in this generality [48].

Let ${\mathcal{D}}$ be a arbitrary divisor on a smooth separated DM stack
${\mathcal{X}}$, and let $r$ be a positive integer. We recall the _root stack_

$\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}$

of [16, §2], [2, App. B]. The divisor ${\mathcal{D}}$ determines a line bundle
${\mathcal{O}}_{{\mathcal{X}}}({\mathcal{D}})$ with section vanishing along
${\mathcal{D}}$, hence a morphism

${\mathcal{X}}\to[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}].$ (3.1)

The root stack is the fiber product

$\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}={\mathcal{X}}\times_{[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]}[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}],$

where
$[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\to[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]$ is
the morphism induced by the $r$th power maps on ${\mathbb{A}}^{1}$ and on
${\mathbb{G}}_{m}$. The substack

$\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}\times_{[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]}[\\{0\\}/{\mathbb{G}}_{m}]$

is the _gerbe of the root stack_ ${\mathcal{G}}_{{\mathcal{D}}}$ [16, Def.
2.4.4]; we recall that a gerbe is a morphism of stacks (in this case,
${\mathcal{G}}_{{\mathcal{D}}}\to{\mathcal{D}}$) which admits (étale) local
sections, such that any pair of local sections are locally isomorphic. For
$r>1$, the root stack is smooth if and only if ${\mathcal{D}}$ is smooth.

######  Lemma 3.2.

Let ${\mathcal{X}}$ be a smooth separated irreducible DM stack and
${\mathcal{D}}$ a divisor on ${\mathcal{X}}$ with normal bundle
${\mathcal{N}}={\mathcal{N}}_{{\mathcal{D}}/{\mathcal{X}}}$. We denote by
${\mathcal{Z}}$ the zero-section of ${\mathcal{N}}$. Then we have a canonical
isomorphism

${\mathcal{N}}_{{\mathcal{G}}_{{\mathcal{D}}}/\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}}\cong\sqrt[r]{({\mathcal{N}},{\mathcal{Z}})}.$

In particular, the complement of the zero-section of
${\mathcal{N}}_{{\mathcal{G}}_{{\mathcal{D}}}/\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}}$
is isomorphic to the complement of the zero-section of
${\mathcal{N}}_{{\mathcal{D}}/{\mathcal{X}}}$.

###### Proof.

There is a $2$-commutative diagram of stacks (_not_ a fiber diagram!)

$\textstyle{[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{({-})^{r}}$$\textstyle{[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({-})^{r}}$$\textstyle{[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]}$

where the horizontal arrows are given by the ${\mathbb{G}}_{m}$-equivariant
morphism sending ${\mathbb{A}}^{1}$ to $\\{0\\}\subset{\mathbb{A}}^{1}$ and
the vertical arrows are induced by the $r$th power maps on ${\mathbb{A}}^{1}$
and on ${\mathbb{G}}_{m}$. When we perform base change by the morphism (3.1)
we obtain

$\textstyle{[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\times_{0\circ({-})^{r},[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]}{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\times_{0,[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]}{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}}$

We analyze the terms on the left. Identifying
$[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]$ with the normal bundle to
$[\\{0\\}/{\mathbb{G}}_{m}]$ in $[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]$,
flatness of (3.1) leads to an identification

$[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\times_{0,[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]}{\mathcal{X}}\cong{\mathcal{N}}_{{\mathcal{D}}/{\mathcal{X}}}.$

Analogously,

$[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\times_{0\circ({-})^{r},[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]}{\mathcal{X}}\cong{\mathcal{N}}_{{\mathcal{G}}_{{\mathcal{D}}}/\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}}.$

Now the fiber square

$\textstyle{{\mathcal{N}}_{{\mathcal{G}}_{{\mathcal{D}}}/\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({-})^{r}}$$\textstyle{{\mathcal{N}}_{{\mathcal{D}}/{\mathcal{X}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]}$

gives rise to an identification of
${\mathcal{N}}_{{\mathcal{G}}_{{\mathcal{D}}}/\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}}$
with the $r$th root stack of ${\mathcal{N}}_{{\mathcal{D}}/{\mathcal{X}}}$
along the zero-section. ∎

We return to the setting of an snc divisor
${\mathcal{D}}={\mathcal{D}}_{1}\cup\dots\cup{\mathcal{D}}_{\ell}$ on a smooth
separated DM stack ${\mathcal{X}}$. As mentioned above, to obtain a smooth
root stack $\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}$ (with $r>1$) we require
${\mathcal{D}}$ to be smooth, equivalently,
${\mathcal{D}}_{i}\cap{\mathcal{D}}_{j}=\emptyset$ for $i\neq j$. A different
construction, the _iterated root stack_ [16, Defn. 2.2.4], preserves
smoothness when applied to an snc divisor. Each ${\mathcal{D}}_{i}$
individually determines a morphism to $[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]$,
so together we have a morphism

${\mathcal{X}}\to[{\mathbb{A}}^{\ell}/{\mathbb{G}}_{m}^{\ell}].$

Let an $\ell$-tuple of positive integers $\mathbf{r}=(r_{1},\dots,r_{\ell})$
be given. The product of the morphisms $({-})^{r_{i}}$ is a morphism

$({-})^{\mathbf{r}}\colon[{\mathbb{A}}^{\ell}/{\mathbb{G}}_{m}^{\ell}]\to[{\mathbb{A}}^{\ell}/{\mathbb{G}}_{m}^{\ell}].$

The iterated root stack is

$\sqrt[\mathbf{r}]{({\mathcal{X}},{\mathcal{D}})}={\mathcal{X}}\times_{[{\mathbb{A}}^{\ell}/{\mathbb{G}}_{m}^{\ell}],({-})^{\mathbf{r}}}[{\mathbb{A}}^{\ell}/{\mathbb{G}}_{m}^{\ell}].$

Another sort of root stack, given in [16, §2], [2, App. B], is the $r$th root
$\sqrt[r]{L/{\mathcal{X}}}$ of a line bundle $L$ on ${\mathcal{X}}$. This is
defined as

$\mathcal{X}\times_{B{\mathbb{G}}_{m}}B{\mathbb{G}}_{m},$ (3.2)

the fiber product of $\mathcal{X}\to B{\mathbb{G}}_{m}$ associated with $L$,
and the $r$th power map $B{\mathbb{G}}_{m}\to B{\mathbb{G}}_{m}$.

######  Lemma 3.3.

Let ${\mathcal{X}}$ be a smooth separated DM stack, ${\mathcal{D}}$ a divisor
on ${\mathcal{X}}$, and $r$ and $s$ positive integers. Let
${\mathcal{X}}^{\prime}=\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}$, with gerbe
of the root stack ${\mathcal{G}}_{{\mathcal{D}}}$ and morphism
$\pi\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$. Then:

  * $\mathrm{(i)}$

One has

$\sqrt[s]{({\mathcal{X}}^{\prime},{\mathcal{G}}_{{\mathcal{D}}})}\cong\sqrt[rs]{({\mathcal{X}},{\mathcal{D}})}.$

  * $\mathrm{(ii)}$

Let $L$ be a line bundle on ${\mathcal{X}}$ and $L^{\prime}=\pi^{*}L$. Then

$\sqrt[s]{L^{\prime}/{\mathcal{X}}^{\prime}}\cong\sqrt[r]{\big{(}\sqrt[s]{L/{\mathcal{X}}},{\mathcal{D}}\times_{{\mathcal{X}}}\sqrt[s]{L/{\mathcal{X}}}\big{)}}$

###### Proof.

Statement (i) follows from the definition, since the composite

$[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\stackrel{{\scriptstyle({-})^{s}}}{{\longrightarrow}}[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\stackrel{{\scriptstyle({-})^{r}}}{{\longrightarrow}}[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]$

is equal (up to $2$-isomorphism) to

$({-})^{rs}\colon[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\to[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}].$

For (ii), we argue similarly, using the two composite morphisms

$({-})^{r}\times 1\circ 1\times({-})^{s}\quad\text{ and }\quad
1\times({-})^{s}\circ({-})^{r}\times 1$

from $[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\times B{\mathbb{G}}_{m}$ to
$[{\mathbb{A}}^{1}/{\mathbb{G}}_{m}]\times B{\mathbb{G}}_{m}$. ∎

######  Proposition 3.4.

Let ${\mathcal{D}}$ be a divisor on a smooth separated DM stack
${\mathcal{X}}$, and $r$ a positive integer. Then

$\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}\to{\mathcal{X}}$

is a coarsening morphism. The same holds when
${\mathcal{D}}={\mathcal{D}}_{1}\cup\dots\cup{\mathcal{D}}_{\ell}$ is an snc
divisor for the iterated root stack

$\sqrt[\mathbf{r}]{({\mathcal{X}},{\mathcal{D}})}\to{\mathcal{X}},$

for any $\mathbf{r}$, and for the $r$th root

$\sqrt[r]{L/X}\to{\mathcal{X}}$

of any line bundle $L$.

###### Proof.

A root stack is a coarsening morphism, by [16, Cor. 2.3.7]. The case of an
iterated root stack is taken care of by the observation [16, Rmk. 2.2.5] that
an iterated root stack may be obtained as a sequence of individual root
stacks. ∎

##  4\. Birationality

Interest in birational geometry of stacks was stimulated by progress in the
Minimal Model Program for moduli stacks, see e.g. [23], from which we have
taken the notions in Definition 4.1, below. All stacks in this section are DM
stacks, separated and of finite type over $k$. (In several places, clearly
indicated, there are DM stacks which are separated and of finite type over a
different field.)

######  Definition 4.1.

Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be stacks.

  * (i)

A morphism $f\colon{\mathcal{X}}\to{\mathcal{Y}}$ is _birational_ if $f$
restricts to an isomorphism ${\mathcal{U}}\to{\mathcal{V}}$ of dense open
substacks of ${\mathcal{X}}$, respectively ${\mathcal{Y}}$.

  * (ii)

A _rational map_ ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is an object of
the category

$\operatorname{MAP}({\mathcal{X}},{\mathcal{Y}})=\operatorname{colim}_{{\mathcal{U}}\subset{\mathcal{X}}}\operatorname{HOM}({\mathcal{U}},{\mathcal{Y}}),$

where the colimit is taken over dense open substacks
${\mathcal{U}}\subset{\mathcal{X}}$. So, a rational map is represented by
$f\colon{\mathcal{U}}\to{\mathcal{Y}}$ for some such ${\mathcal{U}}$, with
$({\mathcal{U}},f)\cong({\mathcal{U}}^{\prime},f^{\prime})$ if and only if $f$
and $f^{\prime}$ become $2$-isomorphic after restriction to common dense open.

  * (iii)

A rational map ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is _dominant_ if,
for some or equivalently every representative $({\mathcal{U}},f)$,
$f({\mathcal{U}})$ is dense in ${\mathcal{Y}}$.

  * (iv)

A rational map ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is _birational_ if,
for some representative $({\mathcal{U}},f)$, $f$ maps ${\mathcal{U}}$
isomorphically to a dense open substack of ${\mathcal{Y}}$.

  * (v)

A rational map ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is _proper_ if
there exist a representative $({\mathcal{U}},f)$, a proper birational morphism
${\mathcal{X}}^{\prime}\to{\mathcal{X}}$, and a proper morphism
${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$ such that $f$ factors up to
$2$-isomorphism as

${\mathcal{U}}\cong{\mathcal{X}}^{\prime}\times_{{\mathcal{X}}}{\mathcal{U}}\to{\mathcal{Y}}.$
(4.1)

######  Proposition 4.2.

Let $f\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ be a representable proper
birational morphism of stacks, with ${\mathcal{X}}$ normal and
${\mathcal{X}}^{\prime}$ reduced. Then the maximal open substack
${\mathcal{U}}\subset{\mathcal{X}}$, over which $f$ restricts to an
isomorphism $f^{-1}({\mathcal{U}})\to{\mathcal{U}}$, has complement everywhere
of codimension $\geq 2$.

###### Proof.

For any finite-type morphism of locally Noetherian schemes, the set of points
of the source, where the morphism is flat, is open by [21, (11.1.1)]. By the
local nature of flatness, we have the same for any morphism of stacks. Letting
${\mathcal{U}}^{\prime}$ denote the locus where $f$ is open, we claim that
${\mathcal{U}}$ is the complement of
$f({\mathcal{X}}^{\prime}\setminus{\mathcal{U}}^{\prime})$. A representable
proper birational morphism, if additionally flat, must be an isomorphism. So

${\mathcal{X}}\setminus
f({\mathcal{X}}^{\prime}\setminus{\mathcal{U}}^{\prime})\subset{\mathcal{U}},$

and we have equality, since ${\mathcal{U}}^{\prime}$ contains
$f^{-1}({\mathcal{U}})$.

This observation shows that the formation of ${\mathcal{U}}$ from
$f\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ commutes with étale base
change. Thus we are reduced to proving the statement when ${\mathcal{X}}$ is a
scheme and ${\mathcal{X}}^{\prime}$ an algebraic space. Then we obtain the
statement by a straightforward application of the valuative criterion for
properness. ∎

######  Proposition 4.3.

Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be stacks, with ${\mathcal{X}}$
reduced, and ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ a rational map.

  * $\mathrm{(i)}$

The closure $\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$ in
${\mathcal{X}}\times{\mathcal{Y}}$ of the image of the graph

${\mathcal{U}}\stackrel{{\scriptstyle(j,f)}}{{\to}}{\mathcal{X}}\times{\mathcal{Y}},$

where $({\mathcal{U}},f)$ is a representative of the rational map and
$j\colon{\mathcal{U}}\to{\mathcal{X}}$ denotes the inclusion, is independent
of the choice of representative. The rational map is proper if and only if
$\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})\to{\mathcal{X}}$ and
$\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})\to{\mathcal{Y}}$ are
proper.

  * $\mathrm{(ii)}$

Suppose $({\mathcal{U}},f)$ is a representative of the rational map
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$. Then there exists a reduced stack
${\mathcal{X}}^{\prime}$ with birational morphism to ${\mathcal{X}}$ and
morphism to ${\mathcal{Y}}$, such that $f$ factors up to $2$-isomorphism as
(4.1) and the induced morphism
${\mathcal{X}}^{\prime}\to\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$
is projective.

###### Proof.

The independence of choice of representative follows from the fact that
${\mathcal{X}}$ is reduced. Suppose
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is proper, i.e., there exist
${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ and
${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$ as in Definition 4.1(v); we may
suppose ${\mathcal{X}}^{\prime}$ reduced. The corresponding morphism
${\mathcal{X}}^{\prime}\to{\mathcal{X}}\times{\mathcal{Y}}$ factors through
${\mathcal{Z}}=\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$. Since
${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is proper and
${\mathcal{Z}}\to{\mathcal{X}}$ is separated,
${\mathcal{X}}^{\prime}\to{\mathcal{Z}}$ is proper. Since, also,
${\mathcal{X}}^{\prime}\to{\mathcal{Z}}$ is surjective, properness of
${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ implies that
${\mathcal{Z}}\to{\mathcal{X}}$ is proper, and the same holds with
${\mathcal{Y}}$ in place of ${\mathcal{X}}$.

