# MMP for locally stable families and wall crossing for moduli of stable pairs

Fanjun Meng  Department of Mathematics, Johns Hopkins University, 3400 N.
Charles Street, Baltimore, MD 21218, USA<EMAIL_ADDRESS>and  Ziquan Zhuang
Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street,
Baltimore, MD 21218, USA<EMAIL_ADDRESS>
###### Abstract.

We construct reduction and wall-crossing morphisms between the moduli spaces
of stable pairs as the coefficients vary, generalizing the earlier work of
Ascher, Bejleri, Inchiostro and Patakfalvi which deals with the klt case.
Along the proof, we show that one can run the MMP with scaling on normal
locally stable families over a normal base, and that the existence of good
minimal models is preserved when reducing coefficients away from zero.

2020 _Mathematics Subject Classification_ : 14E30, 14J10, 14J17.
_Keywords_ : minimal model program, moduli spaces, stable pairs.

##  1\. Introduction

In [Has03], Hassett shows that there are natural reduction and wall-crossing
morphisms between the moduli spaces of weighted pointed stable curves as the
weights vary. The higher-dimensional generalization of weighted pointed stable
curves, whose study goes back to [KSB88], is the notion of stable pairs. These
are semi-log-canonical (slc) pairs $(X,D=\sum a_{i}D_{i})$ (the basic examples
are simple normal crossing pairs $(X,D)$ with coefficients $a_{i}\in(0,1]$)
such that $K_{X}+D$ is ample. The moduli theory of stable pairs has been
established in the past several decades; we refer to [Kol23] for a
comprehensive discussion. In particular, for any dimension $d$, any
coefficient vector $\mathbf{a}$ for the boundary divisor $D$, and any volume
$v=(K_{X}+D)^{d}>0$, there exists a proper moduli space that parameterizes
stable pairs with the given numerical invariants.

It is natural to compare these moduli spaces as the coefficients vary, and one
may ask whether the reduction and wall-crossing morphisms generalize to the
moduli of stable pairs. Such morphisms play an important role in the study of
explicit moduli compactifications, and have been worked out in several
explicit examples, see e.g. [Ale15, Inc20, AB21]. The general question is
taken up recently in [ABIP23] under a klt assumption. The goal of this article
is to remove this klt condition and to construct, in the general setting, the
reduction and wall-crossing morphisms between the moduli spaces of stable
pairs.

These morphisms are typically obtained through an ample model construction: if
we denote by $\mathcal{K}_{\mathbf{a}}$ the moduli space of stable pairs with
coefficient vector $\mathbf{a}$, and let $\mathbf{b}$ be another coefficient
vector, then the expected natural morphism
$\mathcal{K}_{\mathbf{a}}\to\mathcal{K}_{\mathbf{b}}$ should send a stable
pair $(X,\sum a_{i}D_{i})$ to the semi-log-canonical model of $(X,\sum
b_{i}D_{i})$, which is roughly the ample model of $K_{X}+\sum b_{i}D_{i}$.
Since semi-log-canonical models of non-normal semi-log-canonical pairs do not
always exist (see e.g. [Kol11-slc-no-MMP, AK19] or Section 7.1 for some
examples), this imposes restrictions on the locus where the morphism
$\mathcal{K}_{\mathbf{a}}\to\mathcal{K}_{\mathbf{b}}$ can be defined. Another
related restriction, observed in [ABIP23], is that the reduction morphisms can
only be defined on the normalization of the moduli spaces. With these in mind,
our main results give some optimal statements that one may hope for. Roughly
speaking, we show that wall-crossing morphisms exist on the locus that
parameterizes degenerations of normal (hence log canonical) stable pairs,
while the reduction morphisms exist on the normalization of the same locus.

###  1.1. Main theorems

To state our main theorems more precisely, we need to introduce some notation.
Fix a locally stable family (see Definition 2.16)
$f\colon(X,\sum_{i=1}^{n}a_{i}D_{i})\to B$ over a normal (but not necessarily
proper) variety. A special case is when $(X,\sum D_{i})$ is log smooth over
$B$ and $0<a_{i}\leq 1$ for all $i$. For ease of notation, we write
$\mathbf{a}=(a_{1},\dots,a_{n})$. For any coefficient vector
$\mathbf{b}=(b_{1},\dots,b_{n})$, we denote by $\mathbf{b}D$ the divisor
$\sum_{i=1}^{n}b_{i}D_{i}$. Following [Has03, ABIP23], we say that a
coefficient vector $\mathbf{b}=(b_{1},\dots,b_{n})$ is admissible if
$0<b_{i}\leq a_{i}$ for all $i$ and $K_{X/B}+\mathbf{b}D$ is big over $B$.

######  Theorem 1.1 (reduction).

Assume that the generic fiber of $(X,\mathbf{a}D)\to B$ is normal and has a
log canonical model. Then for any admissible coefficient vector $\mathbf{b}$,
there exists a $($unique$)$ stable family
$(Z_{\mathbf{b}},\mathbf{b}\Delta_{\mathbf{b}})\to B$ that is the log
canonical model of $(X,\mathbf{b}D)\to B$ $($i.e. the ample model of
$K_{X/B}+\mathbf{b}D$ over $B)$. In particular, it induces a unique morphism
$\Phi_{\mathbf{b}}\colon B\to\mathcal{K}_{\mathbf{b}}$.

In particular, if we denote by
$\overline{\mathcal{K}}_{\mathbf{a}}^{\operatorname{lc}}$ the normalized
closure of the locus in the moduli space $\mathcal{K}_{\mathbf{a}}$ that
parameterizes log canonical stable pairs with coefficient vector $\mathbf{a}$,
then the above theorem gives the reduction morphism
$\overline{\mathcal{K}}_{\mathbf{a}}^{\operatorname{lc}}\to\overline{\mathcal{K}}_{\mathbf{b}}^{\operatorname{lc}}$
for every admissible coefficient vector $\mathbf{b}$. When the generic fiber
of $(X,\mathbf{a}D)\to B$ is klt and the base $B$ is smooth (this is one of
the cases considered in [ABIP23]), Theorem 1.1 follows from [BCHM10].

We say that a polytope $\mathcal{P}\subseteq{\mathbb{R}}^{n}$ is admissible if
every coefficient vector in $\mathcal{P}$ is admissible. Under the assumptions
of Theorem 1.1, as the coefficient $\mathbf{b}$ varies in an admissible
polytope $\mathcal{P}$, we also have a wall-and-chamber decomposition of
$\mathcal{P}$ corresponding to the reduced closure
$\mathcal{M}_{\mathbf{b}}:=\overline{\Phi_{\mathbf{b}}(B)}$ of the images of
the reduction morphisms $\Phi_{\mathbf{b}}\colon
B\to\mathcal{K}_{\mathbf{b}}$.

######  Theorem 1.2 (wall crossing).

Under the assumptions of Theorem 1.1, let $\mathcal{P}$ be an admissible
rational polytope. Then there exists a finite rational polyhedral
decomposition $\mathcal{P}=\cup_{i=1}^{p}\mathcal{P}_{i}$ such that:

  1. (1)

In any open chamber $\mathcal{P}_{i}^{\circ}$, the family
$(Z_{\mathbf{b}},\Delta_{\mathbf{b}})\to B$ of marked pairs is independent of
$\mathbf{b}\in\mathcal{P}_{i}^{\circ}$, and
$\mathcal{M}_{\mathbf{b}}\cong\mathcal{M}_{\mathbf{b}^{\prime}}$ for any
$\mathbf{b},\mathbf{b}^{\prime}\in\mathcal{P}_{i}^{\circ}$.

  2. (2)

For any $i$, we denote $\mathcal{M}_{\mathbf{b}}$ by $\mathcal{M}_{i}$ for any
$\mathbf{b}\in\mathcal{P}_{i}^{\circ}$ $($this is well-defined by part $(1))$.
Then for any $i,j$ such that $\mathcal{P}_{j}$ is a face of $\mathcal{P}_{i}$,
there exists a $($unique$)$ morphism
$\alpha_{ij}\colon\mathcal{M}_{i}\to\mathcal{M}_{j}$ making the following
diagram commute.

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{M}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{ij}}$$\textstyle{\mathcal{M}_{j}}$

The morphisms $\alpha_{ij}$ are what we refer to as the wall-crossing
morphisms. Moreover, the morphism $\alpha_{ij}$ lifts to a morphism between
the universal families over $\mathcal{M}_{i}$ and $\mathcal{M}_{j}$. Putting
together Theorems 1.1 and 1.2, we recover Hassett’s result in the curve case.
When the generic fiber of $(X,\mathbf{a}D)\to B$ is klt, the above theorem
also improves upon [ABIP23]*Theorem 1.8, where the wall-crossing morphisms are
constructed on the seminormalization of the corresponding moduli locus.

There are also reduction and wall-crossing morphisms in the semi-log-canonical
case if we impose additional assumptions. The following is an example. It is a
generalization of [ABIP23]*Theorem 1.1 to the semi-log-canonical setting. For
any $\mathbf{a}=(a_{1},\ldots,a_{n})$ and $\mathbf{b}=(b_{1},\ldots,b_{n})$,
we write $\mathbf{b}\leq\mathbf{a}$ if $b_{i}\leq a_{i}$ for all $i$.

######  Theorem 1.3.

Let $(X,\sum D_{i})\to B$ be a family of marked pairs over a normal variety
and let $\mathcal{P}$ be a rational polytope such that $(X,\mathbf{a}D)\to B$
is a stable family for every $\mathbf{a}\in\mathcal{P}$. Let
$\mathcal{N}_{\mathbf{a}}$ be the normalized closure of the image of the
morphism $\Phi_{\mathbf{a}}\colon B\to\mathcal{K}_{\mathbf{a}}$. Then there is
a canonical birational reduction morphism
$r_{\mathbf{a},\mathbf{b}}\colon\mathcal{N}_{\mathbf{a}}\to\mathcal{N}_{\mathbf{b}}$
for any $\mathbf{a}\geq\mathbf{b}$ in $\mathcal{P}$. Moreover, we have
$r_{\mathbf{b},\mathbf{c}}\circ
r_{\mathbf{a},\mathbf{b}}=r_{\mathbf{a},\mathbf{c}}$ for any
$\mathbf{a}\geq\mathbf{b}\geq\mathbf{c}$ in $\mathcal{P}$, and there is a
wall-and-chamber and wall-crossing structure on $\mathcal{P}$ corresponding to
$\mathcal{N}_{\mathbf{a}}$ as in Theorem 1.2.

###  1.2. Other results

The proof of the main theorems relies on several other results that might be
of independent interest. Some of these results are already known in the klt
case, but the extension to the log canonical case is more subtle.

A famous conjecture in birational geometry is that every log canonical pair of
nonnegative Kodaira dimension has a good minimal model (Definition 2.6). For
klt pairs of log general type this follows from [BCHM10], but if the pair is
only log canonical then the conjecture is wide open even in the log general
type case. This presents one of the difficulties in the construction of the
reduction morphisms in the log canonical setting. The first ingredient that
goes into our proof of the main theorems is the existence of good minimal
models for those special log canonical pairs that come up in Theorem 1.1. In
fact we prove a more general statement that abundance is preserved when
reducing coefficients away from zero.