The morphism $(j,f)$ factors as the composition of $j\times 1_{{\mathcal{Y}}}$
with the graph of $f$. The latter is obtained by base change from the diagonal
of ${\mathcal{Y}}$, hence is representable and finite and thus is projective.
So we have a projective morphism to the open substack
${\mathcal{U}}\times{\mathcal{Y}}\subset{\mathcal{X}}\times{\mathcal{Y}}$, and
this may be compactified, by applying [35, (15.5)] to realize the coherent
sheaf on the open substack in the definition of projective morphism as
restriction of a coherent sheaf on the ambient stack, and forming the closure
of ${\mathcal{U}}$ in the projective fibration over the ambient stack. This
yields a stack ${\mathcal{X}}^{\prime\prime}$, with projective morphism to
${\mathcal{Z}}$ and
${\mathcal{U}}\cong{\mathcal{X}}^{\prime\prime}\times_{{\mathcal{X}}}{\mathcal{U}}$.
Now if ${\mathcal{Z}}\to{\mathcal{X}}$ and ${\mathcal{Z}}\to{\mathcal{Y}}$ are
proper, then also ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ and
${\mathcal{X}}^{\prime\prime}\to{\mathcal{Y}}$ are proper, and the conditions
in Definition 4.1(v) are met. We have established (i) and, by the construction
of ${\mathcal{X}}^{\prime\prime}$, also (ii). ∎

######  Lemma 4.4.

Let ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ be a proper birational map of
stacks, and let ${\mathcal{S}}\subset{\mathcal{X}}$ and
${\mathcal{T}}\subset{\mathcal{Y}}$ be dense open substacks. Then the
corresponding birational map ${\mathcal{S}}\dashrightarrow{\mathcal{T}}$ is
proper if and only if for one or equivalently every factorization (4.1), the
pre-images of ${\mathcal{S}}$ and ${\mathcal{T}}$ in ${\mathcal{X}}^{\prime}$
are equal.

###### Proof.

First we show that equality of pre-images in a factorization (4.1) implies
properness of ${\mathcal{S}}\dashrightarrow{\mathcal{T}}$. Let
${\mathcal{S}}^{\prime}\subset{\mathcal{X}}^{\prime}$ denote the common pre-
image. Then the proper birational morphisms
${\mathcal{S}}^{\prime}\to{\mathcal{S}}$ and
${\mathcal{S}}^{\prime}\to{\mathcal{T}}$ exhibit
${\mathcal{S}}\dashrightarrow{\mathcal{T}}$ as proper.

It remains to deduce equality of pre-images from the properness of
${\mathcal{S}}\dashrightarrow{\mathcal{T}}$. For this we may suppose without
loss of generality that ${\mathcal{X}}$ is reduced. Then we have
$\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$ as in Proposition 4.3, and

$\Gamma({\mathcal{S}}\dashrightarrow{\mathcal{T}})=p^{-1}({\mathcal{S}})\cap
q^{-1}({\mathcal{T}}),$

where
$p\colon\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})\to{\mathcal{X}}$ and
$q\colon\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})\to{\mathcal{Y}}$
denote the projections. If ${\mathcal{S}}\dashrightarrow{\mathcal{T}}$ is
proper, then by Proposition 4.3(i), the composite morphisms

$\Gamma({\mathcal{S}}\dashrightarrow{\mathcal{T}})\to
p^{-1}({\mathcal{S}})\to{\mathcal{S}}$

and

$\Gamma({\mathcal{S}}\dashrightarrow{\mathcal{T}})\to
q^{-1}({\mathcal{T}})\to{\mathcal{T}}$

are proper. So in each composition the first map, a dense open immersion, is
proper, hence an isomorphism. Thus
$p^{-1}({\mathcal{S}})=q^{-1}({\mathcal{T}})$, and from this follows the
equality of pre-images in ${\mathcal{X}}^{\prime}$ for any factorization
(4.1). ∎

For schemes, the canonical representative on the maximal domain of definition
of a rational map requires the source to be reduced. For stacks, even more is
required: with “ghost automorphisms” (see, e.g., [1, §1.2.1]), we have
examples of pairs of isomorphisms $f$,
$f^{\prime}\colon{\mathcal{X}}\to{\mathcal{Y}}$ of reduced stacks, such that
$f$ and $f^{\prime}$ are not $2$-isomorphic, but become $2$-isomorphic upon
restriction to a dense open substack of ${\mathcal{X}}$.

The role of the residue field at a point of a scheme is, for a stack, played
by the _residual gerbe_ [44, Thm. B.2]: to a point $x$ of a stack
${\mathcal{X}}$ there is a monomorphism ${\mathcal{G}}_{x}\to{\mathcal{X}}$
and a gerbe ${\mathcal{G}}_{x}\to\operatorname{Spec}(\kappa(x))$. There is a
finite surjective morphism to ${\mathcal{G}}_{x}$ from
$\operatorname{Spec}(\lambda)$ for some finite extension field
$\lambda/\kappa(x)$, so by Chevalley’s theorem the morphism
${\mathcal{G}}_{x}\to{\mathcal{X}}$ is affine.

######  Proposition 4.5.

Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be stacks, with ${\mathcal{X}}$
normal. Every rational map ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ has a
well-defined maximal domain of definition ${\mathcal{U}}$, with the property
that up to $2$-isomorphism there is a representative $({\mathcal{U}},f)$, and
for any $({\mathcal{U}}^{\prime},f^{\prime})$ with dense open
${\mathcal{U}}^{\prime\prime}\subset{\mathcal{U}}\cap{\mathcal{U}}^{\prime}$
and $2$-isomorphism $\alpha\colon f|_{{\mathcal{U}}^{\prime\prime}}\Rightarrow
f^{\prime}|_{{\mathcal{U}}^{\prime\prime}}$, we have
${\mathcal{U}}^{\prime}\subset{\mathcal{U}}$, and $\alpha$ extends uniquely to
$f|_{{\mathcal{U}}^{\prime}}\Rightarrow f^{\prime}$.

###### Proof.

Let ${\mathcal{U}}$ be a dense open substack of ${\mathcal{X}}$. We start by
showing that, for $f$, $f^{\prime}\colon{\mathcal{X}}\to{\mathcal{Y}}$, any
$2$-isomorphism $f|_{{\mathcal{U}}}\Rightarrow f^{\prime}|_{{\mathcal{U}}}$
extends uniquely to a $2$-isomorphism $f\Rightarrow f^{\prime}$. A given
$2$-isomorphism $\alpha\colon f|_{{\mathcal{U}}}\Rightarrow
f^{\prime}|_{{\mathcal{U}}}$ determines a section over ${\mathcal{U}}$ of the
representable finite morphism

$\textstyle{{\mathcal{X}}\times_{{\mathcal{Y}}\times{\mathcal{Y}}}{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}_{1}}$$\textstyle{\mathcal{X}}$

where the morphisms to ${\mathcal{Y}}\times{\mathcal{Y}}$ are $(f,f^{\prime})$
and the diagonal $\Delta_{{\mathcal{Y}}}$. Since ${\mathcal{X}}$ is normal,
the closure of the section is isomorphic to ${\mathcal{X}}$. Thus $\alpha$
extends uniquely.

Since ${\mathcal{X}}$ is Noetherian, there is a maximal domain of definition
${\mathcal{U}}$ of ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$. Let
$({\mathcal{U}},f)$ be a representative, and let
$({\mathcal{U}}^{\prime},f^{\prime})$ be another representative, with dense
open
${\mathcal{U}}^{\prime\prime}\subset{\mathcal{U}}\cap{\mathcal{U}}^{\prime}$
and $2$-isomorphism $\alpha\colon f|_{{\mathcal{U}}^{\prime\prime}}\Rightarrow
f^{\prime}|_{{\mathcal{U}}^{\prime\prime}}$. By what we have shown above,
$\alpha$ extends uniquely to $\alpha^{\prime}\colon
f|_{{\mathcal{U}}\cap{\mathcal{U}}^{\prime}}\Rightarrow
f^{\prime}|_{{\mathcal{U}}\cap{\mathcal{U}}^{\prime}}$. With
$({\mathcal{U}},f)$, $({\mathcal{U}}^{\prime},f^{\prime})$, and
$\alpha^{\prime}$ we see that there is a representative with domain of
definition ${\mathcal{U}}\cup{\mathcal{U}}^{\prime}$, so by maximality we must
have ${\mathcal{U}}^{\prime}\subset{\mathcal{U}}$, and we have unique
$\alpha^{\prime}\colon f|_{{\mathcal{U}}^{\prime}}\Rightarrow f^{\prime}$
extending $\alpha$. ∎

######  Remark 4.6.

Let the notation be as in Proposition 4.5. Then
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ has maximal domain of definition
equal to ${\mathcal{X}}$ if and only if the rational map lies in the essential
image of the fully faithful functor

$\operatorname{HOM}({\mathcal{X}},{\mathcal{Y}})\to\operatorname{MAP}({\mathcal{X}},{\mathcal{Y}}).$

To indicate this situation we permit ourselves to employ the descriptive
phrase _is a morphism_ , and also to stipulate a further property, e.g., _is a
proper morphism_.

######  Corollary 4.7.

Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be stacks, with ${\mathcal{X}}$
normal, ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ a rational map,
${\mathcal{U}}$ the maximal domain of definition, and
$g\colon{\mathcal{Z}}\to{\mathcal{X}}$ an étale morphism of stacks. Then
$g^{-1}({\mathcal{U}})$ is the maximal domain of definition of the composite
rational map ${\mathcal{Z}}\dashrightarrow{\mathcal{Y}}$.

###### Proof.

There is no loss of generality in supposing $g$ surjective. The maximal domain
of definition of ${\mathcal{Z}}\dashrightarrow{\mathcal{Y}}$ contains
$g^{-1}({\mathcal{U}})$. So it suffices to show that if
${\mathcal{Z}}\dashrightarrow{\mathcal{Y}}$ is a morphism, then so is
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$.

First, we suppose ${\mathcal{Z}}=Z$, a scheme, so, an étale atlas for
${\mathcal{X}}$. Then $R=Z\times_{{\mathcal{X}}}Z$ is also a scheme, and with
its projection morphisms we have the structure of groupoid in schemes
$R\rightrightarrows Z$ that recovers ${\mathcal{X}}$ in the sense that a
naturally defined prestack $[R\rightrightarrows Z]^{\mathrm{pre}}$ has
${\mathcal{X}}$ as stackification [51]. The two composite morphisms
$R\rightrightarrows Z\to{\mathcal{Y}}$, restricted to
${\mathcal{U}}\times_{{\mathcal{X}}}R$, are canonically identified by the
identification of the latter with
$g^{-1}({\mathcal{U}})\times_{{\mathcal{U}}}g^{-1}({\mathcal{U}})$, hence by
Proposition 4.5 are themselves canonically $2$-isomorphic. The object of
${\mathcal{Y}}$ over $Z$, corresponding to $Z\to{\mathcal{Y}}$, and the
isomorphism of the two pullbacks to $R$, given by the $2$-isomorphism, satisfy
the compatibility condition (commutative diagram of isomorphisms in
${\mathcal{Y}}$ over a fiber product $R\times_{Z}R$) to determine a morphism
$[R\rightrightarrows Z]^{\mathrm{pre}}\to{\mathcal{Y}}$ (again by the
uniqueness assertion of Proposition 4.5). We obtain
${\mathcal{X}}\to{\mathcal{Y}}$ by the stackification property.

The general case follows by taking an étale atlas $V\to{\mathcal{Z}}$ and
applying the previous case to the composite $V\to{\mathcal{X}}$. ∎

######  Corollary 4.8.

Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be stacks, with ${\mathcal{X}}$
normal. Every birational map ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ has a
well-defined maximal open ${\mathcal{U}}\subset{\mathcal{X}}$, mapped
isomorphically by a representative morphism to a dense open substack of
${\mathcal{Y}}$. If ${\mathcal{U}}^{\prime}\subset{\mathcal{X}}$ is open and a
representive morphism maps ${\mathcal{U}}^{\prime}$ isomorphically to a dense
open substack of ${\mathcal{Y}}$, then
${\mathcal{U}}^{\prime}\subset{\mathcal{U}}$.

###### Proof.

Nothing changes if we replace ${\mathcal{Y}}$ by its normal locus, so we may
suppose ${\mathcal{Y}}$ normal as well. Since ${\mathcal{X}}$ is Noetherian, a
maximal ${\mathcal{U}}$ exists. Let $({\mathcal{U}},f)$ be a representative,
where $f$ maps ${\mathcal{U}}$ isomorphically to
${\mathcal{V}}\subset{\mathcal{Y}}$, and let
$({\mathcal{U}}^{\prime},f^{\prime})$ be another representative, such that
$f^{\prime}$ maps ${\mathcal{U}}^{\prime}$ isomorphically to
${\mathcal{V}}^{\prime}\subset{\mathcal{Y}}$. We apply Proposition 4.5 to
obtain a representative with domain of definition
${\mathcal{U}}\cup{\mathcal{U}}^{\prime}$ extending $f$ and $f^{\prime}$, for
the inverse rational map a representative with domain of definition
${\mathcal{V}}\cup{\mathcal{V}}^{\prime}$ extending $f^{-1}$ and
$f^{\prime-1}$, and $2$-isomorphisms from the composite in either order to the
identity. So, by maximality we have
${\mathcal{U}}^{\prime}\subset{\mathcal{U}}$. ∎

######  Lemma 4.9.

Let ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ be a proper birational morphism
of stacks that is a coarsening morphism, with ${\mathcal{X}}$ smooth, and let
${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$ be a morphism of stacks. Suppose that
the corresponding rational map ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$
admits a representative $({\mathcal{U}},f)$, such that the complement of
${\mathcal{U}}$ has everywhere codimension $\geq 2$. Then
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is a morphism.

###### Proof.

First we treat the case that ${\mathcal{X}}=X$ is a scheme, and
${\mathcal{Y}}=[W/G]$ for some affine $W$ and finite group $G$. The morphism
$f$ determines a $G$-torsor $E\to{\mathcal{U}}$ and $G$-equivariant morphism
$E\to W$. By Zariski-Nagata purity, $E$ extends to a $G$-torsor on $X$, and
the regular functions defining the morphism to $W$ extend as well, so in this
case the result is established without even requiring to have a morphism
${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$.

Next, we treat the case that ${\mathcal{X}}=X$ is a scheme and ${\mathcal{Y}}$
is general. Without loss of generality we may suppose that ${\mathcal{U}}$ is
the maximal domain of definition and argue by contradiction, taking $x$ to be
a point of the complement of ${\mathcal{U}}$, above $x$ the point $x^{\prime}$
of ${\mathcal{X}}^{\prime}$, mapping to the point $y$ of ${\mathcal{Y}}$.
Letting $Y$ denote the coarse moduli space of ${\mathcal{Y}}$, there is an
étale neighborhood $Z\to Y$ of the image in $Y$ of $y$, such that
${\mathcal{Y}}\times_{Y}Z\cong[W/G]$ for some $W$ and $G$ as above (by
construction of $Y$ [26]). Since $X$ is the coarse moduli space of
${\mathcal{X}}^{\prime}$ the morphism ${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$
determines $X\to Y$. By the previous case,
$X\times_{Y}Z\dashrightarrow{\mathcal{Y}}\times_{Y}Z$ is a morphism, hence as
well $X\times_{Y}Z\dashrightarrow{\mathcal{Y}}$ is a morphism. By Corollary
4.7, the maximal domain of definition of $X\dashrightarrow{\mathcal{Y}}$
contains $x$, and we have a contradiction.

Finally, we treat the general case. Let $V\to{\mathcal{X}}$ be an étale atlas,
so $V$ is the coarse moduli space of
$V\times_{{\mathcal{X}}}{\mathcal{X}}^{\prime}$. By the previous case,
$V\dashrightarrow{\mathcal{Y}}$ is a morphism. We conclude by Corollary 4.7. ∎

The composition of a pair of rational maps

${\mathcal{X}}\dashrightarrow{\mathcal{Y}}\qquad\text{and}\qquad{\mathcal{Y}}\dashrightarrow{\mathcal{Z}}$

is defined, provided ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is dominant.