######  Theorem 1.4 (=Theorem 3.1).

Let $(X,D)$ be a log canonical pair over $B$ and let $0\leq\Delta\leq D$ be an
${\mathbb{R}}$-divisor. Assume that $(X,D)$ has a good minimal model over $B$,
$K_{X}+\Delta$ is ${\mathbb{R}}$-Cartier, and
$\operatorname{Supp}(\Delta)=\operatorname{Supp}(D)$. Then $(X,\Delta)$ has a
good minimal model or a Mori fiber space over $B$.

This implies Theorem 1.1 when the base $B$ is a single point. In general, we
need to show that the log canonical model construction works in locally stable
families. Thus the next ingredient is a minimal model program (MMP) for such
families.

######  Theorem 1.5 (=Theorem 4.1).

Let $f\colon(X,D)\to B$ be a locally stable family over a normal variety with
normal generic fiber. Then for any $f$-ample ${\mathbb{R}}$-divisor $H$ on
$X$, we may run the $(K_{X/B}+D)$-MMP over $B$ with scaling of $H$. Moreover,
the MMP terminates with a good minimal model over $B$ if the generic fiber has
a good minimal model.

Although MMP with scaling for log canonical pairs is known by [Amb03, Fuj11,
Bir12-lc-flip, HX13, HH20], the singularities of the pair $(X,D)$ in the above
theorem can be much worse, because the base $B$ may have very bad
singularities. On the other hand, $(X,D)$ can be made log canonical after a
base change. The above theorem is therefore a descent statement: in
particular, we show that extremal contractions (resp. flips) descend along
arbitrary reduced (resp. normal) base change. Ultimately, these rely on, among
other things, the invariance of plurigenera for good minimal models of locally
stable families. We remark that when the generic fiber of the family is klt,
some special cases of Theorem 1.5 are implicit in [ABIP23] when they analyze
the invertibility of certain special wall-crossing morphisms. Using Theorem
1.5 we are thus able to give a more direct and intrinsic construction of the
reduction morphisms (that is, without analyzing the aforementioned wall-
crossing morphisms) even in the klt case.

Finally, to obtain the wall-and-chamber structure in Theorem 1.2, the third
ingredient is the finiteness of ample models as the coefficients vary, see
Theorem 5.1. For klt pairs of log general type, this again follows from
[BCHM10]. We refer to Section 5 for the relevant setup and the precise
statement.

###  1.3. Outline

This paper is organized as follows. Section 2 covers some preliminary
definitions and statements. Section 3 studies the existence of good minimal
models for (some) log canonical pairs, Section 4 establishes the MMP with
scaling for locally stable families, while Section 5 deals with the finiteness
of ample models. The main theorems are proved in Section 6, and Section 7
collects some examples to illustrate why some of the assumptions in our main
results are optimal.

### Acknowledgements

The authors would like to thank Dori Bejleri, Giovanni Inchiostro, Vyacheslav
Shokurov and Chenyang Xu for helpful discussions and comments. The second
author is partially supported by a Clay research fellowship, a Sloan
fellowship, and the NSF (DMS-2240926, DMS-2234736).

##  2\. Preliminaries

We work over an algebraically closed field $k$ of characteristic $0$. All
schemes are of finite type over $k$, unless otherwise stated.

###  2.1. Notation and conventions

If $f\colon Y\dashrightarrow X$ is a birational map and $D$ is an
${\mathbb{R}}$-divisor on $X$, we denote its strict transform $f_{*}^{-1}D$ on
$Y$ by $D_{Y}$. If $D$ is ${\mathbb{R}}$-Cartier, its _birational pullback_
$f^{*}D$ is defined as $p_{*}q^{*}D$ where $p\colon Z\to Y$, $q\colon Z\to X$
is a common resolution (the definition is independent of the choice of the
common resolutions).

If $X\to B$ is a projective morphism of normal quasi-projective varieties and
$D\geq 0$ is an ${\mathbb{R}}$-divisor on $X$, we call $(X,D)$ a _pair over
$B$_.

###  2.2. Singularities of pairs

For basic definitions of singularities of pairs, we refer to [KM98] and
[Kol13]. Occasionally we make use of generalized pairs, and we refer to
[BZ16]*Section 4 for their definition and basic properties. In this
subsection, we recall several definitions regarding stable pairs.

######  Definition 2.1.

A scheme $X$ is called _demi-normal_ if it is $S_{2}$ and its codimension $1$
points are either regular points or nodes.

######  Definition 2.2.

Let $\pi\colon\bar{X}\to X$ be the normalization of a demi-normal scheme $X$.
The _conductor ideal_

$\mathcal{H}om_{\mathcal{O}_{X}}(\pi_{*}\mathcal{O}_{\bar{X}},\mathcal{O}_{X})\subset\mathcal{O}_{X}$

defines reduced, pure codimension $1$ and closed subschemes $\Gamma\subset X$
and $\bar{\Gamma}\subset\bar{X}$ which are called the _conductors_ or the
_double loci_.

######  Definition 2.3.

Let $X$ be a demi-normal scheme with normalization $\pi\colon\bar{X}\to X$ and
conductors $\Gamma\subset X$ and $\bar{\Gamma}\subset\bar{X}$. Let $D$ be an
effective ${\mathbb{R}}$-divisor on $X$ whose support does not contain any
irreducible component of $\Gamma$ and $\bar{D}$ the divisorial part of
$\pi^{-1}(D)$. The pair $(X,D)$ is called _semi-log-canonical_ (or _slc_) if

  1. (1)

$K_{X}+D$ is ${\mathbb{R}}$-Cartier, and

  2. (2)

$(\bar{X},\bar{D}+\bar{\Gamma})$ is log canonical.

Moreover, if $X$ is proper and $K_{X}+D$ is ample, $(X,D)$ is called a
_stable_ pair.

###  2.3. Models

We need to deal with different kinds of models of log pairs for our purpose.
In this subsection, we recall their definitions.

######  Definition 2.4.

Let $f\colon X\to B$ be a projective morphism of quasi-projective varieties.
Let $D$ be an ${\mathbb{R}}$-divisor on $X$ such that $X$ is smooth at the
generic points of $D$. We say $D$ is _big over $B$_ or _$f$ -big_ if

$\limsup\frac{h^{0}(F,\mathcal{O}_{F}(\llcorner mD\lrcorner))}{m^{\dim F}}>0$

for the generic fiber $F$ of $f$. Equivalently, $D$ is big over $B$ if
$D\sim_{{\mathbb{R}},B}A+E$, where $A$ is ample over $B$ and $E\geq 0$.

First we recall the existence of dlt modifications which is useful.

######  Theorem 2.5 ([KK10]*Theorem 3.1,[Fuj11]*Theorem 10.5).

Let $(X,D)$ be a log canonical pair where $X$ is a normal quasi-projective
variety. Then there exists a projective birational morphism $f\colon Y\to X$
from a normal quasi-projective variety $Y$ such that

  1. (1)

$Y$ is ${\mathbb{Q}}$-factorial.

  2. (2)

Let $E$ be the sum of all the $f$-exceptional divisors. Then $(Y,D_{Y}+E)$ is
dlt and $K_{Y}+D_{Y}+E=f^{*}(K_{X}+D)$.

The morphism $f$ in the above theorem is called a _dlt modification_ of
$(X,D)$, and $(Y,D_{Y}+E)$ is called a _dlt model_ of $(X,D)$.

Next, we recall the definition of good minimal models and log canonical
models.

######  Definition 2.6.

Let $(X,D)$ be a log canonical pair over $B$. Suppose that $\pi\colon
X\dashrightarrow Y$ is a birational contraction over $B$. Then $(Y,D_{Y})$ is
called a _minimal model of $(X,D)$ over $B$_ if $K_{Y}+D_{Y}$ is
${\mathbb{R}}$-Cartier and nef over $B$, and

$a(E,X,D)<a(E,Y,D_{Y})$

for every $\pi$-exceptional divisor $E$ on $X$. It is called a _good minimal
model of $(X,D)$ over $B$_ if $K_{Y}+D_{Y}$ is semiample over $B$. If in
addition $K_{Y}+D_{Y}$ is also big over $B$, the ample model of $K_{Y}+D_{Y}$
over $B$ is called the _log canonical model of $(X,D)$ over $B$_.

For ease of notation, we will sometimes simply say $Y$ is a minimal (resp. log
canonical) model of $(X,D)$ over $B$, as the boundary divisor $D_{Y}$ is
uniquely determined as the strict transform of $D$.

######  Remark 2.7.

If in the above definition we only require that $a(E,X,D)\leq a(E,Y,D_{Y})$,
then the model $(Y,D_{Y})$ is called a _weak log canonical model of $(X,D)$
over $B$_. Every minimal model is clearly a weak log canonical model. The log
canonical model, if exists, is also a weak log canonical model (as the name
suggests). Any two weak log canonical models are $K$-equivalent (i.e. the
pullbacks of $K_{Y}+D_{Y}$ are the same on any common resolution of the
models, see e.g. [KM98]*Proof of Theorem 3.52), thus the semiampleness of
$K_{Y}+D_{Y}$ over $B$ is independent of the weak log canonical model. In
particular, if $(X,D)$ has a good minimal model over $B$, then every minimal
model of $(X,D)$ over $B$ is good, and the log canonical model is unique, c.f.
[KM98]*Corollary 3.53, [Bir12-lc-flip]*Remark 2.7, [HX13]*Lemma 2.4,
[LM21]*Lemma 2.6.

######  Remark 2.8.

There are several different definitions of minimal models in the literature.
When $(X,D)$ is dlt, the above definition coincides with [KM98]*Definition
3.50. Many authors (e.g. [HX13]*p.165) further require that the minimal models
be $\mathbb{Q}$-factorial. In many applications this is automatic since the
MMP starts with a $\mathbb{Q}$-factorial dlt pair, but in order to run the MMP
for locally stable families in Section 4 we need to drop the
$\mathbb{Q}$-factoriality assumption (c.f. [Fuj11] and [Kol21-MMP-non-Q-
factorial]). In [Bir12-lc-flip] and [HH20], the authors consider
($\mathbb{Q}$-factorial dlt) minimal models $Y$ over $B$ for which the
birational map $X\dashrightarrow Y$ is not necessarily a contraction. By
[HH20]*Theorem 1.7, if $(X,D)$ has a minimal model over $B$ under their
definition, then it also has a minimal model over $B$ under our definition. On
the other hand, if $(X,D)$ has a minimal model $(Y,D_{Y})$ over $B$ under our
definition, then any dlt modification of $(Y,D_{Y})$ is a minimal model of
$(X,D)$ over $B$ under their definition. The minimal models under these
various definitions are all $K$-equivalent, hence if one of them is good then
so are the others (as in the previous remark). Since we are mainly interested
in the existence of (good) minimal models and log canonical models, these
different definitions are essentially equivalent for our purpose.

######  Lemma 2.9.

Let $(X,D)$ and $(Y,\Gamma)$ be log canonical pairs over $B$, and $\pi\colon
Y\dashrightarrow X$ a birational contraction over $B$ such that

(2.1) $K_{Y}+\Gamma=\pi^{*}(K_{X}+D)+E$

for an effective $\pi$-exceptional divisor $E$. Then $(Y,\Gamma)$ has a good
minimal model (resp. log canonical model) over $B$ if and only if $(X,D)$
does, and the two pairs have the same log canonical model over $B$.

In particular, if $Y\dashrightarrow X$ is a $(K_{Y}+\Gamma)$-MMP over $B$ then
$(Y,\Gamma)$ has a good minimal model over $B$ if and only if $(X,\Gamma_{X})$
has a good minimal model over $B$, c.f. [Has19]*Remark 2.16. The same also
applies if $(X,D)$ is a weak log canonical model of $(Y,\Gamma)$ over $B$.

###### Proof.

This is well-known to experts. The assumption (2.1) implies that $K_{X}+D$ is
big over $B$ if and only if $K_{Y}+\Gamma$ is big over $B$. Hence if $\pi$ is
a morphism, the statement follows from [KM98]*Corollary 3.53 and [Has19]*Lemma
2.15 (see also [HX13]*Lemma 2.10 and [Bir12-lc-flip]*Remark 2.8). In the
general case, choose a common log resolution

$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\psi}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X}$

and consider the divisor $\Delta=\Gamma_{Z}+F$ on $Z$, where $F$ is the sum of
the $\varphi$-exceptional divisors. Since $(Y,\Gamma)$ is log canonical, we
have

$K_{Z}+\Delta=\varphi^{*}(K_{Y}+\Gamma)+E_{1}$

for some $\varphi$-exceptional divisor $E_{1}\geq 0$. By (2.1), this also
implies that

$K_{Z}+\Delta=\psi^{*}(K_{X}+D)+E_{2}$

for some $\psi$-exceptional divisor $E_{2}\geq 0$. By [Has19]*Lemma 2.15, we
see that $(Y,\Gamma)$ (resp. $(X,D)$) has a good minimal model over $B$ if and
only if $(Z,\Delta)$ has a good minimal model over $B$, and all three pairs
have the same log canonical model over $B$ by [KM98]*Corollary 3.53. ∎

###  2.4. MMP for log canonical pairs

We need two fundamental results on the MMP for log canonical pairs. The first
result is the existence of good minimal models for pairs of log Calabi-Yau
type.

######  Theorem 2.10 ([Bir12-lc-flip]*Theorem 1.1, [HX13]*Theorem 1.6,
[Has19]*Theorem 1.1).

Let $(X,D+A)$ be a log canonical pair over $B$ where $D,A\geq 0$ are
${\mathbb{R}}$-divisors. Assume that

$K_{X}+D+A\sim_{{\mathbb{R}},B}0$

and that $K_{X}+D$ is ${\mathbb{R}}$-Cartier. Then $(X,D)$ has a good minimal
model or a Mori fiber space over $B$.

The second result is related to the MMP with scaling. In fact, by taking $X=Y$
in the statement below it implies that any log canonical pair $(X,D+A)$ over
$B$ where $A$ is ample over $B$ has a good minimal model over $B$ as long as
$K_{X}+D+A$ is pseudo-effective over $B$.

######  Theorem 2.11 ([HH20]*Theorem 5.9).

Let $(X,D)$ be a log canonical pair over $B$ and let $\pi\colon X\to Y$ be a
projective morphism of normal quasi-projective varieties over $B$. Let $A\geq
0$ be an ${\mathbb{R}}$-divisor on $Y$ that is ample over $B$ such that
$(X,D+\pi^{*}A)$ is log canonical. Assume that $(X,D)$ has a good minimal
model over $Y$, and $K_{X}+D+\pi^{*}A$ is pseudo-effective over $B$, then
$(X,D+\pi^{*}A)$ has a good minimal model over $B$.

We also need the following statement, which is essentially a consequence of
the negativity lemma (and the existence of MMP with scaling).

######  Theorem 2.12 ([Bir12-lc-flip]*Theorem 3.5).

Let $(X,D)$ be a $\mathbb{Q}$-factorial dlt pair over $Z$ such that $X\to Z$
is birational. Assume that

$K_{X}+D\sim_{{\mathbb{R}},Z}M^{+}-M^{-}$

where $M^{+},M^{-}\geq 0$ have no common components and $M^{+}$ is exceptional
over $Z$. Then any $(K_{X}+D)$-MMP over $Z$ with scaling of a relatively ample
divisor contracts $M^{+}$ after finitely many steps.

We should remark that the theorem only implies that after finitely many steps
of the $(K_{X}+D)$-MMP over $Z$ we get a model $X\dashrightarrow Y$ such that
$M^{+}_{Y}=0$. However, $(Y,D_{Y})$ is not necessarily a minimal model over
$Z$ (this would require the termination of the $(K_{X}+D)$-MMP over $Z$, which
is still open).

###  2.5. Families of stable pairs

In this subsection, we recall some definitions and results on families of
stable pairs in [Kol23]. In general, it is quite subtle to give the correct
definition of families of stable pairs. Here we mainly focus on the case when
the bases of the families are reduced.

We start with the definition of relative Mumford divisors. We only consider
the case when the morphism is flat.

######  Definition 2.13 ([Kol23]*Definition 4.68).

Let $f\colon X\to B$ be a flat morphism with reduced, connected and $S_{2}$
fibers of pure dimension $d$ over a reduced scheme $B$. A _relative Mumford
divisor_ on $X$ is a divisor $D$ on $X$ such that

  1. (1)

(Equidimensionality) Every irreducible component of $\operatorname{Supp}D$
dominates an irreducible component of $B$, and all nonempty fibers of
$\operatorname{Supp}D\to B$ are of pure dimension $d-1$.

  2. (2)

(Mumford condition) The morphism $f$ is smooth at the generic points of
$X_{b}\cap\operatorname{Supp}D$ for every $b\in B$.

  3. (3)

(Generic Cartier condition) The divisor $D$ is Cartier at the generic points
of $X_{b}\cap\operatorname{Supp}D$ for every $b\in B$.

Note that when $B$ is normal, conditions (1) and (2) imply condition (3) by
[Kol23]*Theorem 4.4.

We will usually fix a marking when we deal with stable pairs. So next we
recall the definition of families of marked pairs. Fix a coefficient vector
$\mathbf{a}=(a_{1},\ldots,a_{n})$ such that $0\leq a_{i}\leq 1$ for every
$1\leq i\leq n$.

######  Definition 2.14.

A family of varieties marked with $n$ divisors, or an _$n$ -marked family_
over a reduced scheme $B$ is $f\colon(X;D_{1},\ldots,D_{n})\to B$ such that
$f$ is flat with demi-normal and connected fibers, and $D_{i}$ is an
effective, relative Mumford divisor for every $1\leq i\leq n$.

######  Definition 2.15 ([Kol23]*Definition 8.4).

A _family of marked pairs with coefficient vector
$\mathbf{a}=(a_{1},\ldots,a_{n})$_ over a reduced scheme $B$ is a family of
pairs $f\colon(X,\sum^{n}_{i=1}a_{i}D_{i})\to B$ such that
$f\colon(X;D_{1},\ldots,D_{n})\to B$ is an $n$-marked family.

Next, we define locally stable families of pairs.

######  Definition 2.16 ([Kol23]*Theorem-Definition 4.7).

A projective family of marked pairs
$f\colon(X,\Delta=\sum^{n}_{i=1}a_{i}D_{i})\to B$ with coefficient vector
$\mathbf{a}=(a_{1},\ldots,a_{n})$ over a reduced scheme $B$ is _locally
stable_ if $K_{X/B}+\Delta$ is ${\mathbb{R}}$-Cartier, and
$(X_{b},\Delta_{b})$ is slc for every $b\in B$. We also say that
$f\colon(X,\Delta)\to B$ is a _locally stable family of pairs with coefficient
vector $\mathbf{a}=(a_{1},\ldots,a_{n})$_, or simply that $f$ is _locally
stable_.

Note that our definition is slightly different from [Kol23]*Theorem-Definition
4.7, since here we also have a marking (the definition in _loc. cit._ should
be thought of as the unmarked version). Since almost all the families of pairs
we consider in this paper are families of marked pairs, we insist on the
marking in all the definitions.

When the base is smooth, we have the following simple characterization of
locally stable families.

######  Theorem 2.17 ([Kol23]*Theorem 4.54 and Corollary 4.55).

A projective family of pairs $f\colon(X,\Delta)\to B$ over a smooth scheme $B$
is locally stable if and only if for every simple normal crossing divisor $D$
on $B$, the pair $(X,\Delta+f^{*}D)$ is slc.

Similarly, we can define stable families.

######  Definition 2.18.

A family of marked pairs $f\colon(X,\Delta=\sum^{n}_{i=1}a_{i}D_{i})\to B$
with coefficient vector $\mathbf{a}=(a_{1},\ldots,a_{n})$ over a reduced
scheme $B$ is _stable_ if $f\colon(X,\Delta)\to B$ is locally stable, and
$K_{X/B}+\Delta$ is ample over $B$. We also say that $f\colon(X,\Delta)\to B$
is a _stable family of pairs with coefficient vector
$\mathbf{a}=(a_{1},\ldots,a_{n})$_, or simply that $f$ is _stable_.

The foundational theorem in the moduli theory of higher-dimensional varieties
is that stable pairs form a proper moduli.

######  Theorem 2.19 ([Kol23]*Theorems 4.1 and 4.8).

Fix a vector $\mathbf{a}=(a_{1},\ldots,a_{n})$ such that $0<a_{i}\leq 1$ for
every $1\leq i\leq n$, a positive integer $d$, and a positive number $v$. Then
there exists a proper Deligne–Mumford stack $\mathcal{K}_{\mathbf{a},d,v}$
which represents the moduli problem of stable families
$f\colon(X,\sum^{n}_{i=1}a_{i}D_{i})\to B$ with fibers of dimension $d$ and
volume $v$ for reduced bases $B$.

For ease of notation we will omit $d$ in $\mathcal{K}_{\mathbf{a},d,v}$ as the
dimension is often irrelevant in our discussion. We define
$\mathcal{K}_{\mathbf{a}}:=\cup_{v}\mathcal{K}_{\mathbf{a},v}$. By convention,
we extend the definition of $\mathcal{K}_{\mathbf{a}}$ to coefficient vectors
$\mathbf{a}$ where some of the $a_{i}$ are allowed to be zero: in this case,
we simply set $\mathcal{K}_{\mathbf{a}}$ to be
$\mathcal{K}_{\mathbf{a}^{\prime}}$ where $\mathbf{a}^{\prime}$ is obtained by
removing the zero entries in $\mathbf{a}$.

A direct consequence of the representability of the moduli problem is the
following descent property.

######  Lemma 2.20.

Let $\varphi\colon B^{\prime}\to B$ be a proper morphism between normal
varieties with connected fibers, and let $(X^{\prime},D^{\prime})\to
B^{\prime}$ be a stable family. Assume that for every $b\in B$, the induced
family $(X^{\prime},D^{\prime})\times_{B^{\prime}}\varphi^{-1}(b)$ is a
trivial family. Then there exists a stable family $(X,D)\to B$ such that
$(X^{\prime},D^{\prime})\cong(X,D)\times_{B}B^{\prime}$.

Here we equip $\varphi^{-1}(b)$ with the reduced scheme structure. We say that
a stable family $(X,D)\to B$ is trivial if $(X,D)\cong(X_{b},D_{b})\times B$
over $B$ for some (or any) $b\in B$.