######  Proposition 4.10.

Let ${\mathcal{X}}$, ${\mathcal{Y}}$, and ${\mathcal{Z}}$ be stacks,
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ a dominant rational map,
${\mathcal{Y}}\dashrightarrow{\mathcal{Z}}$ a rational map, and
${\mathcal{X}}\dashrightarrow{\mathcal{Z}}$ the composite rational map. If any
two of the rational maps

${\mathcal{X}}\dashrightarrow{\mathcal{Y}},\qquad{\mathcal{Y}}\dashrightarrow{\mathcal{Z}},\qquad{\mathcal{X}}\dashrightarrow{\mathcal{Z}}$

are proper, then so is the third.

###### Proof.

Suppose, first, ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ and
${\mathcal{Y}}\dashrightarrow{\mathcal{Z}}$ are proper. Let
$({\mathcal{U}},f)$ and ${\mathcal{X}}^{\prime}$ be as in Definition 4.1(v),
and let $({\mathcal{V}},g)$ and ${\mathcal{Y}}^{\prime}$ play the analogous
role for ${\mathcal{Y}}\dashrightarrow{\mathcal{Z}}$; without loss of
generality we have $f({\mathcal{U}})\subset{\mathcal{V}}$. Then, with the
stack-theoretic closure of ${\mathcal{U}}$ in
${\mathcal{X}}^{\prime}\times_{{\mathcal{Y}}}{\mathcal{Y}}^{\prime}$ (closed
substack, defined in an altas by the scheme-theoretic closure), we see that
${\mathcal{X}}\dashrightarrow{\mathcal{Z}}$ is proper.

For the remainder of the proof, we impose the additional hypothesis that
${\mathcal{X}}$ and ${\mathcal{Y}}$ are reduced, which we may do without loss
of generality.

Suppose that ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ and
${\mathcal{X}}\dashrightarrow{\mathcal{Z}}$ are proper. We take
${\mathcal{X}}^{\prime}$ as above, let ${\mathcal{X}}^{\prime\prime}$ play the
analogous role for ${\mathcal{X}}\dashrightarrow{\mathcal{Z}}$, and suppose
that respective representatives have a common domain of definition
${\mathcal{U}}$. Now we consider the closure of ${\mathcal{U}}$ in
${\mathcal{X}}^{\prime}\times_{{\mathcal{X}}}{\mathcal{X}}^{\prime\prime}$,
which we denote by ${\mathcal{X}}^{\prime\prime\prime}$. The composite
morphisms from ${\mathcal{X}}^{\prime\prime\prime}$ to ${\mathcal{Y}}$ and
${\mathcal{Z}}$ induce a morphism to ${\mathcal{Y}}\times{\mathcal{Z}}$, which
factors through $\Gamma({\mathcal{Y}}\dashrightarrow{\mathcal{Z}})$. We see as
in the proof of Proposition 4.3(i) that
$\Gamma({\mathcal{Y}}\dashrightarrow{\mathcal{Z}})\to{\mathcal{Y}}$ and
$\Gamma({\mathcal{Y}}\dashrightarrow{\mathcal{Z}})\to{\mathcal{Z}}$ are
proper. So ${\mathcal{Y}}\dashrightarrow{\mathcal{Z}}$ is proper by
Proposition 4.3(i).

It remains to show that if ${\mathcal{X}}\dashrightarrow{\mathcal{Z}}$ and
${\mathcal{Y}}\dashrightarrow{\mathcal{Z}}$ are proper, then
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is proper. It suffices to treat
the case that we have proper morphisms $h\colon{\mathcal{X}}\to{\mathcal{Z}}$
and $g\colon{\mathcal{Y}}\to{\mathcal{Z}}$. Let $({\mathcal{U}},f)$ be a
representative for ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$, for which we
have a $2$-isomorphism from $g\circ f$ to
$h|_{{\mathcal{U}}}\colon{\mathcal{U}}\to{\mathcal{Z}}$. The latter determines

${\mathcal{U}}\to{\mathcal{X}}\times_{{\mathcal{Z}}}{\mathcal{Y}}\cong({\mathcal{X}}\times{\mathcal{Y}})\times_{{\mathcal{Z}}\times{\mathcal{Z}}}{\mathcal{Z}}.$

The diagonal of ${\mathcal{Z}}$ is finite, hence so is
${\mathcal{X}}\times_{{\mathcal{Z}}}{\mathcal{Y}}\to{\mathcal{X}}\times{\mathcal{Y}}$,
and the closure ${\mathcal{V}}$ of the image of ${\mathcal{U}}$ in
${\mathcal{X}}\times_{{\mathcal{Z}}}{\mathcal{Y}}$ has a finite surjective
morphism to
$\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})\subset{\mathcal{X}}\times{\mathcal{Y}}$.
The morphisms ${\mathcal{V}}\to{\mathcal{X}}$ and
${\mathcal{V}}\to{\mathcal{Y}}$ are proper, hence so are
$\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})\to{\mathcal{X}}$ and
$\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})\to{\mathcal{Y}}$, and
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is proper by Proposition 4.3(i). ∎

######  Lemma 4.11.

Let $f\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ be a representable proper
birational morphism of smooth stacks, restricting to an isomorphism
${\mathcal{U}}^{\prime}\cong{\mathcal{U}}$ of dense open substacks. Then there
exist smooth stacks $\mathcal{X}_{0}$, $\dots$, $\mathcal{X}_{m}$, for some
$m$, such that ${\mathcal{X}}_{0}={\mathcal{X}}^{\prime}$,
${\mathcal{X}}_{m}={\mathcal{X}}$, and for every $i$ there is a morphism
$\mathcal{X}_{i+1}\to\mathcal{X}_{i}$ or $\mathcal{X}_{i}\to\mathcal{X}_{i+1}$
given by blowing up a smooth substack:

${\mathcal{X}}_{i+1}\cong
B\ell_{{\mathcal{Z}}_{i}}{\mathcal{X}}_{i}\to{\mathcal{X}}_{i},\quad\text{resp.}\quad{\mathcal{X}}_{i}\cong
B\ell_{{\mathcal{W}}_{i+1}}{\mathcal{X}}_{i+1}\to{\mathcal{X}}_{i+1}.$

We require compatibility with the isomorphism $f$: starting with
${\mathcal{U}}_{0}={\mathcal{U}}^{\prime}$ and continuing to
${\mathcal{U}}_{m}={\mathcal{U}}$, we have

${\mathcal{U}}_{i}\cap{\mathcal{Z}}_{i}=\emptyset,{\mathcal{U}}_{i+1}={\mathcal{X}}_{i+1}\times_{{\mathcal{X}}_{i}}{\mathcal{U}}_{i},\,\text{resp.}\,{\mathcal{U}}_{i+1}\cap{\mathcal{W}}_{i+1}=\emptyset,{\mathcal{U}}_{i}={\mathcal{X}}_{i}\times_{{\mathcal{X}}_{i+1}}{\mathcal{U}}_{i+1},$

for all $i$, and the composite isomorphism

${\mathcal{U}}^{\prime}={\mathcal{U}}_{0}\cong\dots\cong{\mathcal{U}}_{m}={\mathcal{U}}$

is $2$-isomorphic to the restriction of $f$. Furthermore, for every $i$ the
rational map ${\mathcal{X}}_{i}\dashrightarrow{\mathcal{X}}$represented by
$({\mathcal{U}}_{i},{\mathcal{U}}_{i}\cong\dots\cong{\mathcal{U}})$ is a
representable proper morphism.

###### Proof.

If $f$ is an isomorphism, then we are done with $m=1$ and
${\mathcal{Z}}_{0}=\emptyset$, so we suppose the contrary. We get the result
by applying _weak factorization_ of Abramovich and Temkin [4] to $f$. By [4,
§1.6] there is a coherent sheaf of ideals on ${\mathcal{X}}$, defining a
closed substack disjoint from ${\mathcal{U}}$, such that the associated blow-
up ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ factors up to
$2$-isomorphism through $f$, and ${\mathcal{X}}^{\prime\prime}$ is also the
blow-up of a coherent sheaf of ideals on ${\mathcal{X}}^{\prime}$. We use
resolution of singularities to obtain
${\mathcal{X}}^{\prime\prime\prime}\to{\mathcal{X}}^{\prime\prime}$, with
${\mathcal{X}}^{\prime\prime\prime}$ smooth. The morphism
${\mathcal{X}}^{\prime\prime\prime}\to{\mathcal{X}}$ is a composite of blow-
ups, so by [42, (5.1.4)] (really, its proof, carried over to the setting of
stacks using [35, (15.5)] for the needed existence of coherent sheaves), there
is a coherent sheaf of ideals on ${\mathcal{X}}$ whose blow-up yields
${\mathcal{X}}^{\prime\prime\prime}$, and the same holds with
${\mathcal{X}}^{\prime}$ in place of ${\mathcal{X}}$. We apply [4, Thm. 6.1.3]
to obtain weak factorizations ${\mathcal{X}}^{\prime}=\mathcal{X}_{0}$,
$\dots$, $\mathcal{X}_{c}={\mathcal{X}}^{\prime\prime\prime}$ and
$\mathcal{X}_{c}$, $\dots$, $\mathcal{X}_{m}={\mathcal{X}}$. By condition (5)
of [4, §1.2], the $\mathcal{X}_{i}\dashrightarrow{\mathcal{X}}^{\prime}$ for
$i\leq c$ and the $\mathcal{X}_{i}\dashrightarrow{\mathcal{X}}$ for $i\geq c$
are projective morphisms. ∎

######  Lemma 4.12.

Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be stacks, with ${\mathcal{X}}$
smooth, and let

${\mathcal{X}}^{\prime}\stackrel{{\scriptstyle
f}}{{\longrightarrow}}{\mathcal{X}}\stackrel{{\scriptstyle
g}}{{\longrightarrow}}{\mathcal{Y}}$

be morphisms, such that $f$ is representable, proper, and birational. Then $g$
is representable if and only if $g\circ f$ is representable.

###### Proof.

Since a composite of representable morphisms is representable, the only
nontrivial assertion is that representability of $g\circ f$ implies
representability of $g$. By resolution of singularities, we may suppose that
${\mathcal{X}}^{\prime}$ is smooth. We apply Lemma 4.11 to $f$ and, with the
factorization via ${\mathcal{X}}_{i}$, $i=0$, $\dots$, $m$, we see that it
suffices to deduce representability of ${\mathcal{X}}_{i+1}\to{\mathcal{Y}}$
from representability of ${\mathcal{X}}_{i}\to{\mathcal{Y}}$, for every $i$.
This is an instance of the result we are trying to prove, specifically when
$f$ is the blowing up of a smooth substack of ${\mathcal{X}}$. So it suffices
to treat the case $\mathcal{X}^{\prime}\cong B\ell_{\mathcal{Z}}\mathcal{X}$,
which we do by contradiction. Suppose $g$ is not representable. Then, for some
$k$-point $x$ of $\mathcal{X}$, mapping to $k$-point $y$ of $\mathcal{Y}$,
there is $1\neq\gamma\in G_{x}$, such that $\gamma$ is sent by $g$ to the
identity element of $G_{y}$. If $x\notin\mathcal{Z}$, then
$G_{x}=G_{x^{\prime}}$, where $f(x^{\prime})=x$, and $g\circ f$ is not
representable. If $x\in\mathcal{Z}$, then the normal bundle
$\mathcal{N}_{\mathcal{Z}/\mathcal{X}}$ gives rise to a representation of the
cyclic group $\langle\gamma\rangle$, which splits as a sum of one-dimensional
representations. The fiber under $f$ is the projectivization, and to each one-
dimensional representation there corresponds a point whose stabilizer group
contains $\gamma$. So $g\circ f$ is not representable. ∎

######  Definition 4.13.

Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be stacks, with ${\mathcal{X}}$
smooth. We say that a rational map ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$
is _representable_ if there exist a representative $({\mathcal{U}},f)$, a
representable birational morphism ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$,
and a representable morphism ${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$, such
that $f$ factors up to $2$-isomorphism as (4.1), and the induced morphism
${\mathcal{X}}^{\prime}\to\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$
is proper.

######  Example 4.14.

Let ${\mathcal{X}}$, ${\mathcal{Y}}$, and ${\mathcal{Z}}$ be stacks, and let

$g\colon{\mathcal{X}}\to{\mathcal{Z}}\qquad\text{and}\qquad
h\colon{\mathcal{Y}}\to{\mathcal{Z}}$

be representable morphisms. Suppose that ${\mathcal{X}}$ is smooth. Then any
rational map ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$, such that the
composite with $h$ is $2$-isomorphic to $g$, is representable. (For a
representative $({\mathcal{U}},f)$, the closure of the graph in
${\mathcal{X}}\times_{{\mathcal{Z}}}{\mathcal{Y}}$ supplies
${\mathcal{X}}^{\prime}$ as in Definition 4.13.) The particular case
${\mathcal{Z}}=BG$ tells us that $G$-equivariant rational maps of smooth
varieties determine representable maps of quotient stacks.

######  Proposition 4.15.

Let ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ be a rational map of stacks,
with ${\mathcal{X}}$ smooth. Then ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$
is representable if and only if the following two conditions are satisfied:

  * $\mathrm{(i)}$

There exist a representative $({\mathcal{U}},f)$, representable birational
morphism ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$, and morphism
${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$, such that ${\mathcal{X}}^{\prime}$
is smooth, $f$ factors up to $2$-isomorphism as (4.1), and the induced
morphism
${\mathcal{X}}^{\prime}\to\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$
is representable and proper.

  * $\mathrm{(ii)}$

For one or, equivalently, every representative and pair of morphisms in
$\mathrm{(i)}$, the morphism ${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$ is
representable.

###### Proof.

Suppose that the rational map ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is
representable. In Definition 4.13, there is no loss of generality in supposing
${\mathcal{X}}^{\prime}$ to be reduced, and by resolution of singularities, we
may further suppose that ${\mathcal{X}}^{\prime}$ is smooth. Conditions (i)
and (ii) are satisfied.

If we have $({\mathcal{U}},f)$, ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ and
${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$ satisfying (i) and (ii), then
Definition 4.13 is satisfied, so ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$
is representable. It remains only to verify that if statement (ii) holds for
one representative and pair of morphisms in (i), then it holds for every
representative and pair of morphisms. Given a representative
$({\mathcal{U}},f)$ and pair of morphisms
${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ and
${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$ as in (i), with
${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$ representable, let
${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ and
${\mathcal{X}}^{\prime\prime}\to{\mathcal{Y}}$ be another pair of morphisms as
in (i), where the representative may be taken as well to be
$({\mathcal{U}},f)$, after possibly shrinking ${\mathcal{U}}$. We define
${\mathcal{X}}^{\prime\prime\prime}$ by applying resolution of singularities
to the closure of ${\mathcal{U}}$ in
${\mathcal{X}}^{\prime}\times_{{\mathcal{X}}}{\mathcal{X}}^{\prime\prime}$.
Now ${\mathcal{X}}^{\prime\prime\prime}\to{\mathcal{Y}}$ is representable,
since this factors up to $2$-isomorphism through ${\mathcal{X}}^{\prime}$.
Applying Lemma 4.12 to
${\mathcal{X}}^{\prime\prime\prime}\to{\mathcal{X}}^{\prime\prime}$ and
${\mathcal{X}}^{\prime\prime}\to{\mathcal{Y}}$, we deduce that the latter is
representable. ∎

######  Remark 4.16.

We give another criterion for representability of
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$. Let $({\mathcal{U}},f)$ be a
representative, where ${\mathcal{U}}$ is chosen to have affine coarse moduli
space. From the fact that ${\mathcal{X}}$ and ${\mathcal{Y}}$ have finite
diagonal, we deduce that the inclusion $j\colon{\mathcal{U}}\to{\mathcal{X}}$
and the induced
$(j,f)\colon{\mathcal{U}}\to\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$
are affine morphisms. The integral closure of
$\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$ relative to
$(j,f)_{*}{\mathcal{O}}_{{\mathcal{U}}}$ (a local construction for stacks [35,
Chap. 14]) is a normal stack ${\mathcal{W}}$ with finite morphism to
$\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$ and dense open substack
isomorphic to ${\mathcal{U}}$. We may see:

  * •

A different choice of ${\mathcal{U}}$ leads to the same ${\mathcal{W}}$ (up to
canonical isomorphism).