###### Proof.

This follows from Theorem 2.19 but can also be checked directly. By assumption
and standard Hilbert scheme argument, the flat morphism $X^{\prime}\to
B^{\prime}$ descends to $X\to B$. Since $D^{\prime}$ has no variations along
the fibers of $\varphi$, we see that each component of $D^{\prime}$ maps onto
a divisor in $X$. Let $D$ be the natural pushforward of $D^{\prime}$. Since
the base $B$ is normal, the relative Mumford divisor condition can be checked
on the fibers; since it is true for $D^{\prime}$, the same holds for $D$.
Similarly, we know that $(X_{b},D_{b})$ is slc for any $b\in B$. By
construction, we have $(X^{\prime},D^{\prime})\cong(X,D)\times_{B}B^{\prime}$,
and the relatively ample divisor $K_{X^{\prime}/B^{\prime}}+D^{\prime}$
descends to $X$. Its descent is necessarily $K_{X/B}+D$, hence $K_{X/B}+D$ is
${\mathbb{R}}$-Cartier and ample over $B$. These imply that the family
$(X,D)\to B$ is stable. ∎

##  3\. Existence of log canonical models

In this section, we prove several results on the existence of good minimal
models and log canonical models for log canonical pairs. They form the first
main ingredient in the construction of the reduction morphisms in Theorem 1.1.

###  3.1. Existence of good minimal models

First we show that abundance is preserved when reducing coefficients away from
zero.

######  Theorem 3.1.

Let $(X,D)$ be a log canonical pair over $B$ and let $0\leq\Delta\leq D$ be an
${\mathbb{R}}$-divisor. Assume that $(X,D)$ has a good minimal model over $B$,
$K_{X}+\Delta$ is ${\mathbb{R}}$-Cartier, and
$\operatorname{Supp}(\Delta)=\operatorname{Supp}(D)$. Then $(X,\Delta)$ has a
good minimal model or a Mori fiber space over $B$.

######  Remark 3.2.

Here the condition that $\operatorname{Supp}(\Delta)=\operatorname{Supp}(D)$
is crucial. Without this assumption, the statement becomes equivalent to the
full MMP conjecture (including abundance) that any log canonical pair has
either a good minimal model or a Mori fiber space. This is because for any log
canonical pair $(X,\Delta)$ we can always choose some sufficiently ample
divisor $H\geq 0$ such that $(X,D:=\Delta+H)$ is a stable pair. In particular,
it satisfies all the other conditions of the above theorem.

Before we prove this theorem, we first consider a special case.

######  Lemma 3.3.

In the setting of Theorem 3.1, assume that $K_{X}+D$ is semiample over $B$.
Then $(X,\Delta)$ has a good minimal model or a Mori fiber space over $B$.

###### Proof.

If $K_{X}+\Delta$ is not pseudo-effective over $B$, then it has a Mori fiber
space over $B$ by [HH20]*Theorem 1.7. Thus we may assume that $K_{X}+\Delta$
is pseudo-effective over $B$. We need to show that it has a good minimal model
over $B$.

Let $\pi\colon X\to Z$ be the ample model of $K_{X}+D$ over $B$. Then
$K_{X}+D\sim_{\mathbb{R}}\pi^{*}A$ for some ${\mathbb{R}}$-divisor $A$ that is
ample over $B$. Since $0\leq\Delta\leq D$ and
$\operatorname{Supp}(\Delta)=\operatorname{Supp}(D)$ by assumption, there
exists some ${\mathbb{R}}$-divisor $0\leq\Gamma\leq\Delta$ on $X$ such that
$\Delta$ is a nontrivial convex combination of $\Gamma$ and $D$, i.e.

$\Delta=t\Gamma+(1-t)D$

for some $t\in(0,1)$. We then have

$K_{X}+\Delta=t(K_{X}+\Gamma)+(1-t)(K_{X}+D)\sim_{{\mathbb{R}},B}t(K_{X}+\Gamma+\pi^{*}H)$

where $0\leq H\sim_{{\mathbb{R}},B}\frac{1-t}{t}A$ such that
$(X,\Gamma+\pi^{*}H)$ is log canonical. Therefore, it suffices to show that
$(X,\Gamma+\pi^{*}H)$ has a good minimal model over $B$. As
$K_{X}+\Gamma+(D-\Gamma)\sim_{{\mathbb{R}},Z}0$, by Theorem 2.10 we see that
$(X,\Gamma)$ has a good minimal model over $Z$. Since $H$ is ample over $B$,
the statement we want follows from Theorem 2.11. ∎

We now consider the general case of Theorem 3.1.

###### Proof of Theorem 3.1.

As in the proof of Lemma 3.3, we may assume that $K_{X}+\Delta$ is pseudo-
effective over $B$ (otherwise it has a Mori fiber space over $B$). We first
reduce to the case when the birational contraction to some weak log canonical
model of $(X,D)$ over $B$ is a morphism. By assumption, $(X,D)$ has a good
minimal model $(Z,D_{Z})$ over $B$. Denote by $\pi\colon X\dashrightarrow Z$
the induced birational contraction over $B$. Choose a common log resolution

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\psi}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{Z}$

and let $F$ be the sum of the $\varphi$-exceptional divisors. Since $(X,D)$ is
log canonical, we have $K_{W}+D_{W}+F-\varphi^{*}(K_{X}+D)$ is effective and
$\varphi$-exceptional. In particular, we see that $(Z,D_{Z})$ is a weak log
canonical model of $(W,D_{W}+F)$ over $B$. By Lemma 2.9, we also know that
$(X,\Delta)$ has a good minimal model over $B$ if and only if
$(W,\Delta_{W}+F)$ has a good minimal model over $B$. Thus in order to prove
the theorem we may replace $(X,D)$ by $(W,D_{W}+F)$ and $\Delta$ by
$\Delta_{W}+F$. In particular, we may assume that the induced birational map
$\pi\colon X\dashrightarrow Z$ to the weak log canonical model $(Z,D_{Z})$
over $B$ is a morphism. Moreover, we may assume that $(X,D)$ is
${\mathbb{Q}}$-factorial dlt.

We write

$K_{X}+D=\pi^{*}(K_{Z}+D_{Z})+E\sim_{{\mathbb{R}},Z}E$

where $E$ is $\pi$-exceptional. As $(Z,D_{Z})$ is a weak log canonical model
of $(X,D)$ over $B$, we also know that $E\geq 0$. Recall that $0\leq\Delta\leq
D$, thus from

$K_{X}+\Delta=K_{X}+D-(D-\Delta)\sim_{{\mathbb{R}},Z}E-(D-\Delta)$

we get

$K_{X}+\Delta\sim_{{\mathbb{R}},Z}M^{+}-M^{-}$

where $M^{+},M^{-}\geq 0$ have no common components, $M^{-}\leq D-\Delta$ and
$M^{+}\leq E$. In particular, $M^{+}$ is exceptional over $Z$. Since $(X,D)$
is ${\mathbb{Q}}$-factorial dlt, $(X,\Delta)$ is ${\mathbb{Q}}$-factorial dlt
as well. By Theorem 2.12, the $(K_{X}+\Delta)$-MMP over $Z$ with scaling of
some ample divisor contracts $M^{+}$ after finitely many steps. Let
$X\dashrightarrow Y$ be the corresponding birational contraction over $Z$, let
$\varphi\colon Y\to Z$ be the induced morphism and let
$\Delta^{\prime}=M^{-}_{Y}\geq 0$. Then $M^{+}_{Y}=0$ and by pushing forward
the previous identities to $Y$ we obtain

$K_{Y}+\Delta_{Y}+\Delta^{\prime}=\varphi^{*}(K_{Z}+D_{Z}).$

This implies that $(Z,D_{Z})$ is also a weak log canonical model of
$(Y,D^{\prime}:=\Delta_{Y}+\Delta^{\prime})$ over $B$, and moreover,
$K_{Y}+D^{\prime}$ is semiample over $B$. From $\Delta_{Y}\leq D^{\prime}\leq
D_{Y}$ and $\operatorname{Supp}(\Delta)=\operatorname{Supp}(D)$ we also see
that $\operatorname{Supp}(\Delta_{Y})=\operatorname{Supp}(D^{\prime})$, hence
Lemma 3.3 applies to $(Y,D^{\prime})$ and $\Delta_{Y}$, and we deduce that
$(Y,\Delta_{Y})$ has a good minimal model over $B$. Since $X\dashrightarrow Y$
is part of a $(K_{X}+\Delta)$-MMP over $Z$, it follows from Lemma 2.9 (and the
remark thereafter) that $(X,\Delta)$ has a good minimal model over $B$ as
well. The proof is now complete. ∎

###  3.2. Log canonical models

When we decrease coefficients from a locally stable family of pairs, we often
end up with pairs $(X,\Delta)$ such that $K_{X}+\Delta$ is no longer
${\mathbb{R}}$-Cartier. Nonetheless, they are of log canonical type (sometimes
also called potentially log canonical), i.e. there exists some
${\mathbb{R}}$-divisor $D\geq\Delta$ on $X$ such that $(X,D)$ is log
canonical. In this subsection, we define and study the log canonical models of
such pairs. A key input is the following construction.

######  Lemma 3.4 ([Kol18-log-plurigenera]*Proposition 19).

Let $(X,D)$ be a log canonical pair and let $\Gamma$ be an effective
${\mathbb{R}}$-divisor supported on $D$. Then there exists a small birational
projective morphism $\pi\colon Y\to X$ such that $-\Gamma_{Y}$ is
${\mathbb{R}}$-Cartier and $\pi$-ample.

###### Proof.

We reproduce the proof for the reader’s convenience. Let $W\to X$ be a dlt
modification of $(X,D)$ and let $E$ be the sum of its exceptional divisors.
Then $(W,D_{W}+E)$ is dlt and

$K_{W}+D_{W}+E\sim_{{\mathbb{R}},X}0.$

Choose some $0<\varepsilon\ll 1$ such that $D\geq\varepsilon\Gamma$. By
Theorem 2.10, the pair $(W,D_{W}-\varepsilon\Gamma_{W}+E)$ has a good minimal
model over $X$. Let $\pi\colon Y\to X$ be the corresponding log canonical
model over $X$. Note that $K_{Y}+D_{Y}+E_{Y}\sim_{{\mathbb{R}},X}0$, hence

$K_{Y}+D_{Y}-\varepsilon\Gamma_{Y}+E_{Y}\sim_{{\mathbb{R}},X}-\varepsilon\Gamma_{Y}$

is ${\mathbb{R}}$-Cartier and $\pi$-ample by construction. Let us show that
$\pi$ is small; in other words, $E_{Y}=0$. To see this, we observe that as
$-\Gamma_{Y}$ is $\pi$-ample, the $\pi$-exceptional locus is necessarily
contained in the support of $\Gamma_{Y}$. Since $E_{Y}$ and $\Gamma_{Y}$ do
not have any common component, we deduce that $E_{Y}=0$. This finishes the
proof. ∎

Repeating this construction for each component of $D$, we get

######  Corollary 3.5.

Let $(X,D)$ be a log canonical pair. Then there exists a small birational
projective morphism $\pi\colon Y\to X$ such that $K_{Y}$ and every irreducible
component of $D_{Y}$ are all ${\mathbb{Q}}$-Cartier.

By [BCHM10], every variety that underlies a klt pair has a small
${\mathbb{Q}}$-factorial modification, but this is no longer true in the log
canonical case. The construction above thus provides a small “partial
${\mathbb{Q}}$-factorial” modification for log canonical pairs. We should
point out that, if the pair $(X,D)$ is only slc, then it may not be possible
to find a small modification as in Corollary 3.5. In fact, if $X$ is a
surface, then the only small birational modification is the identity, but not
every irreducible component of $D$ is $\mathbb{Q}$-Cartier (e.g. if the
component intersects some double locus where two components of $X$ meet).

We next define log canonical models and good minimal models for pairs of log
canonical type.

######  Definition 3.6.

Let $(X,\Delta)$ be a pair of log canonical type over $B$. Suppose that
$X\dashrightarrow Z$ is a birational contraction over $B$. Then
$(Z,\Delta_{Z})$ is called a log canonical model (resp. good minimal model) of
$(X,\Delta)$ over $B$ if there exists a small birational projective morphism
$\pi\colon Y\to X$ such that $K_{Y}+\Delta_{Y}$ is ${\mathbb{R}}$-Cartier and
$(Z,\Delta_{Z})$ is a log canonical model (resp. good minimal model) of
$(Y,\Delta_{Y})$ over $B$.

Similarly, we can define weak log canonical models for pairs of log canonical
type. We leave the details to the reader.

######  Remark 3.7.

The existence of such a small birational projective morphism is guaranteed by
Lemma 3.4. Since $K_{Y}+\Delta_{Y}$ is ${\mathbb{R}}$-Cartier,
$(Y,\Delta_{Y})$ is a log canonical pair. Thus we can talk about the log
canonical model (resp. good minimal models) of $(Y,\Delta_{Y})$ over $B$. By
Lemma 2.9, the existence of the log canonical model (resp. good minimal
models) is independent of the choice of the small birational projective
morphisms, and the log canonical model of $(X,\Delta)$ over $B$, if exists, is
unique. Moreover, it coincides with the standard definition of the log
canonical model (resp. good minimal models) of $(X,\Delta)$ over $B$ if
$K_{X}+\Delta$ is ${\mathbb{R}}$-Cartier.

######  Remark 3.8.

Let $g\colon Y\to X$ be any birational projective morphism such that
$(Y,\Delta_{Y}+E)$ is a log canonical pair where $E$ is the sum of all the
$g$-exceptional divisors. Then by Lemma 2.9, the log canonical model of
$(X,\Delta)$ over $B$ defined above is the same as the log canonical model of
$(Y,\Delta_{Y}+E)$ over $B$. Since $(X,\Delta)$ is a pair of log canonical
type over $B$, there exists an ${\mathbb{R}}$-divisor $D\geq\Delta$ on $X$
such that $(X,D)$ is log canonical. In practice, we usually choose $g$ to be a
dlt modification of $(X,D)$ or a log resolution of $(X,D)$. It is convenient
to work with these special birational projective morphisms sometimes.

The results from the previous subsection then have straightforward
generalizations to the non-${\mathbb{R}}$-Cartier setting. In particular, we
have

######  Theorem 3.9.

Let $(X,D)$ be a pair of log canonical type over $B$ and let $0\leq\Delta\leq
D$ be an ${\mathbb{R}}$-divisor. Assume that $(X,D)$ has a good minimal model
over $B$, $K_{X}+\Delta$ is pseudo-effective (resp. big) over $B$, and
$\operatorname{Supp}(\Delta)=\operatorname{Supp}(D)$. Then $(X,\Delta)$ has a
good minimal model (resp. log canonical model) over $B$.

Here we say that $K_{X}+\Delta$ is pseudo-effective over $B$ if there exists a
small birational projective morphism $Y\to X$ such that $K_{Y}+\Delta_{Y}$ is
${\mathbb{R}}$-Cartier and pseudo-effective over $B$. This notion is
independent of the choice of the small birational projective morphisms.

###### Proof.

By Corollary 3.5, there exists a small birational projective morphism
$\pi\colon Y\to X$ such that $K_{Y}+D_{Y}$ and $K_{Y}+\Delta_{Y}$ are both
${\mathbb{R}}$-Cartier. By Remark 3.7, we see that $(Y,D_{Y})$ has a good
minimal model over $B$. Thus our assumptions imply that $(Y,D_{Y})$ and
$\Delta_{Y}$ satisfy the assumptions of Theorem 3.1, which then implies that
$(Y,\Delta_{Y})$ has a good minimal model over $B$. If $K_{X}+\Delta$ is big
over $B$, so is $K_{Y}+\Delta_{Y}$ (since $\pi$ is small), hence we also get a
log canonical model of $(Y,\Delta_{Y})$ over $B$. By definition, it follows
that $(X,\Delta)$ has a good minimal model (resp. log canonical model if
$K_{X}+\Delta$ is big over $B$) over $B$. ∎

##  4\. MMP for locally stable families

###  4.1. MMP with scaling

The goal of this section is to develop an MMP theory for locally stable
families. The main result is the following statement. This is another main
ingredient in the construction of the reduction morphisms.

######  Theorem 4.1.

Let $f\colon(X,D)\to B$ be a locally stable family over a normal variety with
normal generic fiber. Then for any $f$-ample ${\mathbb{R}}$-divisor $H$ on
$X$, we may run the $(K_{X/B}+D)$-MMP over $B$ with scaling of $H$. Moreover,
the MMP terminates with a good minimal model over $B$ if the generic fiber has
a good minimal model.

When $K_{X/B}+D$ is pseudo-effective over $B$ and the $(K_{X/B}+D)$-MMP over
$B$ terminates, we get a minimal model $(X_{m},D_{m})\to B$ which is locally
stable such that $K_{X_{m}/B}+D_{m}$ is nef over $B$. We say that it is a good
minimal model of $(X,D)\to B$ if $K_{X_{m}/B}+D_{m}$ is semiample over $B$. If
in addition $K_{X/B}+D$ is big over $B$, we may also define the log canonical
model of $(X,D)\to B$ as the ample model of $K_{X_{m}/B}+D_{m}$ over $B$. It
is necessarily also the ample model of $K_{X/B}+D$ over $B$. All these are
analogous to Definition 2.6.

To prove Theorem 4.1, we shall first take a resolution $\widetilde{B}\to B$ of
singularities, run the MMP after the base change, and descend every step of
the MMP to $B$. As such, we need to analyze how maps such as extremal
contractions and flips descend. The next several lemmas are designed to take
care of these.

The first ingredient is the following invariance of plurigenera for good
minimal models of locally stable families. By cohomology and base change, it
eventually translates into the descent property of the extremal contractions.

######  Lemma 4.2.

Let $f\colon(X,D)\to B$ be a locally stable family over a connected reduced
scheme with normal generic fibers. Assume that $D$ is a
${\mathbb{Q}}$-divisor, $K_{X/B}+D$ is nef over $B$ and every generic fiber of
$f$ has a good minimal model. Then for any sufficiently divisible positive
integer $m$, the dimension of
$H^{0}(X_{b},\mathcal{O}_{X_{b}}(m(K_{X_{b}}+D_{b})))$ is independent of $b\in
B$.

###### Proof.

We may freely shrink $B$ around any chosen point as needed. In particular, we
may assume that $B$ is quasi-projective. Since the assumptions and the
statement are preserved by any birational proper base change, we may also
replace $B$ by a resolution of singularities and assume that $B$ is a smooth
variety. Note that this implies that $X$ is normal, since the fibers of $X\to
B$ are all reduced and $S_{2}$ while the generic fiber is normal, and together
with the smoothness of the base these properties imply that $X$ is smooth in
codimension one and $S_{2}$. Then $(X,D)$ is log canonical by Theorem 2.17,
and every log canonical center of $(X,D)$ dominates $B$ by [Kol23]*Corollary
4.56. By [HX13]*Theorem 1.1 and [Has19]*Theorem 1.2, we deduce that
$K_{X/B}+D$ is semiample over $B$. Let $b\in B$ and let
$\Gamma=\sum\Gamma_{i}$ be any simple normal crossing divisor on $B$
containing $b$ as a $0$-dimensional stratum where $\Gamma_{i}$ are the
irreducible components of $\Gamma$. By Theorem 2.17, the pair
$(X,D+f^{*}\Gamma)$ is log canonical.

Let $\pi\colon X\to Y$ be the ample model of $K_{X/B}+D$ over $B$ and let
$g\colon Y\to B$ be the induced morphism. In particular,
$K_{X}+D\sim_{{\mathbb{Q}},Y}0$. By the canonical bundle formula
[Kol07-canonical-bundle-formula]*Theorem 8.5.1, we may write

$K_{X}+D\sim_{{\mathbb{Q}}}\pi^{*}(K_{Y}+\Delta+M)$

where $(Y,\Delta+M)$ is a generalized pair over $B$ and
$(Y,\Delta+g^{*}\Gamma+M)$ is generalized log canonical. By [LX23]*Theorem
4.10 and [Kol13]*Theorem 9.26, any union (with the reduced scheme structure)
of the generalized log canonical centers of $(Y,\Delta+g^{*}\Gamma+M)$ is
seminormal. By [LX23]*Lemma 3.17(3), any intersection of unions of generalized
log canonical centers is a union of generalized log canonical centers. Putting
all these together, we deduce that any scheme-theoretic intersection of unions
of generalized log canonical centers is reduced by [Kol13]*Lemma 10.21 and
thus seminormal. Since $(Y,\Delta+g^{*}\Gamma+M)$ is generalized log
canonical, each $g^{*}\Gamma_{i}$ is reduced and is a union of generalized log
canonical centers of $(Y,\Delta+g^{*}\Gamma+M)$. Since $Y_{b}$ is a connected
component of the scheme-theoretic intersection of all $g^{*}\Gamma_{i}$, we
deduce that $Y_{b}$ is reduced and seminormal.

We claim that $g$ is flat. Morally this follows from a generalized log
canonical version of Theorem 2.17. As such a version does not seem available
in the literature, we will deduce flatness using some results in [Kol23].
Since $Y_{b}$ is geometrically reduced for every closed point $b\in B$, so is
$Y_{b}$ for every (possibly non-closed) $b\in B$ since the set $\\{b\in
B\,|\,Y_{b}\text{ is geometrically reduced over }k(b)\\}$ is constructible in
$B$. Since $f\colon X\to B$ is flat, every fiber of $f$ is of pure dimension
$\dim X-\dim B$. Since the dimensions of the fibers of $X\to Y$ and $Y\to B$
are upper semicontinuous and $(\dim X_{b}-\dim Y_{b})+\dim Y_{b}=\dim X_{b}$
is constant, we deduce that every fiber of $g\colon Y\to B$ is of pure
dimension $\dim Y-\dim B$. By [Kol23]*2.71.2, $g$ is universally open. Thus
the morphism $g$ is pure dimensional by [Kol23]*Claim 2.71.1. Since $g$ is
pure dimensional and has geometrically reduced fibers, $g$ is flat by
[Kol23]*Lemma 10.58.

Let $L=K_{Y}+\Delta+M$. Then by Lemma 4.3, we have
$(\pi_{b})_{*}\mathcal{O}_{X_{b}}\cong\mathcal{O}_{Y_{b}}$ and hence the
projection formula gives

(4.1) $H^{0}(X_{b},\mathcal{O}_{X_{b}}(m(K_{X_{b}}+D_{b})))\cong
H^{0}(X_{b},\mathcal{O}_{X_{b}}(m\pi_{b}^{*}L_{b}))\cong
H^{0}(Y_{b},\mathcal{O}_{Y_{b}}(mL_{b}))$

for any sufficiently divisible integer $m$. Note that $L$ is ample over $B$ by
construction, hence $R^{i}g_{*}\mathcal{O}_{Y}(mL)=0$ for any $i>0$ and any
sufficiently divisible positive integer $m$. By cohomology and base change, it
follows that $h^{0}(Y_{b},\mathcal{O}_{Y_{b}}(mL_{b}))$ is independent of
$b\in B$, hence the lemma follows from (4.1). ∎

The following lemma is used in the above proof.