  * •

For any normal stack ${\mathcal{X}}^{\prime}$ with birational morphism to
${\mathcal{X}}$ and morphism to ${\mathcal{Y}}$, such that $f$ factors up to
$2$-isomorphism as (4.1), the induced morphism
${\mathcal{X}}^{\prime}\to\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$
factors up to $2$-morphism through ${\mathcal{W}}$.

With these facts, we see: ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is
representable if and only if, for some or, equivalently, any smooth
$\widetilde{{\mathcal{W}}}$ with representable proper birational morphism to
${\mathcal{W}}$, the composite morphisms

$\widetilde{{\mathcal{W}}}\to\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})\to{\mathcal{X}}\qquad\text{and}\qquad\widetilde{{\mathcal{W}}}\to\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})\to{\mathcal{Y}}$

are representable.

######  Example 4.17.

In Definition 4.13 it is not possible to drop the smoothness hypothesis, or
even relax smoothness to normality. Fix $\mathcal{Y}=\operatorname{Spec}(k)$,
and consider an elliptic curve $(E,\infty)$ with a $2$-torsion point $P\in
E(k)$. Let translation by $P$ define an action of
$G={\mathbb{Z}}/2{\mathbb{Z}}$ on $E$, and let
$L={\mathcal{O}}_{E}(-2[P]-2[\infty])$. The $G$-action extends to an action on
$L$ and its compactification
$X^{\prime}={\mathbb{P}}(L\oplus{\mathcal{O}}_{E})$, and to an action on the
variety $X$ with elliptic singularity that we obtain by contracting the zero-
section of $L$ in $X^{\prime}$. The $G$-action on $X^{\prime}$ is free, so

${\mathcal{X}}^{\prime}=[X^{\prime}/G]$

is isomorphic to a smooth projective variety. The contraction morphism is
$G$-equivariant and thus determines a representable proper morphism from
${\mathcal{X}}^{\prime}$ to

${\mathcal{X}}=[X/G].$

The stack ${\mathcal{X}}$ is normal and has stabilizer $G$ at its singular
point. Thus we have a morphism ${\mathcal{X}}\to{\mathcal{Y}}$ that is not
representable but admits a factorization (4.1), satisfying the conditions of
Proposition 4.15(i), with ${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$
representable.

######  Proposition 4.18.

Let ${\mathcal{X}}$, ${\mathcal{Y}}$, and ${\mathcal{Z}}$ be stacks, with
${\mathcal{X}}$ and ${\mathcal{Y}}$ smooth. Let
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ be a dominant rational map and
${\mathcal{Y}}\dashrightarrow{\mathcal{Z}}$ a rational map.

  * $\mathrm{(i)}$

If ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ and
${\mathcal{Y}}\dashrightarrow{\mathcal{Z}}$ are representable and at least one
of them is proper, then the composite
${\mathcal{X}}\dashrightarrow{\mathcal{Z}}$ is representable.

  * $\mathrm{(ii)}$

If ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is a representable morphism and
${\mathcal{Y}}\dashrightarrow{\mathcal{Z}}$ is representable, then the
composite ${\mathcal{X}}\dashrightarrow{\mathcal{Z}}$ is representable.

  * $\mathrm{(iii)}$

If the composite ${\mathcal{X}}\dashrightarrow{\mathcal{Z}}$ is proper and
representable, and ${\mathcal{Y}}\dashrightarrow{\mathcal{Z}}$ is
representable, then ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is
representable.

###### Proof.

For (i), we take $({\mathcal{U}},f)$ and ${\mathcal{X}}^{\prime}$ with
morphisms to ${\mathcal{X}}$ and ${\mathcal{Y}}$ as in Definition 4.13 and,
analogously, $({\mathcal{V}},g)$ and ${\mathcal{Y}}^{\prime}$ with morphisms
to ${\mathcal{Y}}$ and ${\mathcal{Z}}$, for
${\mathcal{Y}}\dashrightarrow{\mathcal{Z}}$. Without loss of generality, we
have $f({\mathcal{U}})\subset{\mathcal{V}}$. Then we have the stack-theoretic
closure of ${\mathcal{U}}$ in
${\mathcal{X}}^{\prime}\times_{{\mathcal{Y}}}{\mathcal{Y}}^{\prime}$, with
composite morphisms to ${\mathcal{X}}$ and ${\mathcal{Z}}$, exhibiting
${\mathcal{X}}\dashrightarrow{\mathcal{Z}}$ as representable.

Essentially the same argument, with ${\mathcal{X}}^{\prime}={\mathcal{X}}$,
establishes (ii).

We reduce (iii) to the case of morphisms ${\mathcal{X}}\to{\mathcal{Z}}$ and
${\mathcal{Y}}\to{\mathcal{Z}}$, which is Example 4.14. The reduction to the
case of a morphism ${\mathcal{X}}\to{\mathcal{Z}}$ is straightforward. The
further reduction to the case of a morphism ${\mathcal{Y}}\to{\mathcal{Z}}$
makes use of the $2$-commutative diagram

$\textstyle{{\mathcal{X}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}\times{\mathcal{Y}}\times{\mathcal{Z}}}$

where ${\mathcal{X}}^{\prime}$, taken smooth, is as in Definition 4.13 applied
to ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}^{\prime}$, and the bottom
morphism is induced by the graph of ${\mathcal{X}}\to{\mathcal{Z}}$ and the
projection to ${\mathcal{Y}}$. ∎

######  Example 4.19.

In the setting of Proposition 4.18, even when all the stacks are proper it is
not possible to deduce from the representability of
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ and
${\mathcal{X}}\dashrightarrow{\mathcal{Z}}$, that
${\mathcal{Y}}\dashrightarrow{\mathcal{Z}}$ is representable. (Take, e.g.,
${\mathcal{X}}={\mathcal{Z}}=\operatorname{Spec}(k)$ and ${\mathcal{Y}}=BG$
for a nontrivial finite group $G$.) Also, statements (i) and (iii) may fail if
we drop the respective properness hypothesis, and (ii) may fail if instead of
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ it is
${\mathcal{Y}}\dashrightarrow{\mathcal{Z}}$ that is assumed to be a morphism.
(For (i) and (ii), consider ${\mathcal{X}}={\mathbb{P}}(1,2)$,
${\mathcal{Y}}={\mathbb{A}}^{1}$, and ${\mathcal{Z}}={\mathbb{P}}^{1}$, for
(iii), the same with ${\mathcal{Y}}$ and ${\mathcal{Z}}$ swapped.)

######  Proposition 4.20.

Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be stacks, with ${\mathcal{X}}$
smooth, and ${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ a representable
rational map, with representative $({\mathcal{U}},f)$. Then there exist a
smooth stack ${\mathcal{X}}^{\prime}$, a representable birational morphism
${\mathcal{X}}^{\prime}\to{\mathcal{X}}$, and a representable morphism
${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$, such that $f$ factors up to
$2$-isomorphism as (4.1) and the induced morphism
${\mathcal{X}}^{\prime}\to\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$
is projective.

###### Proof.

We let ${\mathcal{Z}}=\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$ and
apply Proposition 4.3(ii) to obtain ${\mathcal{X}}^{\prime\prime}$ with
${\mathcal{X}}^{\prime\prime}\to{\mathcal{Z}}$ projective, and
${\mathcal{U}}\cong{\mathcal{X}}^{\prime\prime}\times_{{\mathcal{X}}}{\mathcal{U}}$.
Let $\mathsf{X}^{\prime\prime}\to{\mathcal{X}}$, with
${\mathcal{X}}^{\prime\prime}\to\mathsf{X}^{\prime\prime}$ and $2$-isomorphism
from ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ to the composite, be the
relative coarse moduli space. So,
${\mathcal{X}}^{\prime\prime}\to\mathsf{X}^{\prime\prime}$ is quasi-finite and
proper, and $\mathsf{X}^{\prime\prime}\to{\mathcal{X}}$ is representable. We
apply resolution of singularities to $\mathsf{X}^{\prime\prime}$ to obtain
smooth $\widetilde{\mathsf{X}}^{\prime\prime}$. We have the bottom two rows of
the diagram

$\textstyle{\widehat{{\mathcal{X}}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}^{\prime\prime}\times_{\mathsf{X}^{\prime\prime}}\widetilde{\mathsf{X}}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathsf{X}}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{X}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{Y}}$

By [3, Prop. 3.4], $\widetilde{\mathsf{X}}^{\prime\prime}\to{\mathcal{X}}$ is
the relative coarse moduli space of

${\mathcal{X}}^{\prime\prime}\times_{\mathsf{X}^{\prime\prime}}\widetilde{\mathsf{X}}^{\prime\prime}\to{\mathcal{X}}.$

The composite
$\widetilde{\mathsf{X}}^{\prime\prime}\dashrightarrow{\mathcal{Y}}$ is
representable by Proposition 4.18(ii). So, there exists a smooth stack
${\mathcal{X}}^{\prime}$ with representable birational morphism to
$\widetilde{\mathsf{X}}^{\prime\prime}$ and representable morphism to
${\mathcal{Y}}$, such that a suitable representative factors up to
$2$-isomorphism as in (4.1) and the induced
${\mathcal{X}}^{\prime}\to\widetilde{\mathsf{X}}^{\prime\prime}\times{\mathcal{Y}}$
is proper. In fact, we claim that
${\mathcal{X}}^{\prime}\to\widetilde{\mathsf{X}}^{\prime\prime}$ is proper.
For this, letting $\widehat{{\mathcal{X}}}^{\prime}$ denote the normalization
of
${\mathcal{X}}^{\prime\prime}\times_{\mathsf{X}^{\prime\prime}}{\mathcal{X}}^{\prime}$,
it suffices to show that the composite
$\widehat{{\mathcal{X}}}^{\prime}\to\widetilde{\mathsf{X}}^{\prime\prime}$ is
proper. We apply Proposition 4.5 to see that we have a $2$-commutative square

$\textstyle{\widehat{{\mathcal{X}}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}^{\prime\prime}\times_{\mathsf{X}^{\prime\prime}}\widetilde{\mathsf{X}}^{\prime\prime}\times{\mathcal{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}^{\prime\prime}\times_{\mathsf{X}^{\prime\prime}}\widetilde{\mathsf{X}}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{X}}\times{\mathcal{Y}}\times{\mathcal{Y}}}$

where the bottom morphism factors through ${\mathcal{Z}}$. The top morphism is
proper, since it is a morphism, obtained from
${\mathcal{X}}^{\prime}\to\widetilde{\mathsf{X}}^{\prime\prime}\times{\mathcal{Y}}$
via base change, composed with a normalization morphism. Also, the right-hand
morphism is proper. So the left-hand morphism is proper and remains so when
followed by the second projection morphism. The claim is justified.

By Proposition 4.2, the maximal open substack over which
${\mathcal{X}}^{\prime}\to\widetilde{\mathsf{X}}^{\prime\prime}$ is an
isomorphism, has complement everywhere of codimension $\geq 2$. This open
substack is a domain of definition for
$\widetilde{\mathsf{X}}^{\prime\prime}\dashrightarrow{\mathcal{Y}}$. So, by
Lemma 4.9, $\widetilde{\mathsf{X}}^{\prime\prime}\dashrightarrow{\mathcal{Y}}$
is a morphism. We are done, since
$\widetilde{\mathsf{X}}^{\prime\prime}\to{\mathcal{X}}$ restricts to an
isomorphism over ${\mathcal{U}}$ and
$\widetilde{\mathsf{X}}^{\prime\prime}\to{\mathcal{Z}}$ is projective. Indeed,
$\widetilde{\mathsf{X}}^{\prime\prime}\to{\mathcal{Z}}$ is proper, since
${\mathcal{X}}^{\prime\prime}\times_{\mathsf{X}^{\prime\prime}}\widetilde{\mathsf{X}}^{\prime\prime}\to{\mathcal{Z}}$
is, and projectivity follows from the fact that by [33, Lemma 2], some power
of a relatively ample line bundle for
${\mathcal{X}}^{\prime\prime}\times_{\mathsf{X}^{\prime\prime}}\widetilde{\mathsf{X}}^{\prime\prime}$
over ${\mathcal{Z}}$ descends to $\widetilde{\mathsf{X}}^{\prime\prime}$ and
is relatively ample on $\widetilde{\mathsf{X}}^{\prime\prime}$ by [20,
(2.6.2)]. ∎

Looking at Definition 4.1, we can recognize several notions of birational
equivalence of stacks and match these to notions that have appeared in the
literature, e.g., in [52], [23], [10], [5], [29]. Our choice is compatible
with [29], see also [30, Defn. 6].

######  Definition 4.21.

Two smooth stacks ${\mathcal{X}}$ and ${\mathcal{Y}}$ are _birationally
equivalent_ , written

${\mathcal{X}}\sim{\mathcal{Y}},$

if there exists a representable proper birational map from ${\mathcal{X}}$ to
${\mathcal{Y}}$. Concretely, this means that there exists a stack
${\mathcal{X}}^{\prime}$ with representable proper birational morphisms
$\sigma$ and $\tau$,

$\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{X}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
6.75002pt\raise-27.9475pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{$\textstyle{\mathcal{X}}$}}}}}}}{\hbox{\kern
23.27pt\raise-37.61108pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
45.79001pt\raise-37.61108pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{Y}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(4.2)

Smooth stacks, representable proper birational maps, and $2$-isomorphisms of
representable proper birational maps form a $2$-groupoid ($2$-category in
which every morphism is an equivalence and every $2$-morphism is invertible).
We denote by

$\operatorname{BIR}({\mathcal{X}},{\mathcal{Y}})$

the category of representable proper birational maps from ${\mathcal{X}}$ to
${\mathcal{Y}}$ and the case ${\mathcal{Y}}={\mathcal{X}}$ by

$\operatorname{BIR\,AUT}({\mathcal{X}}),$

the birational automorphism $2$-group (monoidal category in which every object
is weakly invertible and every morphism is invertible) of ${\mathcal{X}}$.

######  Remark 4.22.

By Propositions 4.10 and 4.18(i), birational equivalence in Definition 4.21 is
an equivalence relation. In the setting of (4.2), we have already observed
that without loss of generality ${\mathcal{X}}^{\prime}$ may be taken to be
smooth (by resolution of singularities). By Chow’s lemma,
${\mathcal{X}}^{\prime}$ may further be taken to be quasi-projective. We also
record the observation, that when ${\mathcal{X}}^{\prime}$ is a smooth quasi-
projective stack and ${\mathcal{Z}}$ is a stack, the following conditions on a
morphism $g\colon{\mathcal{X}}^{\prime}\to{\mathcal{Z}}$ are equivalent:

  * •

$g$ is representable and proper;

  * •

$g$ is projective.

Generally, projective morphisms are representable and proper. Now suppose that
$g$ is representable and proper. Letting $X^{\prime}$ denote the coarse moduli
space of ${\mathcal{X}}^{\prime}$, we factor $g$ as

${\mathcal{X}}^{\prime}\to{\mathcal{X}}^{\prime}\times{\mathcal{Z}}\to
X^{\prime}\times{\mathcal{Z}}\to{\mathcal{Z}}.$

The first morphism, the graph of $g$, is representable and finite. The second
morphism is quasi-finite and proper. Their composite is quasi-finite and
proper, yet is representable (since $g$ is representable), hence is a
representable finite morphism. The last morphism is quasi-projective. So $g$
is as well, and we have what we want, since a proper quasi-projective morphism
is projective. In conclusion, Definition 4.21 admits an equivalent
reformulation, where ${\mathcal{X}}^{\prime}$ is a smooth quasi-projective
stack and $\sigma$ and $\tau$ are birational projective morphisms.