######  Lemma 4.3.

Let $f\colon X\to Y$ be a proper surjective morphism of reduced schemes of
finite type over an algebraically closed field of characteristic $0$. Assume
that $f$ has connected fibers and $Y$ is seminormal. Then
$f_{*}\mathcal{O}_{X}\cong\mathcal{O}_{Y}$.

###### Proof.

This is well known. We take a Stein factorization $f=h\circ g$, where $g\colon
X\to Z$ is proper, $h\colon Z\to Y$ is finite, and
$g_{*}\mathcal{O}_{X}\cong\mathcal{O}_{Z}$. In particular, $g$ is surjective.
Since $f$ has connected fibers and is surjective, we deduce that $h$ is both
injective and surjective. Since $X$ is reduced and
$g_{*}\mathcal{O}_{X}\cong\mathcal{O}_{Z}$, $Z$ is also reduced. Thus $h\colon
Z\to Y$ is a partial seminormalization by [Kol13]*Definition 10.11 and the
remark thereafter, since we are working over an algebraically closed field of
characteristic $0$. Since $Y$ is seminormal, we deduce that $h$ is an
isomorphism and thus $f_{*}\mathcal{O}_{X}\cong\mathcal{O}_{Y}$. ∎

A direct consequence of Lemma 4.2 is the following statement. Among other
things, it guarantees that the extremal contractions in the MMP descend. For a
related result when the log canonical divisor is relatively semiample and big,
see [Kol23]*Proposition 8.35.

######  Lemma 4.4.

Let $f\colon(X,D)\to B$ be a locally stable family over a reduced scheme with
normal generic fibers. Let $\widetilde{B}\to B$ be a proper surjective
morphism from a reduced scheme and let
$\widetilde{f}\colon(\widetilde{X},\widetilde{D})\to\widetilde{B}$ be the
locally stable family obtained by base change via $\widetilde{B}\to B$. Assume
that $K_{\widetilde{X}/\widetilde{B}}+\widetilde{D}$ is nef over
$\widetilde{B}$ and every normal generic fiber of $\widetilde{f}$ has a good
minimal model. Then $K_{X/B}+D$ is semiample over $B$, and the formation of
its ample model commutes with base change.

###### Proof.

We may assume that every irreducible component of $\widetilde{B}$ dominates an
irreducible component of $B$. In particular, every generic fiber of
$\widetilde{f}$ is normal and has a good minimal model. After replacing
$\widetilde{B}$ by a resolution of singularities, we may further assume that
$\widetilde{B}$ is smooth. By the same argument as in the proof of Lemma 4.2,
we deduce that $K_{\widetilde{X}/\widetilde{B}}+\widetilde{D}$ is semiample
over $\widetilde{B}$. By Lemma 4.5, we may write $D$ as a convex combination
$D=\sum c_{i}D_{i}$ for some $c_{i}>0$, $\sum_{i}c_{i}=1$ and some effective
${\mathbb{Q}}$-divisors $D_{i}$ such that for each $i$, $(X,D_{i})\to B$ is
locally stable, $K_{\widetilde{X}/\widetilde{B}}+\widetilde{D}_{i}$ is
semiample over $\widetilde{B}$, and the ample model of
$K_{\widetilde{X}/\widetilde{B}}+\widetilde{D}_{i}$ over $\widetilde{B}$
coincides with that of $K_{\widetilde{X}/\widetilde{B}}+\widetilde{D}$. If the
lemma holds for all the families $(X,D_{i})\to B$, then it also holds for
$(X,D)\to B$. Therefore, in order to prove the lemma we may assume that $D$
has rational coefficients.

In particular, we may choose a positive integer $r$ such that
$r(K_{\widetilde{X}/\widetilde{B}}+\widetilde{D})$ is Cartier and base point
free over $\widetilde{B}$. It follows that $r(K_{X_{b}}+D_{b})$ is Cartier and
base point free for any $b\in B$, thus the semiampleness of $K_{X/B}+D$ over
$B$ follows if we can show that the formation of
$f_{*}\mathcal{O}_{X}(m(K_{X/B}+D))$ commutes with base change for any
sufficiently divisible positive integer $m$. Since $B$ is reduced, by
cohomology and base change it suffices to show that
$h^{0}(X_{b},\mathcal{O}_{X_{b}}(m(K_{X_{b}}+D_{b})))$ is locally constant
with respect to $b\in B$ (see e.g. [Har77]*Proof of Corollary 12.9). Since the
family $(X,D)\to B$ is locally stable, this condition can be checked after the
base change via the proper surjective morphism $\widetilde{B}\to B$, hence the
result follows from Lemma 4.2. ∎

The following result is used in the above proof.

######  Lemma 4.5.

Let $f\colon(X,D)\to B$ be a locally stable family over a reduced scheme.
Assume that $K_{X/B}+D$ is semiample over $B$. Then we can write
$D=\sum_{i}c_{i}D_{i}$ as a convex combination for some $c_{i}>0$,
$\sum_{i}c_{i}=1$ and some effective ${\mathbb{Q}}$-divisors $D_{i}$ such that
for each $i$, the family $(X,D_{i})\to B$ is locally stable, $K_{X/B}+D_{i}$
is semiample over $B$, and the ample model of $K_{X/B}+D_{i}$ over $B$
coincides with that of $K_{X/B}+D$.

###### Proof.

Let $g\colon X\to Y$ be the ample model of $K_{X/B}+D$ over $B$. By
[Kol23]*Proposition 11.47, there exist convex approximations $D=\sum
c_{i}^{n}D_{i}^{n}$ ($n=1,2,\dots$) where $c_{i}^{n}>0$, $\sum c_{i}^{n}=1$,
each $D_{i}^{n}$ is an effective ${\mathbb{Q}}$-divisor such that
$(X,D_{i}^{n})\to B$ is locally stable, and $\left\lVert
D-D_{i}^{n}\right\rVert\to 0$ as $n\to\infty$ (choose any norm on the vector
space spanned by the components of $D$). Moreover, for each fixed $n$, one may
also make the coefficients $c_{i}^{n}$ linearly independent over
${\mathbb{Q}}$. Since $K_{X/B}+D\sim_{\mathbb{R}}g^{*}H$ for some
${\mathbb{R}}$-Cartier ${\mathbb{R}}$-divisor $H$ that is ample over $B$, by
[Kol23]*Claim 11.43.2 we first see that
$K_{X/B}+D_{i}^{n}\sim_{\mathbb{Q}}g^{*}H_{i}^{n}$ for some
${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $H_{i}^{n}$ on $Y$; as
$n\to\infty$, we then have $\left\lVert H-H_{i}^{n}\right\rVert\to 0$, and
thus $H_{i}^{n}$ is ample over $B$ for sufficiently large $n$. In particular
$Y$ is also the ample model of $K_{X/B}+D_{i}^{n}$ over $B$, and the lemma
holds for $c_{i}:=c_{i}^{n}$ and $D_{i}:=D_{i}^{n}$ by taking some
sufficiently large $n$. ∎

We next need to show that flips descend. The main ingredient is the following
lemma.

######  Lemma 4.6.

Let $f\colon(X,D)\to B$ be a locally stable family over a normal variety with
normal generic fiber. Then for any effective ${\mathbb{R}}$-divisor $\Gamma$
supported on $D$, there exists a small birational projective morphism
$\pi\colon Y\to X$ such that:

  1. (1)

$(Y,D_{Y})\to B$ is locally stable with normal generic fiber, and

  2. (2)

$-\Gamma_{Y}$ is ${\mathbb{R}}$-Cartier and $\pi$-ample.

######  Remark 4.7.

We remark that this is the only place in the MMP where we need the base $B$ to
be normal; in fact, the examples in [ABIP23]*Section 8.1 can be used to show
that flips do not descend if the base is only reduced (or even seminormal). On
the other hand, we expect the lemma to hold when $X\to B$ is only quasi-
projective, although we do not have a proof.

###### Proof.

First note that the small birational modification $\pi$, if exists, is
uniquely determined by the given conditions through a Proj construction, thus
the question is of local nature and we may freely shrink $B$ as needed. Thus
we may assume that $B$ is quasi-projective. Let $\varphi\colon B^{\prime}\to
B$ be a resolution of singularities and let $(X^{\prime},D^{\prime})\to
B^{\prime}$ denote the base change of the family $(X,D)\to B$. Let
$\Gamma^{\prime}$ be the (Weil-divisor) pullback of $\Gamma$ (in the sense of
[Kol23]*Definition 4.2.7).

Since $B^{\prime}$ is smooth, the pair $(X^{\prime},D^{\prime})$ is log
canonical by Theorem 2.17. Thus by Lemma 3.4, there exists a small birational
projective morphism $\pi^{\prime}\colon Y^{\prime}\to X^{\prime}$ such that
$-\Gamma^{\prime}_{Y^{\prime}}$ is ${\mathbb{R}}$-Cartier and
$\pi^{\prime}$-ample. Since

(4.2)
$K_{Y^{\prime}}+D^{\prime}_{Y^{\prime}}=\pi^{\prime*}(K_{X^{\prime}}+D^{\prime}),$

the family $(Y^{\prime},D^{\prime}_{Y^{\prime}})\to B^{\prime}$ is also
locally stable by another application of Theorem 2.17. The plan is to descend
the family $(Y^{\prime},D^{\prime}_{Y^{\prime}})\to B^{\prime}$ to the
original base $B$.

Choose some $0<\varepsilon\ll 1$ and some general relative hyperplane section
$H$ on $X$ that is sufficiently ample over $B$. Let
$\Delta=D-\varepsilon\Gamma\geq 0$ and let $H^{\prime}$ (resp.
$\Delta^{\prime}$) be the pullback of $H$ (resp. $\Delta$) to $X^{\prime}$.
Then it is not hard to see that

$K_{Y^{\prime}/B^{\prime}}+\Delta^{\prime}_{Y^{\prime}}+H^{\prime}_{Y^{\prime}}=\pi^{\prime*}(K_{X^{\prime}/B^{\prime}}+D^{\prime}+H^{\prime})-\varepsilon\Gamma^{\prime}_{Y^{\prime}}$

is ample over $B^{\prime}$ and by Bertini theorem we may also assume that
$(Y^{\prime},\Delta^{\prime}_{Y^{\prime}}+H^{\prime}_{Y^{\prime}})\to
B^{\prime}$ is a stable family (i.e. the fibers are slc). If we can show that
for any closed point $b\in B$, the induced stable family over
$\varphi^{-1}(b)\subseteq B^{\prime}$ is a trivial family (as before we treat
$\varphi^{-1}(b)$ as a reduced scheme), then by Lemma 2.20 the family
$(Y^{\prime},\Delta^{\prime}_{Y^{\prime}}+H^{\prime}_{Y^{\prime}})\to
B^{\prime}$ descends to a stable family $(Y,\Delta_{Y}+H_{Y})\to B$ making the
diagram commutative with Cartesian squares:

$\textstyle{(Y^{\prime},\Delta^{\prime}_{Y^{\prime}}+H^{\prime}_{Y^{\prime}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{\prime}}$$\textstyle{(Y,\Delta_{Y}+H_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{(X^{\prime},D^{\prime}+H^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\prime}}$$\textstyle{(X,D+H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{B.}$

The induced map $\pi\colon Y\to X$ is then the small birational modification
we sought after. Note that $(Y,D_{Y})\to B$ is automatically locally stable
once we know its existence. Indeed, the morphism $Y\to B$ is flat by
construction, $K_{Y/B}+D_{Y}$ is ${\mathbb{R}}$-Cartier as it is the pullback
of $K_{X/B}+D$, and over a normal base the relative Mumford divisor condition
can be checked on the fibers (see the remark after Definition 2.13). Moreover,
the fibers of $(Y,D_{Y})\to B$ are also fibers of
$(Y^{\prime},D^{\prime}_{Y^{\prime}})\to B^{\prime}$, hence they are slc.
Hence $(Y,D_{Y})\to B$ is locally stable.