######  Example 4.23.

If $\mathcal{X}$ and $\mathcal{Y}$ have dimension $1$, then they are
birationally equivalent if and only if they are isomorphic. In particular, the
birational type of an orbifold curve is the same as its isomorphism type.

######  Example 4.24.

Suppose that $k$ is algebraically closed, and let $X$ and $Y$ be nonsingular
projective varieties with faithful $G$-actions, for a finite group $G$. If $X$
and $Y$ are $G$-birationally equivalent then the corresponding orbifolds

$\mathcal{X}=[X/G]\qquad\text{and}\qquad\mathcal{Y}=[Y/G]$

are birationally equivalent. It is possible, however, for $\mathcal{X}$ and
$\mathcal{Y}$ to be birationally equivalent, even when $X$ and $Y$ are not
$G$-birationally equivalent. For instance, if $G$ is abelian and we consider
two faithful $n$-dimensional representations of $G$ and the corresponding
actions on $X=Y={\mathbb{P}}^{n}$, Reichstein and Youssin [43, Thm. 7.1] have
shown that $X$ and $Y$ are $G$-birationally equivalent if and only if the
determinants of the representations

$\omega_{X},\omega_{Y}\in\operatorname{{\textstyle\bigwedge}}^{n}G^{\vee}$

satisfy

$\omega_{X}=\pm\omega_{Y}.$

But there is always an automorphism $\iota\in\operatorname{Aut}(G)$ satisfying

$(\operatorname{{\textstyle\bigwedge}}^{n}\iota)^{*}\omega_{X}=\omega_{Y}.$

Changing a $G$-action by an automorphism does not change the isomorphism type
of the stack quotient. Consequently, for this class of actions $\mathcal{X}$
and $\mathcal{Y}$ are birationally equivalent even when $X$ and $Y$ are not
$G$-birationally equivalent.

######  Theorem 4.25.

Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be smooth stacks and
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ a representable proper birational
map, represented by $({\mathcal{U}},f)$ where $f$ maps dense open
${\mathcal{U}}\subset{\mathcal{X}}$ isomorphically to dense open
${\mathcal{V}}\subset{\mathcal{Y}}$. Then there exist smooth stacks
${\mathcal{X}}_{0}$, $\dots$, ${\mathcal{X}}_{m}$ for some $m$, such that
${\mathcal{X}}_{0}={\mathcal{X}}$, ${\mathcal{X}}_{m}={\mathcal{Y}}$, and for
every $i$ there is a morphism ${\mathcal{X}}_{i+1}\to{\mathcal{X}}_{i}$ or
${\mathcal{X}}_{i}\to{\mathcal{X}}_{i+1}$, given by blowing up a smooth
substack

${\mathcal{X}}_{i+1}\cong
B\ell_{{\mathcal{Z}}_{i}}{\mathcal{X}}_{i}\to{\mathcal{X}}_{i},\quad\text{resp.}\quad{\mathcal{X}}_{i}\cong
B\ell_{{\mathcal{W}}_{i+1}}{\mathcal{X}}_{i+1}\to{\mathcal{X}}_{i+1}.$

We require compatibility with the isomorphism $f$: starting with
${\mathcal{U}}_{0}={\mathcal{U}}$ and continuing to
${\mathcal{U}}_{m}={\mathcal{V}}$, we have

${\mathcal{U}}_{i}\cap{\mathcal{Z}}_{i}=\emptyset,{\mathcal{U}}_{i+1}={\mathcal{X}}_{i+1}\times_{{\mathcal{X}}_{i}}{\mathcal{U}}_{i},\,\text{resp.}\,{\mathcal{U}}_{i+1}\cap{\mathcal{W}}_{i+1}=\emptyset,{\mathcal{U}}_{i}={\mathcal{X}}_{i}\times_{{\mathcal{X}}_{i+1}}{\mathcal{U}}_{i+1},$

for all $i$, and the composite isomorphism

${\mathcal{U}}={\mathcal{U}}_{0}\cong\dots\cong{\mathcal{U}}_{m}={\mathcal{V}}$

is $2$-isomorphic to $f$. Furthermore, there exists $0\leq c\leq m$ such that
for every $i\leq c$ the rational map
${\mathcal{X}}_{i}\dashrightarrow{\mathcal{X}}$ represented by
$({\mathcal{U}}_{i},{\mathcal{U}}_{i}\cong\dots\cong{\mathcal{U}})$ is a
projective morphism, and for every $i\geq c$ the rational map
${\mathcal{X}}_{i}\dashrightarrow{\mathcal{Y}}$ represented by
$({\mathcal{U}}_{i},{\mathcal{U}}_{i}\cong\dots\cong{\mathcal{V}})$ is a
projective morphism.

###### Proof.

We start by applying Proposition 4.20 to
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ and $({\mathcal{U}},f)$ to obtain
smooth ${\mathcal{X}}^{\prime}$, with representable birational proper morphism
to ${\mathcal{X}}$ restricting to an isomorphism over ${\mathcal{U}}$ and
representable birational proper morphism to ${\mathcal{Y}}$ resricting to an
isomorphism over ${\mathcal{V}}$. We claim that such ${\mathcal{X}}^{\prime}$
exists with the additional requirement that
${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ and
${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$ are projective morphisms, given as
the blowing up of respective coherent sheaves of ideals. Indeed, we may apply
the construction from the start of the proof of Lemma 4.11 to obtain a
coherent sheaf of ideals ${\mathcal{I}}$ on ${\mathcal{X}}$, so that the
corresponding blow-up factors up to $2$-isomorphism through
${\mathcal{X}}^{\prime}$ and is identified with the blow-up of
${\mathcal{I}}^{\prime}={\mathcal{I}}{\mathcal{O}}_{{\mathcal{X}}^{\prime}}$.
Analogously, we have a coherent sheaf of ideals ${\mathcal{J}}$ on
${\mathcal{Y}}$ and
${\mathcal{J}}^{\prime}={\mathcal{J}}{\mathcal{O}}_{{\mathcal{X}}^{\prime}}$.
Through both of these blow-ups, we have factorizations up to $2$-isomorphism
of the blow-up associated with ${\mathcal{I}}^{\prime}{\mathcal{J}}^{\prime}$,
by [42, (5.1.2)(v)]. As in the proof of Lemma 4.11 we conclude with resolution
of singularities and two applications of [4, Thm. 6.1.3]. ∎

######  Remark 4.26.

As in [31, Rmk. 3.7] we may consider snc divisors
${\mathcal{D}}\subset{\mathcal{X}}$ and ${\mathcal{E}}\subset{\mathcal{Y}}$,
disjoint from ${\mathcal{U}}$, respectively ${\mathcal{V}}$. Suppose that the
given representable proper birational map
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ restricts to a (necessarily
representable) proper birational map
${\mathcal{X}}\setminus{\mathcal{D}}\dashrightarrow{\mathcal{Y}}\setminus{\mathcal{E}}$.
Following the proof of Theorem 4.25, we have
${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ and
${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$, projective morphisms given by
blowing up coherent sheaves of ideals, and by Lemma 4.4 the pre-images of
${\mathcal{D}}$ and ${\mathcal{E}}$ in ${\mathcal{X}}^{\prime}$ coincide set-
theoretically. Applying embedded resolution of singularities to the common
pre-image, we may suppose that this is an snc divisor. Then by weak
factorization with boundary divisors [4, Thm. 6.1.3], we obtain a sequence of
blow-ups as in Theorem 4.25, where every stack ${\mathcal{X}}_{i}$ is equipped
with a compatible snc divisor ${\mathcal{D}}_{i}$, and every center of blow-up
${\mathcal{Z}}_{i}$ or ${\mathcal{W}}_{i}$ has normal crossing with
${\mathcal{D}}_{i}$.

##  5\. Divisoriality and action construction

One of the ingredients of [29] is a birational modification procedure known as
_divisorialification_ [9], [12], used to bring a given $n$-dimensional
orbifold $\mathcal{X}$ into a form, suitable for the determination of its
class

$[\mathcal{X}]\in\overline{\mathrm{Burn}}_{n}.$

Specifically, a sequence of blow-ups in smooth centers is performed, leading
to an orbifold that satisfies the following condition.

######  Definition 5.1.

We call a DM stack $\mathcal{X}$ _divisorial_ if $\mathcal{X}$ admits a
representable morphism to a stack of the form

$B{\mathbb{G}}_{m}^{r}$

for some positive integer $r$.

In other words, there should exist a finite collection $L_{1}$, $\dots$,
$L_{r}$ of line bundles on $\mathcal{X}$, which induces faithful
representations of the stabilizer groups of $\mathcal{X}$. (It suffices to
consider stabilizers at geometric points of $\mathcal{X}$.)

######  Remark 5.2.

If ${\mathcal{Y}}\to{\mathcal{X}}$ is a representable morphism and
${\mathcal{X}}$ is divisorial then so is ${\mathcal{Y}}$.

######  Proposition 5.3 (Bergh-Rydh).

Let $\mathcal{X}$ be an orbifold over $k$. Then there is a canonical sequence
of blow-ups in smooth centers

$\mathcal{Y}\to\dots\to\ \mathcal{X}$

such that $\mathcal{Y}$ is divisorial.

###### Proof.

This is proved in [12], which supplies an algorithm that takes, as input
$\mathcal{X}$ with an snc divisor
$\mathcal{D}=\mathcal{D}_{1}\cup\dots\cup\mathcal{D}_{\ell}$ and yields a
blow-up in a smooth center $\mathcal{X}^{\prime}\to\mathcal{X}$, with snc
divisor consisting of the proper transforms of the $\mathcal{D}_{i}$ and the
exceptional divisor, which decreases a numerical quantity, the maximal value
of the _divisorial index_. When the divisorial index is identically zero, we
have a divisorial orbifold. ∎

######  Remark 5.4.

Let ${\mathcal{X}}$ be an orbifold with snc divisor
$\mathcal{D}=\mathcal{D}_{1}\cup\dots\cup\mathcal{D}_{\ell}$, such that the
divisorial index is not identically zero. Then the center of blow-up
${\mathcal{Z}}\subset{\mathcal{X}}$, defined by the divisorialification
algorithm as the locus of maximal divisorial index, is smooth and meets
$\mathcal{D}$ transversally. Smoothness is established in [12, Prop.
6.5(iii)]. Transverse intersection with ${\mathcal{D}}$ follows from the
characterization [12, Rmk. 3.4] in terms of smoothness of the associated
morphism to $[{\mathbb{A}}^{\ell}/{\mathbb{G}}_{m}^{\ell}]$ and the
identification of divisorial index with the codimension of stackiness [12,
Defn. 5.8] of $X$ over $[{\mathbb{A}}^{\ell}/{\mathbb{G}}_{m}^{\ell}]$, which
we see by arguing as in the proof of [12, Prop. 6.5].

For the next statement we recall the well-known fact [40, Prop. 2.1] that
every smooth separated irreducible DM stack is a gerbe over an orbifold, i.e.,
to given $\mathcal{X}$ with geometric generic stabilizer $H$ there is an
orbifold $\mathcal{Y}$ with morphism

$\mathcal{X}\to\mathcal{Y},$ (5.1)

which étale locally over $\mathcal{Y}$ is the projection from a product with
$BH$; in particular, the morphism (5.1) is smooth. As recalled in the proof of
[31, Lemma 7.3], a line bundle on $\mathcal{X}$ descends to $\mathcal{Y}$ if
and only if it induces the trivial representation of $H$.

######  Proposition 5.5.

Suppose that $\mathcal{X}$ is a smooth separated irreducible DM stack of
finite type over $k$, and suppose that $k$ contains all roots of unity. The
following are equivalent:

  * (1)

$\mathcal{X}$ is divisorial.

  * (2)

The orbifold $\mathcal{Y}$, associated to $\mathcal{X}$ as in (5.1), is
divisorial, and a dense open substack of $\mathcal{X}$ is isomorphic to
$\mathcal{U}\times BH$, for some dense open $\mathcal{U}\subset\mathcal{Y}$
and finite abelian group $H$.

  * (3)

There exist a divisorial orbifold $\mathcal{Y}$, line bundles $N_{1}$,
$\dots$, $N_{s}$ on $\mathcal{Y}$ and positive integers $n_{1}$, $\dots$,
$n_{s}$, such that

$\mathcal{X}\cong\sqrt[n_{1}]{N_{1}/\mathcal{Y}}\times_{\mathcal{Y}}\dots\times_{\mathcal{Y}}\sqrt[n_{s}]{N_{s}/\mathcal{Y}}.$

###### Proof.

To show (1) $\Rightarrow$ (2), we suppose that $\mathcal{X}$ is divisorial,
with line bundles $L_{1}$, $\dots$, $L_{r}$ inducing faithful representations
of stabilizer groups, and let $\mathcal{X}\to\mathcal{Y}$ and $H$ be as in
(5.1). Since divisorial stacks have abelian stabilizer groups, $H$ is abelian.
Since $\mathcal{Y}$ has trivial generic stabilizer, it has a dense open
substack $\mathcal{U}$, isomorphic to a scheme. We write

$H\cong{\mathbb{Z}}/n_{1}{\mathbb{Z}}\times\dots\times{\mathbb{Z}}/n_{s}{\mathbb{Z}}$
(5.2)

according to the structure theorem of finite abelian groups (that is,
$n_{i}\geq 2$ for all $i$, with $n_{i}\mid n_{i+1}$) and fix a primitive
$n_{i}$th root of unity $\lambda_{i}$ for every $i$. For each $1\leq i\leq s$
some tensor combination $M_{i}$ of $L_{1}$, $\dots$, $L_{r}$ induces the
character

$H\stackrel{{\scriptstyle\mathrm{pr}_{i}}}{{\longrightarrow}}{\mathbb{Z}}/n_{i}{\mathbb{Z}}\stackrel{{\scriptstyle\lambda_{i}}}{{\longrightarrow}}k^{\times}.$
(5.3)

So $M_{i}^{n_{i}}$ induces the trivial character of $H$, hence descends to a
line bundle $N_{i}$ on $\mathcal{Y}$. Shrinking $\mathcal{U}$, we may suppose
that the line bundles $N_{1}$, $\dots$, $N_{s}$ are all trivial. Choosing
trivializations, we get for each $i$ a morphism

$\mathcal{U}\times_{\mathcal{Y}}\mathcal{X}\to
B{\mathbb{Z}}/n_{i}{\mathbb{Z}}$

from the identification of $B{\mathbb{Z}}/n_{i}{\mathbb{Z}}$ with
$B\mu_{n_{i}}$ (using the chosen root of unity $\lambda_{i}$ to identify
${\mathbb{Z}}/n_{i}{\mathbb{Z}}$ with $\mu_{n_{i}}$) and the fiber square

$\textstyle{B\mu_{n_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B{\mathbb{G}}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({-})^{n_{i}}}$$\textstyle{\operatorname{Spec}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B{\mathbb{G}}_{m}}$

Combined, these lead to an isomorphism

$\mathcal{U}\times_{\mathcal{Y}}\mathcal{X}\to\mathcal{U}\times
B{\mathbb{Z}}/n_{1}{\mathbb{Z}}\times\dots\times
B{\mathbb{Z}}/n_{s}{\mathbb{Z}}\cong\mathcal{U}\times BH.$

Furthermore, for each $1\leq j\leq r$ we find a suitable tensor combination
$P_{j}$ of $M_{1}$, $\dots$, $M_{s}$, that induces the same character of $H$
as $L_{j}$. Consequently $L_{j}\otimes P_{j}^{-1}$ descends to $\mathcal{Y}$,
i.e., is isomorphic to the pullback of some line bundle $Q_{j}$ on
$\mathcal{Y}$. Using that $L_{1}$, $\dots$, $L_{r}$ induce faithful
representations of the stabilizer groups of $\mathcal{X}$, we see that

$N_{1},\dots,N_{s},Q_{1},\dots,Q_{r}$

induce faithful representations of the stabilizer groups of $\mathcal{Y}$. So
$\mathcal{Y}$ is divisorial.