Let us now show that
$(Y^{\prime},\Delta^{\prime}_{Y^{\prime}}+H^{\prime}_{Y^{\prime}})\times_{B^{\prime}}\varphi^{-1}(b)$
is a trivial family over $\varphi^{-1}(b)$. In fact, it suffices to show that
$(Y^{\prime},D^{\prime}_{Y^{\prime}})\times_{B^{\prime}}\varphi^{-1}(b)$ is a
trivial family, since then $\Delta^{\prime}_{Y^{\prime}}$ (supported on
$D^{\prime}_{Y^{\prime}}$) and $H^{\prime}_{Y^{\prime}}$ (itself a pullback
from $X$) cannot have any variation over $\varphi^{-1}(b)$.

To simplify the notation we denote $W:=\varphi^{-1}(b)$ and
$(Z,G):=(Y^{\prime},D^{\prime}_{Y^{\prime}})\times_{B^{\prime}}\varphi^{-1}(b)$.
The restriction of the morphism $\pi^{\prime}\colon Y^{\prime}\to X^{\prime}$
to $Z\subseteq Y^{\prime}$ induces a morphism

$\psi\colon Z\to X^{\prime}\times_{B^{\prime}}\varphi^{-1}(b)\cong X_{b}\times
W.$

Since $-\Gamma^{\prime}_{Y^{\prime}}$ is $\pi^{\prime}$-ample, we deduce that

$\operatorname{Exc}(\pi^{\prime})\subseteq\operatorname{Supp}(\Gamma^{\prime}_{Y^{\prime}})\subseteq\operatorname{Supp}(D^{\prime}_{Y^{\prime}})$

and hence $\operatorname{Exc}(\psi)\subseteq\operatorname{Supp}(G)$. By
restricting (4.2) to $Z$ we also see that $(Z,G)$ is the crepant pullback of
$(X_{b},D_{b})\times W$. Note that $\psi$ has connected fibers since
$\pi^{\prime}$ does; moreover, $W$ is connected since $B$ is normal. We may
now conclude that the family $(Z,G)\to W$ is trivial by the following Lemma
4.8. ∎

######  Lemma 4.8.

Consider a birational morphism with connected fibers between locally stable
families over a connected reduced scheme:

$\textstyle{(Z,G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{(X,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B}$

Assume that:

  1. (1)

$\operatorname{Exc}(\psi)\subseteq\operatorname{Supp}(G)$,

  2. (2)

$\psi$ is crepant, i.e. $K_{Z/B}+G=\psi^{*}(K_{X/B}+D)$,

  3. (3)

$-G_{0}$ is $\psi$-ample for some effective ${\mathbb{R}}$-Cartier
${\mathbb{R}}$-divisor $G_{0}$ supported on $G$, and

  4. (4)

$(X,D)\to B$ is a trivial family.

Then $(Z,G)\to B$ is a trivial family.

###### Proof.

The lemma is then proved just as in [ABIP23]*Proposition 6.4 and Lemma 6.6.
The main points are:

  * •

Kollár’s gluing theory [Kol13]*Proposition 5.3 ensures that $(Z,G)$ is
determined by its normalization together with the gluing relation induced by
the involution on the conductor. If the normalization is a trivial family,
then as the exceptional locus of $\psi$ does not contain any generic point of
the non-normal locus of any fiber (by assumption (1)), the gluing relation is
just induced by that of $(X,D)$, hence is also constant in the family and it
follows that $(Z,G)\to B$ is also necessarily a trivial family. So we may
reduce to the case when $Z$ and $X$ are normal.

  * •

Assumption (3) implies that $Z$ is determined, through a Proj construction, by
the exceptional divisors extracted by $\psi$ and the coefficients of $G_{0}$.

  * •

Assumptions (1) and (2) imply that the $\psi$-exceptional divisors have
negative discrepancies with respect to $(X,D)$ (these can be defined e.g. over
the generic points of $B$). The only way to get such divisors is by
successively blowing up strata of a log resolution of $(X,D)$ ([ABIP23]*Lemma
6.6). If $(X,D)\to B$ is a trivial family and the base is connected, then such
$\psi$-exceptional divisors have no variation in the family. Thus the family
$(Z,G)\to B$ is also trivial.

∎

The last ingredient we need is the fact that the family remains locally stable
after each MMP step. We already see this for flips (Lemma 4.6), so it remains
to check the case of extremal contractions. We shall prove a slightly more
general statement, since it is also useful in Section 6 when we study the
wall-crossing morphisms.

######  Lemma 4.9.

Let $f\colon(X,D)\to B$ be a locally stable family over a reduced scheme with
normal generic fibers. Assume that $K_{X/B}+D$ is semiample and big over $B$
and let $\pi\colon X\to Y$ be its ample model over $B$. Then
$g\colon(Y,D_{Y}:=\pi_{*}D)\to B$ is stable.

When the base $B$ is seminormal, this is essentially proved in
[ABIP23]*Theorem 5.1. The stronger result above over a reduced base is one of
the ingredients which allows us to construct the wall-crossing morphisms
without passing to the seminormalization.

###### Proof.

We need to verify the conditions in Section 2.5. Since the question is local
on $B$ we may freely shrink $B$ as needed. Thus we may assume that $B$ is
connected.

The flatness of $Y\to B$ and the equidimensionality of
$\operatorname{Supp}(D_{Y})\to B$ are checked just as in [ABIP23]*Theorem 5.1.
Here we briefly recall the argument for the reader’s convenience. Since the
formation of the ample model commutes with base change by Lemma 4.4, the fiber
$Y_{b}$ of $g$ is the ample model of $K_{X_{b}}+D_{b}$, which is necessarily
reduced and connected. By the same argument as in the proof of Lemma 4.2, $g$
has geometrically reduced fibers of pure dimension $\dim Y-\dim B$. Since $f$
is flat, $f$ is universally open. Thus $g$ is universally open since $\pi$ is
surjective. Thus the morphism $g$ is pure dimensional by [Kol23]*Claim 2.71.1.
Since $g$ is pure dimensional and has geometrically reduced fibers, $g$ is
flat by [Kol23]*Lemma 10.58. Since $K_{X/B}+D$ is big over $B$, the morphism
$\pi$ is birational and thus $\dim Y_{b}=d$ for all $b\in B$, where $d$ is the
relative dimension of $f\colon X\to B$. If $\Delta$ is an irreducible
component of $D_{Y}$, then it is the $\pi$-image of some irreducible component
$\Gamma$ of $D$. Since $\Gamma$ dominates an irreducible component of $B$, so
does $\Delta$. Moreover, $\dim\Delta_{b}\leq\dim\Gamma_{b}=d-1$ with equality
for general $b\in B$. Combined with the upper semicontinuity of fiber
dimensions, we see that $\dim\Delta_{b}=\dim\Gamma_{b}=d-1$ for all $b\in B$.
This gives the equidimensionality condition.

We proceed to check the Mumford and the generic Cartier conditions (Definition
2.13). By [Kol23]*Proposition 4.26, these can be checked on a general relative
hyperplane section on $Y$. By Bertini theorem and [Kol23]*Proposition 10.56,
we also know that the restriction of $(X,D)$ to the general member of a
relative base point free linear system (e.g. the pullback of a general
relative hyperplane section on $Y$) remains locally stable over $B$. Thus we
may assume that $X\to B$ has relative dimension $1$. In particular, each fiber
$X_{b}$ is a nodal curve and $\operatorname{Supp}(D_{b})$ is contained in its
smooth locus. By [Has03]*Theorem 3.6, the ample model of $K_{X/B}+D$ over $B$
is a stable family. In particular, it satisfies the Mumford and the generic
Cartier conditions.

To this end, we conclude that $(Y,D_{Y})\to B$ is a family of marked pairs (in
the sense of Definition 2.15). It remains to check that it is a stable family
(Definition 2.18). By construction, we have $K_{X/B}+D=\pi^{*}(K_{Y/B}+D_{Y})$
and $K_{Y/B}+D_{Y}$ is ${\mathbb{R}}$-Cartier. Since the formation of the
ample model commutes with base change by Lemma 4.4, the remaining conditions
in Definition 2.18 can be checked after a base change. In particular, we may
replace $B$ by a resolution of singularities and assume it is smooth. The
statement then follows from [Kol23]*Corollary 4.57. ∎

We are now in a position to run the MMP for locally stable families over a
normal base.

###### Proof of Theorem 4.1.

Let $\widetilde{B}\to B$ be a resolution of singularities and let
$(\widetilde{X},\widetilde{D})$ be the locally stable family obtained from
$(X,D)$ by base change to $\widetilde{B}$. Our assumption guarantees that
$\widetilde{X}$ is normal, since the fibers of $\widetilde{X}\to\widetilde{B}$
are all reduced and $S_{2}$ while the generic fiber is normal, and these
properties imply that $\widetilde{X}$ is smooth in codimension one and
$S_{2}$.

By Theorem 2.17, the pair $(\widetilde{X},\widetilde{D})$ is log canonical,
hence we may run the $(K_{\widetilde{X}/\widetilde{B}}+\widetilde{D})$-MMP
(equivalently, the $(K_{\widetilde{X}}+\widetilde{D})$-MMP) over
$\widetilde{B}$ with scaling of $\widetilde{H}$ (more details on this below),
where $\widetilde{H}$ is the pullback of $H$ to $\widetilde{X}$. To prove the
theorem, our goal is to show that every step in this MMP descends to $B$.