For (2) $\Rightarrow$ (3), we need the observation that a line bundle on an
open substack of $\mathcal{X}$ can always be extended to $\mathcal{X}$ [36,
Lemma 4.1]. Writing $H$ as in (5.2), the characters (5.3) determine line
bundles on $\mathcal{U}\times BH$, which are the restrictions of line bundles
$M_{1}$, $\dots$, $M_{s}$ on $\mathcal{X}$. Now $M_{i}^{n_{i}}$ descends to
$\mathcal{Y}$. An identification of $M_{i}^{n_{i}}$ with the pullback to
$\mathcal{X}$ of a line bundle $N_{i}$ on $\mathcal{Y}$, for every $i$, leads
to an isomorphism as in statement (3).

For (3) $\Rightarrow$ (1), the pullbacks of line bundles on $\mathcal{Y}$ and
the tautological line bundles on $\sqrt[n_{i}]{N_{i}/\mathcal{Y}}$ (coming
from the identity representation of the second factor $B{\mathbb{G}}_{m}$ in
(3.2)) exhibit $\mathcal{X}$ as divisorial. ∎

######  Corollary 5.6.

Suppose that ${\mathcal{X}}$ is a smooth separated irreducible DM stack,
${\mathcal{D}}$ is a smooth divisor on ${\mathcal{X}}$, and $r$ is a positive
integer. Then ${\mathcal{X}}$ is divisorial if and only if
$\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}$ is divisorial.

###### Proof.

There is no loss of generality in supposing that ${\mathcal{D}}$ is
irreducible. By [16, Cor. 3.1.2], every line bundle on
$\sqrt[r]{({\mathcal{X}},{\mathcal{D}})}$ is a twist by a power of
${\mathcal{O}}({\mathcal{G}}_{{\mathcal{D}}})$ of a line bundle, pulled back
from ${\mathcal{X}}$. The result follows quickly from this fact. ∎

Unlike in the case of orbifolds, a general smooth separated Deligne-Mumford
stack does not become divisorial after suitable blow-ups. Indeed, condition
(2) of Proposition 5.5, if not satisfied already by $\mathcal{X}$, will never
hold after performing blow-ups.

######  Proposition 5.7.

Suppose that $\mathcal{X}$ is a smooth separated irreducible DM stack of
finite type over $k$, where $k$ contains all roots of unity. Then there exists
a smooth separated irreducible DM stack $\mathcal{X}^{\prime}$ that is
divisorial and birationally equivalent to $\mathcal{X}$, if and only if
$\mathcal{X}$ has a dense open substack isomorphic to $U\times BH$, for some
algebraic variety $U$ and finite abelian group $H$.

In this case, suitable $\mathcal{X}^{\prime}$ may be obtained from the
orbifold $\mathcal{Y}$, associated to $\mathcal{X}$, by

  * •

applying the divisorialification procedure (Propositon 5.3) to obtain

$\mathcal{Y}^{\prime}\to\mathcal{Y},$

a composition of blow-ups in smooth centers, with $\mathcal{Y}^{\prime}$
divisorial, and

  * •

forming the fiber diagram

$\textstyle{\mathcal{X}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{Y}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{Y}}$

###### Proof.

If $\mathcal{X}^{\prime}$ is birationally equivalent to $\mathcal{X}$, then
$\mathcal{X}$ and $\mathcal{X}^{\prime}$ have dense open substacks that are
isomorphic. By condition (2) of Proposition 5.5, then, if
$\mathcal{X}^{\prime}$ is divisorial then $\mathcal{X}$ has a dense open
substack isomorphic to $\mathcal{U}^{\prime}\times BH$, for some orbifold
$\mathcal{U}^{\prime}$. There is an algebraic variety $U$, isomorphic to a
dense open in $\mathcal{U}^{\prime}$, and $\mathcal{X}$ has a dense open
substack isomorphic to $U\times BH$.

Now suppose $\mathcal{X}$ has a dense open substack isomorphic to $U\times
BH$. Let $\mathcal{Y}$ be the orbifold, associated to $\mathcal{X}$, and let
us apply Proposition 5.3 to obtain $\mathcal{Y}^{\prime}\to\mathcal{Y}$. Since
$\mathcal{X}\to\mathcal{Y}$ is smooth, $\mathcal{X}^{\prime}$ is smooth over
$\mathcal{Y}^{\prime}$ and therefore itself smooth. The morphism
$\mathcal{Y}^{\prime}\to\mathcal{Y}$ is representable, proper, and birational,
hence so is $\mathcal{X}^{\prime}\to\mathcal{X}$. Since the rigidification in
[40, Prop. 2.1] is canonical, and we have
$\mathcal{X}^{\prime}\to\mathcal{Y}^{\prime}$ étale locally projection from a
product with $BH$ and $\mathcal{Y}^{\prime}$ an orbifold;
$\mathcal{Y}^{\prime}$ is the orbifold, associated to $\mathcal{X}^{\prime}$.
By Proposition 5.5, $\mathcal{X}^{\prime}$ is divisorial. ∎

######  Definition 5.8.

Suppose that the base field $k$ contains all roots of unity. A _linearizable_
DM stack is a smooth separated irreducible DM stack, satisfying the equivalent
conditions of Proposition 5.7.

######  Remark 5.9.

Suppose that ${\mathcal{X}}$ and ${\mathcal{Y}}$ are smooth separated
irreducible DM stacks, and there exists a birational map
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$. Then ${\mathcal{X}}$ is
linearizable if and only if ${\mathcal{Y}}$ is linearizable.

######  Lemma 5.10.

Suppose that the base field $k$ contains all roots of unity. Let $\mathcal{X}$
be a linearizable DM stack, $H$ the generic stabilizer of $\mathcal{X}$, and
$\chi$ a character of $H$. Then there exists a line bundle $L$ on
$\mathcal{X}$, such that the character of $H$ induced by $L$ is $\chi$. If
$L^{\prime}$ is another such line bundle on $\mathcal{X}$, then there exists a
dense open substack $\mathcal{U}\subset\mathcal{X}$, such that the
restrictions of $L$ and $L^{\prime}$ to $\mathcal{U}$ are isomorphic.

###### Proof.

The character $\chi$ determines a line bundle on a dense open substack of
$\mathcal{X}$, isomorphic to $U\times BH$. As noted in the proof of
Proposition 5.5, any line bundle on an open substack of $\mathcal{X}$ is the
restriction of a line bundle $L$ on $\mathcal{X}$, then $L$ is as desired.

Let $L^{\prime}$ be another such line bundle. It suffices to show that
$L^{-1}\otimes L^{\prime}$ is trivial on some dense open substack of
$\mathcal{X}$, i.e., it suffices to treat the case $\chi$ is trivial and
$L^{\prime}={\mathcal{O}}_{\mathcal{X}}$. Let $\mathcal{Y}$ be the orbifold,
associated to $\mathcal{X}$. The triviality of $\chi$ implies that $L$ is
isomorphic to the pullback of a line bundle on $\mathcal{Y}$ (cf. the morphism
(5.1) and subsequent observations). Since $\mathcal{Y}$ is an orbifold there
is a dense open substack, isomorphic to an algebraic variety. We are done,
since a line bundle on an algebraic variety is Zariski locally trivial. ∎

######  Lemma 5.11.

Let $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Y}^{\prime}$ be smooth
separated irreducible DM stacks and let

$\mathcal{Y}\to\mathcal{X}\qquad\text{and}\qquad\mathcal{Y^{\prime}}\to\mathcal{X}$

be representable proper dominant morphisms, such that for some dense open
substack ${\mathcal{U}}$ of ${\mathcal{X}}$ there exists an isomorphism
${\mathcal{Y}}\times_{{\mathcal{X}}}{\mathcal{U}}\to{\mathcal{Y}}^{\prime}\times_{{\mathcal{X}}}{\mathcal{U}}$
over ${\mathcal{U}}$. Then

$\mathcal{Y}\sim\mathcal{Y^{\prime}}.$

###### Proof.

The induced birational map
${\mathcal{Y}}\dashrightarrow{\mathcal{Y}}^{\prime}$ is proper by Proposition
4.10, and representable by Example 4.14. ∎

######  Definition 5.12.

Suppose $k$ contains all roots of unity. Let $\mathcal{K}$ be a linearizable
DM stack with generic stabilizer $H$ and $a_{1}$, $\dots$, $a_{c}\in
H^{\vee}$. We take $L_{i}$ as in Lemma 5.10, for the character $a_{i}$. Now

$\mathcal{Y}:={\mathbb{P}}({\mathcal{O}}_{\mathcal{K}}\oplus
L_{1}\oplus\dots\oplus L_{c})$

is a linearizable DM stack with generic stabilizer

$\overline{H}:=\ker(a_{1})\cap\dots\cap\ker(a_{c}).$

We call this the _action construction_ (A).

Suppose, for some $i$, we replace $L_{i}$ by another line bundle
$L^{\prime}_{i}$, satisfying the requirement of Lemma 5.10 for the character
$a_{i}$. Then there is dense open $\mathcal{U}\subset\mathcal{X}$, such that
$L_{i}|_{\mathcal{U}}$ is isomorphic to $L^{\prime}_{i}|_{\mathcal{U}}$. Then,
by Lemma 5.11, the construction with $L^{\prime}_{i}$ in place of $L_{i}$
yields a birationally equivalent stack $\mathcal{Y}^{\prime}$. The _action
construction_ , performed with $\mathcal{K}$ and $a_{1}$, $\dots$, $a_{c}$,
yields the stack $\mathcal{Y}$, which is well-defined up to birational
equivalence.

######  Remark 5.13.

The birational equivalence class of $\mathcal{Y}$, in the action construction,
only depends on the birational equivalence class of $\mathcal{K}$.

######  Remark 5.14.

Continuing with the theme of Example 2.3, we explain how linearizability
impacts the identifications of stabilizer groups of a smooth separated
irreducible DM stack $\mathcal{X}$. The inertia stack (Remark 2.4)
$\mathcal{I}_{\mathcal{X}}$ is finite over $\mathcal{X}$ and has the structure
of a group object in the category of stacks with representable morphism to
$\mathcal{X}$. Restricting to the maximal open substack $\mathcal{U}$ of
$\mathcal{X}$ where $\mathcal{I}_{\mathcal{X}}\to\mathcal{X}$ has constant
degree we obtain a finite étale group object over $\mathcal{U}$, and
$\mathcal{U}$ is a gerbe, étale locally over its coarse moduli space $U$
isomorphic to a product with $BH$ for some finite group $H$, the geometric
stabilier of $\mathcal{X}$. _Suppose $H$ is abelian._ Then the phenomenon
reported in Example 2.3 at points of $\mathcal{U}$ manifests itself in finite
étale $G\to U$, with $G\times_{U}\mathcal{U}\cong\mathcal{I}_{\mathcal{U}}$.
Generally, $G\to U$ is a _twisted form_ of $H$, e.g., if
${\mathbb{Z}}/2{\mathbb{Z}}$ acts without fixed points on a smooth variety $Y$
with quotient $X$ and we let ${\mathfrak{S}}_{3}$ act on $Y$ via
${\mathfrak{S}}_{3}\to{\mathbb{Z}}/2{\mathbb{Z}}$, then for
$\mathcal{U}=\mathcal{X}=[Y/{\mathfrak{S}}_{3}]$ we obtain $G=X\sqcup Y$, a
twisted form of $H={\mathbb{Z}}/3{\mathbb{Z}}$. If $G\cong U\times H$ then the
gerbe has _trivial band_ and there is a _canonical_ identification of the
geometric stabilizer groups at any pair of points of $\mathcal{U}$. When $k$
has all roots of unity, we have implications

$\text{trivial gerbe (over $U$)}\ \Rightarrow\ \text{$\mathcal{X}$
linearizable}\ \Rightarrow\ \text{trivial band}$

and examples of possibilities for gerbe/band, when
$H={\mathbb{Z}}/m{\mathbb{Z}}$ ($m\geq 2$):

  * •

${\mathbb{P}}(m,m)$ is linearizable but as a gerbe over ${\mathbb{P}}^{1}$ is
nontrivial;

  * •

there is a Brauer group obstruction to linearizability, for a gerbe with
trivial band (cf. [24, Prop. 4]).

######  Remark 5.15.

Quasiprojective smooth DM stacks admit projective compactifications.
Specifically, by combining [28, Thm. 5.3] with Remark 3.1, we see that every
quasiprojective smooth DM stack ${\mathcal{U}}$ may be embedded as an open
substack of a projective smooth DM stack ${\mathcal{X}}$, with complement an
snc divisor. If ${\mathcal{U}}$ is divisorial, then so is ${\mathcal{X}}$ (by
the ability, mentioned in the proof of Proposition 5.5, to extend line bundles
to ${\mathcal{X}}$, and the argument of Remark 5.4).

##  6\. Weak factorization for stacks

Section 4 concluded with a weak factorization result for representable proper
birational maps of stacks. The case of birational maps that are proper but not
necessarily representable is more subtle. Its treatment requires the
_functorial destackification_ construction of Bergh [9] and Bergh-Rydh [12]
and appears in several formulations: Bergh [9, Cor. 1.5] and [10, Thm. 1.2],
Harper [22, Thm. 1.1], and Bergh-Rydh [12, Thm. D]. A proper birational map
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ is factored as a sequence of
_stacky blow-ups_ and their inverses, where a stacky blow-up is by definition
a blow-up of a smooth substack or a root operation along a smooth divisor.

We continue to work with the convention, that all stacks are DM stacks,
separated and of finite type over $k$, a field of characteristic $0$.

As remarked in [12, Sect. 1], Harper’s proof of weak factorization can be
streamlined by applying the _relative_ form of functorial destackification of
Bergh and Rydh [12, Thm. 7.1]. We recall this result, in the case that we
need.

######  Proposition 6.1 (Bergh-Rydh).

Let $f\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ be a birational morphism
of smooth stacks, restricting to an isomorphism
${\mathcal{U}}^{\prime}\cong{\mathcal{U}}$ of dense open substacks. Then there
exist smooth stacks ${\mathcal{X}}_{0}$, $\dots$, ${\mathcal{X}}_{m}$ for some
$m$, and a sequence of stacky blow-ups

$\displaystyle{\mathcal{X}}^{\prime\prime}={\mathcal{X}}_{m}\to\dots\to{\mathcal{X}}_{1}\to{\mathcal{X}}_{0}={\mathcal{X}}^{\prime},$
$\displaystyle{\mathcal{X}}_{i+1}\cong
B\ell_{{\mathcal{Z}}_{i}}{\mathcal{X}}_{i},\quad\text{resp.}\quad\sqrt[q_{i}\\!]{({\mathcal{X}}_{i},{\mathcal{Z}}_{i})},$
$\displaystyle{\mathcal{U}}_{0}={\mathcal{U}}^{\prime},\qquad{\mathcal{U}}_{i+1}={\mathcal{X}}_{i+1}\times_{{\mathcal{X}}_{i}}{\mathcal{U}}_{i},\qquad{\mathcal{U}}^{\prime\prime}={\mathcal{U}}_{m},\qquad{\mathcal{U}}_{i}\cap{\mathcal{Z}}_{i}=\emptyset,$

such that, taking ${\mathcal{X}}^{\prime\prime\prime}\to{\mathcal{X}}$ to be
the relative coarse moduli space of
${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$, the stack
${\mathcal{X}}^{\prime\prime\prime}$ is smooth, with

${\mathcal{X}}^{\prime\prime}\cong\sqrt[\mathbf{s}]{({\mathcal{X}}^{\prime\prime\prime},{\mathcal{D}}^{\prime\prime\prime})},$

iterated root stack of some snc divisor ${\mathcal{D}}^{\prime\prime\prime}$
on ${\mathcal{X}}^{\prime\prime\prime}$ and tuple of positive integers
$\mathbf{s}$, such that
${\mathcal{X}}^{\prime\prime}\times_{{\mathcal{X}}^{\prime\prime\prime}}{\mathcal{D}}^{\prime\prime\prime}$
is disjoint from ${\mathcal{U}}^{\prime\prime}$.

We restate the theorem of Bergh, Harper, and Bergh-Rydh, with attention to the
properties of the morphisms to ${\mathcal{X}}$, respectively ${\mathcal{Y}}$
from the intermediate stacks in the factorization.

######  Theorem 6.2 (Bergh, Harper, Bergh-Rydh).

Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be smooth stacks and
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ a proper birational map,
represented by $({\mathcal{U}},f)$ where $f$ maps dense open
${\mathcal{U}}\subset{\mathcal{X}}$ isomorphically to dense open
${\mathcal{V}}\subset{\mathcal{Y}}$. Then there exist smooth stacks
${\mathcal{X}}_{0}$, $\dots$, ${\mathcal{X}}_{m}$ for some $m$, with
${\mathcal{X}}_{0}={\mathcal{X}}$, ${\mathcal{X}}_{m}={\mathcal{Y}}$, and for
every $i$ a birational proper morphism
${\mathcal{X}}_{i+1}\to{\mathcal{X}}_{i}$ or
${\mathcal{X}}_{i}\to{\mathcal{X}}_{i+1}$ such that, starting from
${\mathcal{U}}_{0}={\mathcal{U}}$ and proceeding to
${\mathcal{U}}_{m}={\mathcal{V}}$, the morphism
${\mathcal{X}}_{i+1}\to{\mathcal{X}}_{i}$ or
${\mathcal{X}}_{i}\to{\mathcal{X}}_{i+1}$ identifies dense open substacks
${\mathcal{U}}_{i}\subset{\mathcal{X}}_{i}$ and
${\mathcal{U}}_{i+1}\subset{\mathcal{X}}_{i+1}$, and the composite isomorphism

${\mathcal{U}}={\mathcal{U}}_{0}\cong\dots\cong{\mathcal{U}}_{m}={\mathcal{V}}$

is $2$-isomorphic to $f$. Furthermore, there are

$0<b<c<d<m$

such that:

  * •

The rational maps ${\mathcal{X}}_{c}\dashrightarrow{\mathcal{X}}$ and
${\mathcal{X}}_{c}\dashrightarrow{\mathcal{Y}}$, represented by
$({\mathcal{U}}_{c},{\mathcal{U}}_{c}\cong\dots\cong{\mathcal{U}})$,
respectively
$({\mathcal{U}}_{c},{\mathcal{U}}_{c}\cong\dots\cong{\mathcal{V}})$, are
morphisms, and the induced morphism
${\mathcal{X}}_{c}\to\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$ is
projective.

  * •

For all $b\leq i<c$ we have ${\mathcal{X}}_{i}\to{\mathcal{X}}_{i+1}$, a
stacky blow-up

${\mathcal{X}}_{i}\cong
B\ell_{{\mathcal{W}}_{i+1}}{\mathcal{X}}_{i+1}\to{\mathcal{X}}_{i+1}\quad\text{or}\quad\sqrt[p_{i}\\!]{({\mathcal{X}}_{i+1},{\mathcal{W}}_{i+1})}\to{\mathcal{X}}_{i+1}.$

  * •

For all $c\leq i<d$ we have ${\mathcal{X}}_{i+1}\to{\mathcal{X}}_{i}$, a
stacky blow-up

${\mathcal{X}}_{i+1}\cong
B\ell_{{\mathcal{Z}}_{i}}{\mathcal{X}}_{i}\to{\mathcal{X}}_{i}\quad\text{or}\quad\sqrt[q_{i}\\!]{({\mathcal{X}}_{i},{\mathcal{Z}}_{i})}\to{\mathcal{X}}_{i}.$

  * •

We have ${\mathcal{X}}_{b}\to{\mathcal{X}}_{b-1}$, appearing in an expression
of ${\mathcal{X}}_{b-1}\to{\mathcal{X}}$ as relative coarse moduli space of
${\mathcal{X}}_{b}\to{\mathcal{X}}$, and

${\mathcal{X}}_{b}\cong\sqrt[\mathbf{r}]{({\mathcal{X}}_{b-1},{\mathcal{D}})}$

for some snc divisor ${\mathcal{D}}$ and tuple of positive integers
$\mathbf{r}$.

  * •

We have ${\mathcal{X}}_{d}\to{\mathcal{X}}_{d+1}$, appearing in an expression
of ${\mathcal{X}}_{d+1}\to{\mathcal{Y}}$ as relative coarse moduli space of
${\mathcal{X}}_{d}\to{\mathcal{Y}}$, and

${\mathcal{X}}_{d}\cong\sqrt[\mathbf{s}]{({\mathcal{X}}_{d+1},{\mathcal{E}})}$

for some snc divisor ${\mathcal{E}}$ and tuple of positive integers
$\mathbf{s}$.

  * •

For every $0\leq i<b-1$ and every $d+1\leq i<m$ the birational morphism
between ${\mathcal{X}}_{i}$ and ${\mathcal{X}}_{i+1}$ is a blow-up of a smooth
substack

${\mathcal{X}}_{i+1}\cong
B\ell_{{\mathcal{Z}}_{i}}{\mathcal{X}}_{i}\to{\mathcal{X}}_{i}\quad\text{or}\quad{\mathcal{X}}_{i}\cong
B\ell_{{\mathcal{W}}_{i+1}}{\mathcal{X}}_{i+1}\to{\mathcal{X}}_{i+1}.$

We define ${\mathcal{Z}}_{b-1}={\mathcal{D}}$ and
${\mathcal{W}}_{d+1}={\mathcal{E}}$ and require
${\mathcal{U}}_{i}\cap{\mathcal{Z}}_{i}=\emptyset$, respectively
${\mathcal{U}}_{i+1}\cap{\mathcal{W}}_{i+1}=\emptyset$ for every $i$.
Additionally, for every $0\leq i\leq b-1$, resp. $d+1\leq i\leq m$ the
rational map ${\mathcal{X}}_{i}\dashrightarrow{\mathcal{X}}$, resp.
${\mathcal{X}}_{i}\dashrightarrow{\mathcal{Y}}$ represented by
$({\mathcal{U}}_{i},{\mathcal{U}}_{i}\cong\dots\cong{\mathcal{U}})$, resp.
$({\mathcal{U}}_{i},{\mathcal{U}}_{i}\cong\dots\cong{\mathcal{V}})$, is a
representable proper morphism.

###### Proof.

We start by applying Proposition 4.3(ii) to
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$. This gives us a reduced stack
${\mathcal{X}}^{\prime}$, which by resolution of singularities we may suppose
to be smooth, with birational morphisms to ${\mathcal{X}}$ and
${\mathcal{Y}}$, restricting to isomorphisms
${\mathcal{U}}^{\prime}\to{\mathcal{U}}$ and
${\mathcal{U}}^{\prime}\to{\mathcal{V}}$, where
${\mathcal{U}}^{\prime}={\mathcal{X}}^{\prime}\times_{{\mathcal{X}}}{\mathcal{U}}$.
The morphism $f$ factors up to $2$-isomorphism as

${\mathcal{U}}\cong{\mathcal{U}}^{\prime}\to{\mathcal{Y}}.$

The induced morphism
${\mathcal{X}}^{\prime}\to\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})$
is projective, and the morphisms
$\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})\to{\mathcal{X}}$ and
$\Gamma({\mathcal{X}}\dashrightarrow{\mathcal{Y}})\to{\mathcal{Y}}$ are
proper, as are the composite morphisms
${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ and
${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$.

We apply Proposition 6.1 to ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ and to
${\mathcal{X}}^{\prime}\to{\mathcal{Y}}$. This yields sequences of stacky
blowups of ${\mathcal{X}}^{\prime}$, terminating in
${\mathcal{X}}^{\prime\prime}$, respectively ${\mathcal{Y}}^{\prime\prime}$,
and relative coarse moduli spaces

${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}^{\prime\prime\prime}\to{\mathcal{X}}\qquad\text{resp.}\qquad{\mathcal{Y}}^{\prime\prime}\to{\mathcal{Y}}^{\prime\prime\prime}\to{\mathcal{Y}}.$

Now we apply Lemma 4.11 to the representable proper birational morphisms
${\mathcal{X}}^{\prime\prime\prime}\to{\mathcal{X}}$ and
${\mathcal{Y}}^{\prime\prime\prime}\to{\mathcal{Y}}$. We relabel
${\mathcal{X}}^{\prime\prime\prime}$ as ${\mathcal{X}}_{b-1}$ and
${\mathcal{X}}^{\prime\prime}$ as ${\mathcal{X}}_{b}$, where $b-1$ is the
number of blow-ups from the application of Lemma 4.11 to
${\mathcal{X}}^{\prime\prime\prime}\to{\mathcal{X}}$, and similarly relabel
the stacky blow-ups from the application of Proposition 6.1 to
${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ as

${\mathcal{X}}_{b}\to\dots\to{\mathcal{X}}_{c}={\mathcal{X}}^{\prime}$

and in an analogous fashion obtain

${\mathcal{X}}_{d}\to\dots\to{\mathcal{X}}_{c}={\mathcal{X}}^{\prime},$

where ${\mathcal{Y}}^{\prime\prime}$ is now denoted by ${\mathcal{X}}_{d}$.
Finally, we relabel the blow-ups from the remaining application of Lemma 4.11
to obtain ${\mathcal{X}}_{d+1}$, $\dots$, ${\mathcal{X}}_{m}$, with
${\mathcal{X}}_{d+1}={\mathcal{Y}}^{\prime\prime\prime}$ and
${\mathcal{X}}_{m}={\mathcal{Y}}$. ∎

######  Proposition 6.3.

Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be smooth stacks,
${\mathcal{X}}\dashrightarrow{\mathcal{Y}}$ a proper birational map
represented by $({\mathcal{U}},f)$ where $f$ maps dense open
${\mathcal{U}}\subset{\mathcal{X}}$ isomorphically to dense open
${\mathcal{V}}\subset{\mathcal{Y}}$, and let smooth stacks
${\mathcal{X}}_{0}$, $\dots$, ${\mathcal{X}}_{m}$, and birational morphism
${\mathcal{X}}_{i+1}\to{\mathcal{X}}_{i}$ or
${\mathcal{X}}_{i}\to{\mathcal{X}}_{i+1}$ for every $i$ be as in Theorem 6.2.
Then:

  * $\mathrm{(i)}$

If ${\mathcal{X}}$ and ${\mathcal{Y}}$ are divisorial, then
${\mathcal{X}}_{i}$ is divisorial for every $i$.

  * $\mathrm{(ii)}$

If ${\mathcal{X}}$ and ${\mathcal{Y}}$ are quasi-projective (resp.
projective), then ${\mathcal{X}}_{i}$ is quasi-projective (resp. projective)
for every $i$.

###### Proof.

Let $0<b<c<d<m$ be as in Theorem 6.2. For (i), if ${\mathcal{X}}$ and
${\mathcal{Y}}$ are divisorial, then so is ${\mathcal{X}}\times{\mathcal{Y}}$,
hence also ${\mathcal{X}}_{c}$, by Remark 5.2. By Remark 5.2 and Corollary
5.6, so are the stacks ${\mathcal{X}}_{i}$ for all $b\leq i\leq d$. We have
projective morphisms to ${\mathcal{X}}$ or ${\mathcal{Y}}$ from the remaining
stacks ${\mathcal{X}}_{i}$, so these are divisorial as well by Remark 5.2.

We obtain (ii) from Lemma 2.2 and Proposition 3.4. ∎

######  Remark 6.4.

In the setting of Theorem 6.2, suppose that the complement of ${\mathcal{U}}$
is an snc divisor in ${\mathcal{X}}$, and the complement of ${\mathcal{V}}$ is
an snc divisor in ${\mathcal{Y}}$. Then the outcome of Theorem 6.2 maybe be
strengthened, by the requirement, that the complement of ${\mathcal{U}}_{i}$
is an snc divisor ${\mathcal{D}}_{i}\subset{\mathcal{X}}_{i}$ for every $i$,
and every ${\mathcal{Z}}_{i}$ and ${\mathcal{W}}_{i}$ that appears as a center
of a blow-up ${\mathcal{X}}_{i+1}\cong
B\ell_{{\mathcal{Z}}_{i}}{\mathcal{X}}_{i}$ respectively
${\mathcal{X}}_{i-1}\cong B\ell_{{\mathcal{W}}_{i}}{\mathcal{X}}_{i}$ has
normal crossing with ${\mathcal{D}}_{i}$. In the proof, we apply embedded
resolution of singularities to achieve that
${\mathcal{X}}^{\prime}\setminus{\mathcal{U}}^{\prime}$ is an snc divisor. We
apply the evident variant of Proposition 6.1, where
${\mathcal{X}}\setminus{\mathcal{U}}$ and
${\mathcal{X}}^{\prime}\setminus{\mathcal{U}}^{\prime}$ are snc divisors and
in the conclusion every ${\mathcal{X}}_{i}\setminus{\mathcal{U}}_{i}$ is an
snc divisor, with which ${\mathcal{Z}}_{i}$ has normal crossing. We invoke a
similarly evident variant of Lemma 4.11, with boundary divisors.

##  7\. Refined Burnside groups for orbifolds

In this section, we assume that the base field $k$ (of characteristic $0$)
contains all roots of unity. We define

$\operatorname{\mathcal{B}\mathit{urn}}_{n}\qquad\text{and}\qquad{\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n},$

abelian groups receiving birational invariants of $n$-dimensional orbifolds;
the second group is a quotient of the first. The definitions are in terms of
generators and relations. A projective, respectively quasi-projective orbifold
of dimension $n$ will determine a class in
$\operatorname{\mathcal{B}\mathit{urn}}_{n}$, respectively
${\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}$.

The following definition is in parallel with the definition of

$\operatorname{Burn}_{n}(G),$

the equivariant Burnside group of $n$-dimensional varieties with generically
free action of a finite group $G$, in [31].

######  Definition 7.1.

The abelian groups $\operatorname{\mathcal{B}\mathit{urn}}_{n}$ and
${\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}$ are generated by
pairs

$(\mathcal{K},\alpha),$

where

  * •

$\mathcal{K}$ is a projective linearizable DM stack of dimension $d\leq n$,
and

  * •

$\alpha=(a_{1},\ldots,a_{n-d})$ is a sequence of length $n-d$ of nonzero
characters of the generic stabilizer $H$ of $\mathcal{K}$, generating
$H^{\vee}$.

The group $\operatorname{\mathcal{B}\mathit{urn}}_{n}$ is defined by the
following relations.

(O) (re)ordering: For all permutations $\sigma\in\mathfrak{S}_{n-d}$ we have

$(\mathcal{K},\alpha)=(\mathcal{K},\alpha^{\prime}),\quad\alpha^{\prime}:=(a_{\sigma(1)},\ldots,a_{\sigma(n-d)}).$

(I) (birational) isomorphisms: if $\mathcal{K}^{\prime}$ is another projective
linearizable DM stack of dimension $d$, with corresponding characters
$\alpha^{\prime}=(a^{\prime}_{1},\dots,a^{\prime}_{n-d})$, and there is a
birational equivalence

$\mathcal{K}\sim\mathcal{K}^{\prime}$

transforming $\alpha$ to $\alpha^{\prime}$, then

$(\mathcal{K},\alpha)=(\mathcal{K}^{\prime},\alpha^{\prime}).$

(V) vanishing: For all $\mathcal{K}$ and $\alpha=(a_{1},a_{2},\ldots,a_{n-d})$
with

$a_{1}+a_{2}=0$

we have

$(\mathcal{K},\alpha)=0.$

(B) blowup: suppose $n-d\geq 2$, set $a:=a_{1}-a_{2}$, then

$(\mathcal{K},\alpha)=\Theta_{1}+\Theta_{2},$

where

$\Theta_{1}=\begin{cases}0,&\text{if $a=0$},\\\
(\mathcal{K},\alpha_{1})+(\mathcal{K},\alpha_{2}),&\text{if $a\neq
0$},\end{cases}$

with

$\alpha_{1}:=(a_{1},a_{2}-a_{1},a_{3},\dots,a_{n-d}),\quad\alpha_{2}:=(a_{2},a_{1}-a_{2},a_{3},\dots,a_{n-d}),$

and

$\Theta_{2}=\begin{cases}0,&\text{if $a_{i}\in\langle a\rangle$ for some
$i$},\\\ (\mathcal{Y},(b_{2},\dots,b_{n-d})),&\text{otherwise},\end{cases}$

with $\mathcal{Y}$ obtained by construction (A) from $\mathcal{K}$ and the
character $a$; the generic stabilizer of $\mathcal{Y}$ is naturally a subgroup
of the generic stabilizer of $\mathcal{K}$, and the character $b_{i}$ is
obtained by restricting $a_{i}$, for $i=2$, $\dots$, $n-d$.

The group ${\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}$ is defined
by the same relations, except that (I) is replaced by

($\overline{\mathbf{I}}$)
$({\mathcal{K}},\alpha)=({\mathcal{K}}^{\prime},\alpha^{\prime})$ if there is
a birational map ${\mathcal{K}}\dashrightarrow{\mathcal{K}}^{\prime}$
transforming $\alpha$ to $\alpha^{\prime}$.

######  Remark 7.2.

Relation ($\overline{\mathbf{I}}$) differs from (I), in that the requirement
${\mathcal{K}}\sim{\mathcal{K}}^{\prime}$ of (I) is weakened to the
requirement, in ($\overline{\mathbf{I}}$), of just a birational map, i.e., an
isomorphism of dense open substacks. Since ${\mathcal{K}}$ always has a dense
open substack isomorphic to $U\times BH$ for some algebraic variety $U$, there
are birational automorphisms of ${\mathcal{K}}$ realizing every automorphism
of $H$. Consequently, in
${\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}$ we have
$({\mathcal{K}},\alpha)=({\mathcal{K}},\alpha^{\prime})$ when
$\alpha^{\prime}$ is obtained from $\alpha$ by applying an automorphism of
$H$.

######  Definition 7.3.

Let ${\mathcal{X}}$ be a projective orbifold of dimension $n$, that is
divisorial. For a finite abelian group $H$, we call a _stabilizer- $H$
component_ of ${\mathcal{X}}$ the closure of a component of the nonsingular
locally closed substack of ${\mathcal{X}}$, defined by the condition of having
geometric stabilizer group isomorphic to $H$; this is a smooth substack of
${\mathcal{X}}$. A _stabilizer component_ of ${\mathcal{X}}$ is a closed
substack, that is a stabilizer-$H$ component for some $H$. The class

$[\mathcal{X}]\in\operatorname{\mathcal{B}\mathit{urn}}_{n}$

is defined to be the sum

$[\mathcal{X}]:=\sum_{\mathcal{K}\subset\mathcal{X}}(\mathcal{K},\beta_{\mathcal{K}}(\mathcal{X}))$
(7.1)

over stabilizer components of ${\mathcal{X}}$, where the normal bundle
${\mathcal{N}}_{\mathcal{K}/\mathcal{X}}$ determines a representation and
hence a sequence of characters (up to order) of the generic stabilizer of
$\mathcal{K}$, denoted above by $\beta_{\mathcal{K}}(\mathcal{X})$.

######  Remark 7.4.

Suppose ${\mathcal{X}}$ is a smooth separated irreducible DM stack of finite
type over $k$, that is divisorial. Then by Proposition 5.5 we know that the
associated orbifold ${\mathcal{Y}}$ is divisorial. We call a stabilizer
component of ${\mathcal{X}}$ the inverse image by
${\mathcal{X}}\to{\mathcal{Y}}$ of any stabilizer component of
${\mathcal{Y}}$. If $H_{0}$ denotes the generic stabilizer of ${\mathcal{X}}$,
and ${\mathcal{K}}$ is a stabilizer component of ${\mathcal{X}}$, inverse
image of stabilizer component ${\mathcal{L}}\subset{\mathcal{Y}}$, then
$H_{0}$ is a subgroup of the generic stabilizer $H$ of ${\mathcal{K}}$, and
$H/H_{0}$ is the generic stabilizer of ${\mathcal{L}}$. We identify the
characters in $\beta_{{\mathcal{L}}}({\mathcal{Y}})$ with characters of $H$,
trivial on $H_{0}$, which we denote by $\beta_{{\mathcal{K}}}({\mathcal{X}})$.

######  Proposition 7.5.

In $\operatorname{\mathcal{B}\mathit{urn}}_{n}$ we have the following
relations:

  * $\mathrm{(i)}$

If $\alpha=(a_{1},\dots,a_{n-d})$ with $a_{1}+\dots+a_{j}=0$ for some $2\leq
j\leq n-d$, then

$(\mathcal{K},\alpha)=0.$

  * $\mathrm{(ii)}$

For any $2\leq j\leq n-d$,

$(\mathcal{K},\alpha)=\sum_{\begin{subarray}{c}\emptyset\neq
I\subseteq\\{1,\dots,j\\}\\\ \text{satisfying
$(*)$}\end{subarray}}(\mathcal{Y}_{I},\beta_{I})$

where a nonempty subset $I\subseteq\\{1,\dots,j\\}$, $i_{0}\in I$, determines
the subgroup $H_{I}\subseteq H$, intersection of the kernels of
$a_{i}-a_{i_{0}}$ for all $i\in I$, we apply construction _(A)_ to the
characters

$(a_{i}-a_{i_{0}})_{i\in I\setminus\\{i_{0}\\}}$

to obtain the linearizable DM stack $\mathcal{Y}_{I}$ with generic stabilizer
$H_{I}$, we define $\alpha_{I}$ to consist of the characters

$a_{i_{0}},\qquad(a_{i}-a_{i_{0}})_{i\in\\{1,\dots,j\\}\setminus
I}\qquad\text{and}\qquad(a_{j+1},\dots,a_{n-d}),$

we define the condition

$(*):\ \text{the character $a$ restricts nontrivially to $H_{I}$, for all
$a\in\alpha_{I}$},$

and we define $\beta_{I}$ to consist of the restrictions to $H_{I}$ of the
characters in $\alpha_{I}$.

###### Proof.

We prove $\mathrm{(i)}$ by induction on $j$. The base case $j=2$ is relation
(V). For the inductive step, we apply relation (B) to
$(\mathcal{K},\alpha^{\prime})$, where

$\alpha^{\prime}=(a_{1},a_{1}+a_{2},a_{3},\dots,a_{n-d}).$

By relation (O) and the induction hypothesis we have
$(\mathcal{K},\alpha^{\prime})=0$. The right-hand side of (B) consists of
terms that vanish for the same reason, along with $(\mathcal{K},\alpha)$. So
$(\mathcal{K},\alpha)=0$.

We also prove $\mathrm{(ii)}$ by induction on $j$. The base case $j=2$ is
relation (B). For the inductive step, we apply the induction hypothesis and
for each resulting $(\mathcal{Y}_{I},\beta_{I})$ we apply relation (B) to the
restrictions of the characters $a_{i_{0}}$ and $a_{j+1}$. The rest of the
argument is combinatorial and follows exactly the proof of [31, Prop. 4.7], on
which this statement is based. ∎

##  8\. Birational invariants of orbifolds

We continue to work over a base field $k$ of characteristic $0$ containing all
roots of unity. The goal of this section is to exhibit birational invariants
of $n$-dimensional orbifolds, with values in
$\operatorname{\mathcal{B}\mathit{urn}}_{n}$ and
${\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}$.

######  Theorem 8.1.

Let ${\mathcal{X}}$ and ${\mathcal{X}}^{\prime}$ be projective orbifolds of
dimension $n$, that are divisorial. If ${\mathcal{X}}$ and
${\mathcal{X}}^{\prime}$ are birationally equivalent, then

$[{\mathcal{X}}]=[{\mathcal{X}}^{\prime}]$

in $\operatorname{\mathcal{B}\mathit{urn}}_{n}$.

###### Proof.

By Theorem 4.25, it suffices to consider a blowup
$\pi\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ of divisorial projective
orbifolds, with center in a smooth substack ${\mathcal{W}}$. We split (7.1)
into two sums, according to whether ${\mathcal{K}}$ is contained in
${\mathcal{W}}$:

$[{\mathcal{X}}]=\sum_{{\mathcal{K}}\not\subset{\mathcal{W}}}(\mathcal{K},\beta_{\mathcal{K}}(\mathcal{X}))+\sum_{{\mathcal{K}}\subset{\mathcal{W}}}(\mathcal{K},\beta_{\mathcal{K}}(\mathcal{X})).$

Note first that ${\mathcal{K}}\cap{\mathcal{W}}$ is smooth; this follows from
the analysis of local stack geometry in [7].

The analogous sum for ${\mathcal{X}}^{\prime}$ is split analogously, according
to containment in the exceptional divisor ${\mathcal{E}}$:

$[{\mathcal{X}}^{\prime}]=\sum_{{\mathcal{K}}^{\prime}\not\subset{\mathcal{E}}}({\mathcal{K}}^{\prime},\beta_{{\mathcal{K}}^{\prime}}({\mathcal{X}}^{\prime}))+\sum_{{\mathcal{K}}^{\prime}\subset{\mathcal{E}}}({\mathcal{K}}^{\prime},\beta_{{\mathcal{K}}^{\prime}}({\mathcal{X}}^{\prime})).$

The respective first sums are equal. For any ${\mathcal{K}}^{\prime}$ in the
second sum for ${\mathcal{X}}^{\prime}$ the image
$\pi({\mathcal{K}}^{\prime})$ is contained in a unique ${\mathcal{K}}$,
appearing in the formula for $[{\mathcal{X}}]$, with the same generic
stabilizer as $\pi({\mathcal{K}}^{\prime})$. We claim that
$\pi({\mathcal{K}}^{\prime})$ is an entire connected component ${\mathcal{V}}$
of ${\mathcal{K}}\cap{\mathcal{W}}$. This follows by an analysis of
${\mathcal{E}}\times_{{\mathcal{X}}}{\mathcal{V}}\cong{\mathbb{P}}({\mathcal{N}}_{{\mathcal{W}}/{\mathcal{X}}}|_{{\mathcal{V}}})$,
which reveals also that when $\mathcal{K}\not\subset\mathcal{W}$ there are
always two characters summing to $0$ in
$\beta_{{\mathcal{K}}^{\prime}}({\mathcal{X}}^{\prime})$, hence
$({\mathcal{K}}^{\prime},\beta_{{\mathcal{K}}^{\prime}}({\mathcal{X}}^{\prime}))=0$
by (V). When $\mathcal{K}\subset\mathcal{W}$ the contribution to
$[{\mathcal{X}}^{\prime}]$ from the
${\mathcal{K}}^{\prime}\subset{\mathcal{E}}$ with
$\pi({\mathcal{K}}^{\prime})={\mathcal{K}}$ is precisely the right-hand side
of the relation from Proposition 7.5(ii), where $j$ is the codimension of
${\mathcal{W}}$, consequently $[{\mathcal{X}}]=[{\mathcal{X}}^{\prime}]$. ∎

In the next definition we allow ourselves to write a symbol
$({\mathcal{K}},\alpha)\in{\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}$
when ${\mathcal{K}}$ is a quasiprojective linearizable DM stack. By the
results about projective compactifications stated in Section 5, such
${\mathcal{K}}$ may be embedded as an open substack in a projective
linearizable DM stack $\overline{{\mathcal{K}}}$. We introduce

$({\mathcal{K}},\alpha):=(\overline{{\mathcal{K}}},\alpha)\in{\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}.$

This, by relation ($\overline{\mathbf{I}}$), is independent of the choice of
$\overline{{\mathcal{K}}}$.

Since ${\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}$ is defined by
a larger set of relations than $\operatorname{\mathcal{B}\mathit{urn}}_{n}$,
there is a canonical homomorphism

$\operatorname{\mathcal{B}\mathit{urn}}_{n}\to{\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}.$

In any formula involving classes from both
$\operatorname{\mathcal{B}\mathit{urn}}_{n}$ and
${\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}$ there is the
implicit use of this homomorphism.

######  Definition 8.2.

Let ${\mathcal{U}}$ be a quasi-projective orbifold of dimension $n$, that is
divisorial. The corresponding naive class in
${\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}$ is

$[{\mathcal{U}}]^{\mathrm{naive}}:=\sum_{{\mathcal{K}}\subset{\mathcal{U}}}({\mathcal{K}},\beta_{{\mathcal{K}}}({\mathcal{U}})),$

where the sum is as in (7.1), but for ${\mathcal{U}}$. If ${\mathcal{U}}$ is
the complement, in a divisorial projective orbifold ${\mathcal{X}}$ of an snc
divisor ${\mathcal{D}}$,

${\mathcal{D}}=\bigcup_{i\in{\mathcal{I}}}{\mathcal{D}}_{i},$

then we define the class in
${\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}$ to be

$[{\mathcal{U}}]:=[{\mathcal{X}}]+\sum_{\emptyset\neq
I\subseteq{\mathcal{I}}}(-1)^{|I|}[{\mathcal{N}}_{{\mathcal{D}}_{I}/{\mathcal{X}}}]^{\mathrm{naive}},$

where ${\mathcal{D}}_{I}$ denotes the intersection of the ${\mathcal{D}}_{i}$
with $i\in I$.

Notice, by Remark 5.2, ${\mathcal{N}}_{{\mathcal{D}}_{I}/{\mathcal{X}}}$ is
divisorial, for every $I$. By the same argument as in the proof of Theorem
8.1, $[{\mathcal{U}}]^{\mathrm{naive}}$ is a birational invariant. It also
remains unchanged upon passage to suitable open substacks.

######  Lemma 8.3.

Let ${\mathcal{U}}$ and ${\mathcal{U}}^{\prime}$ be a quasi-projective
orbifolds of dimension $n$. We suppose that ${\mathcal{U}}^{\prime}$ is
divisorial.

  * $\mathrm{(i)}$

If ${\mathcal{U}}$ is an open substack of ${\mathcal{U}}^{\prime}$, such that
${\mathcal{U}}\cap{\mathcal{K}}\neq\emptyset$ for every stabilizer component
${\mathcal{K}}$ of ${\mathcal{U}}^{\prime}$, then in
${\overline{\operatorname{\mathcal{B}\mathit{urn}}}}_{n}$ we have

$[{\mathcal{U}}]^{\mathrm{naive}}=[{\mathcal{U}}^{\prime}]^{\mathrm{naive}}.$

  * $\mathrm{(ii)}$

Let $\pi\colon{\mathcal{U}}\to{\mathcal{Z}}$ and