The steps in this (non-${\mathbb{Q}}$-factorial)
$(K_{\widetilde{X}/\widetilde{B}}+\widetilde{D})$-MMP with scaling over
$\widetilde{B}$ take the following form (see e.g. [Kol21-MMP-non-Q-
factorial]*Definition 1). Let $(\widetilde{X}_{i},\widetilde{D}_{i})$ be the
birational model obtained at the $i$-th step (when $i=0$ we set
$(\widetilde{X}_{0},\widetilde{D}_{0}):=(\widetilde{X},\widetilde{D})$) and
let $\widetilde{H}_{i}$ be the pushforward of $\widetilde{H}$ to
$\widetilde{X}_{i}$ which remains ${\mathbb{R}}$-Cartier as will be seen from
the construction. Let $t_{i}\geq 0$ be the smallest nonnegative number such
that $K_{\widetilde{X}_{i}}+\widetilde{D}_{i}+t_{i}\widetilde{H}_{i}$ is nef
over $\widetilde{B}$. If $t_{i}=0$, the MMP terminates with a minimal model
over $\widetilde{B}$. Otherwise $t_{i}>0$ and we let $\widetilde{Z}_{i}$ be
the ample model of
$K_{\widetilde{X}_{i}}+\widetilde{D}_{i}+t_{i}\widetilde{H}_{i}$ over
$\widetilde{B}$, whose existence is guaranteed by [Fuj11]*Theorem 1.1(4)
(alternatively, it follows from Theorem 2.11, since
$(\widetilde{X}_{i},\widetilde{D}_{i}+t_{i}\widetilde{H}_{i})$ is a minimal
model of $(\widetilde{X},\widetilde{D}+t_{i}\widetilde{H})$ over
$\widetilde{B}$). If $\dim\widetilde{Z}_{i}<\dim\widetilde{X}_{i}$, the MMP
terminates with a Fano fibration over $\widetilde{B}$. Otherwise, the morphism
$\widetilde{\varphi}_{i}\colon\widetilde{X}_{i}\to\widetilde{Z}_{i}$ is
birational and the next step of the MMP factorizes as

$\textstyle{(\widetilde{X}_{i},\widetilde{D}_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\varphi}_{i}}$$\scriptstyle{\widetilde{f}_{i}}$$\textstyle{\widetilde{Z}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\widetilde{X}_{i+1},\widetilde{D}_{i+1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\psi}_{i}}$$\scriptstyle{\widetilde{f}_{i+1}}$$\textstyle{\widetilde{B}}$

where $\widetilde{\psi}_{i}$ is a small birational projective morphism such
that $-\widetilde{H}_{i+1}$ is $\widetilde{\psi}_{i}$-ample. Its existence is
guaranteed by Lemma 3.4, since
$(\widetilde{Z}_{i},\widetilde{\varphi}_{i*}(\widetilde{D}_{i}+t_{i}\widetilde{H}_{i}))$
is log canonical for a general choice of $H\geq 0$ in its
${\mathbb{R}}$-linear equivalence class over $B$. We shall fix this choice of
$H$ in the rest of the proof. In particular, since

$\displaystyle
K_{\widetilde{X}_{i+1}}+\widetilde{D}_{i+1}+(t_{i}-\varepsilon)\widetilde{H}_{i+1}$
$\displaystyle=\widetilde{\psi}_{i}^{*}\widetilde{\varphi}_{i*}(K_{\widetilde{X}_{i}}+\widetilde{D}_{i}+t_{i}\widetilde{H}_{i})-\varepsilon\widetilde{H}_{i+1}$
$\displaystyle=\widetilde{\psi}_{i}^{*}(\mbox{ample}/\widetilde{B})-\varepsilon\widetilde{H}_{i+1}$

we know that
$K_{\widetilde{X}_{i+1}}+\widetilde{D}_{i+1}+(t_{i}-\varepsilon)\widetilde{H}_{i+1}$
is $\widetilde{f}_{i+1}$-ample for $0<\varepsilon\ll 1$ and hence
$t_{i+1}<t_{i}$. By induction on $i$, this also implies that
$K_{\widetilde{X}_{i}}+\widetilde{D}_{i}+t\widetilde{H}_{i}$ is
$\widetilde{f}_{i}$-ample for $t_{i}<t<t_{i-1}$.

We need to descend this sequence of
$(K_{\widetilde{X}/\widetilde{B}}+\widetilde{D})$-MMP with scaling over
$\widetilde{B}$ to $B$. By induction on $i$, we may assume that
$(\widetilde{X}_{i},\widetilde{D}_{i}+t_{i}\widetilde{H}_{i})\to\widetilde{B}$
descends to a locally stable family $(X_{i},D_{i}+t_{i}H_{i})\to B$. By Lemma
4.4, the ample model
$\widetilde{\varphi}_{i}\colon\widetilde{X}_{i}\to\widetilde{Z}_{i}$ descends
to give the ample model $\varphi_{i}\colon X_{i}\to Z_{i}$ of
$K_{X_{i}/B}+D_{i}+t_{i}H_{i}$ over $B$. Moreover, when $\varphi_{i}$ is
birational, the induced family $(Z_{i},\varphi_{i*}(D_{i}+t_{i}H_{i}))\to B$
is stable by Lemma 4.9. By Lemma 4.6, there exists a small birational
projective morphism $\psi_{i+1}\colon X_{i+1}\to Z_{i}$, which is necessarily
the descent of
$\widetilde{\psi}_{i+1}\colon\widetilde{X}_{i+1}\to\widetilde{Z}_{i}$, such
that $-H_{i+1}$ is $\psi_{i+1}$-ample and $(X_{i+1},D_{i+1}+t_{i}H_{i+1})\to
B$ is a locally stable family, where $D_{i+1}$ (resp. $H_{i+1}$) is the
pushforward of $D_{i}$ (resp. $H_{i}$). This finishes the construction of the
MMP with scaling over $B$.

Finally, if the generic fiber of
$(\widetilde{X},\widetilde{D})\to\widetilde{B}$, which is also the generic
fiber of $(X,D)\to B$, has a good minimal model, then since every log
canonical center of $(\widetilde{X},\widetilde{D})$ dominates $\widetilde{B}$
by [Kol23]*Corollary 4.56, using [HX13]*Theorem 1.1 and [Has19]*Theorem 1.2 we
conclude that $(\widetilde{X},\widetilde{D})$ has a good minimal model
$(\widetilde{X}_{m},\widetilde{D}_{m})$ over $\widetilde{B}$. From the
previous discussion, we know that it descends to a minimal model
$(X_{m},D_{m})$ of $(X,D)$ over $B$. By Lemma 4.4, we see that $(X_{m},D_{m})$
is also a good minimal model over $B$. The proof is now complete. ∎

###  4.2. Semi-log-canonical models

If the locally stable family in Theorem 4.1 does not have a normal generic
fiber, then the MMP may not exist, since MMP does not apply to slc pairs in
general (see e.g. [Kol11-slc-no-MMP, AK19] for some related examples).
However, if we assume that the generic fiber has a semi-log-canonical model,
then the family still has a relative semi-log-canonical model. This is enough
for most applications in our work.

######  Theorem 4.10.

Let $f\colon(X,D)\to B$ be a locally stable family over a normal variety and
let $0\leq\Delta\leq D$ be an ${\mathbb{R}}$-divisor. Assume that the generic
fiber of $(X,\Delta)\to B$ has a semi-log-canonical model. Then the family
$(X,\Delta)\to B$ has a semi-log-canonical model.

To make this precise, we need to define semi-log-canonical models for the
families that appear in the above statement. If the generic fiber is normal
and $K_{X/B}+\Delta$ is ${\mathbb{R}}$-Cartier, then $(X,\Delta)\to B$ is
locally stable and its (semi-)log canonical model is just the log canonical
model defined right after Theorem 4.1. The ${\mathbb{R}}$-Cartier assumption
can be dropped from the definition in a way similar to Definition 3.6.

######  Definition 4.11.

Let $(X,D)\to B$ be a locally stable family over a normal variety with normal
generic fiber and let $0\leq\Delta\leq D$ be an ${\mathbb{R}}$-divisor such
that $K_{X/B}+\Delta$ is big on the generic fiber. Suppose that
$X\dashrightarrow Z$ is a birational contraction over $B$ and
$(Z,\Delta_{Z})\to B$ is a stable family. Then $(Z,\Delta_{Z})\to B$ is called
a log canonical model of $(X,\Delta)\to B$ if there exists a small birational
projective morphism $Y\to X$ such that $(Y,\Delta_{Y})\to B$ is locally stable
with normal generic fiber and $(Z,\Delta_{Z})\to B$ is the log canonical model
of $(Y,\Delta_{Y})\to B$.

######  Remark 4.12.

Such a small birational projective morphism $Y\to X$ always exists by Lemma
4.6. Since a stable family is determined by its generic fiber (thanks to the
separatedness of the moduli of stable pairs), by Remark 3.7 we see that the
log canonical model $(Z,\Delta_{Z})\to B$ is unique if it exists. Moreover, by
applying Theorem 4.1 to the family $(Y,\Delta_{Y})\to B$ we deduce that
$(X,\Delta)\to B$ has a log canonical model if and only if its generic fiber
has one. Thus we can also define the log canonical model as a stable family
over $B$ whose generic fiber is the log canonical model of the generic fiber
of $(X,\Delta)\to B$. This interpretation will be convenient in some of our
subsequent discussions.

To define semi-log-canonical models in the general setting, the following
observation is necessary.

######  Lemma 4.13.

Let $f\colon(X,D)\to B$ be a locally stable family over a normal variety. Then
its normalization $(X^{\nu},\Gamma+D^{\nu})\to B$ (where $\Gamma$ is the
conductor) is a disjoint union of locally stable families over $B$.

The main subtlety here is the flatness of $X^{\nu}\to B$.

###### Proof.

By adding a general relative hyperplane section to $D$ we may assume that the
family is stable. When $B$ is smooth, the lemma is a direct consequence of
Theorem 2.17. In general, let $\varphi\colon\widetilde{B}\to B$ be a
resolution of singularities and let
$(\widetilde{X},\widetilde{D})\to\widetilde{B}$ be the stable family obtained
by base change. Then its normalization
$(\widetilde{X}^{\nu},\widetilde{\Gamma}+\widetilde{D}^{\nu})\to\widetilde{B}$
is stable (more precisely, it is a disjoint union of stable families over
$\widetilde{B}$). We claim that it descends to a stable family over $B$. By
Lemma 2.20, it suffices to check that the family is trivial along the
exceptional fibers of $\varphi$.

Let $b\in B$, $W:=\varphi^{-1}(b)$ and
$Z:=\widetilde{X}^{\nu}\times_{\widetilde{B}}W$. Note that $W$ is connected as
$B$ is normal. As in the proof of Lemma 4.6, in order to prove the above claim
we only need to show that $Z\to W$ is a trivial family, since the boundary
part is supported on the preimage of
$\operatorname{Supp}(D_{b})\cup\operatorname{Sing}(X_{b})$, thus cannot have
any variation. Since
$(\widetilde{X}^{\nu},\widetilde{\Gamma}+\widetilde{D}^{\nu})\to\widetilde{B}$
is stable, the conductor $\widetilde{\Gamma}$ does not contain any irreducible
component of any fiber of $\widetilde{X}^{\nu}\to\widetilde{B}$. As the
normalization morphism $\widetilde{X}^{\nu}\to\widetilde{X}$ is an isomorphism
away from $\widetilde{\Gamma}$, we deduce that it restricts to a partial
normalization $Z_{w}\to X_{b}$ for any $w\in W$. The latter is uniquely
determined if we know which codimension one nodes in $X_{b}$ are normalized,
because $Z_{w}$ is $S_{2}$. Since there are only finitely many codimension one
nodes and $W$ is connected, we see that $Z_{w}\to X_{b}$ is independent of
$w\in W$. Thus the conditions of Lemma 2.20 are verified and we conclude that
$(\widetilde{X}^{\nu},\widetilde{\Gamma}+\widetilde{D}^{\nu})\to\widetilde{B}$
descends to a stable family $(X^{\nu},\Gamma+D^{\nu})\to B$, which is at least
a partial normalization of $(X,D)\to B$.

It remains to check that $X^{\nu}$ is normal. Since the fibers of
$(X^{\nu},\Gamma+D^{\nu})\to B$ are slc, we see that $X^{\nu}\to B$ has
$S_{2}$ fibers. Combined with the normality of $B$, we deduce from [EGA-
IV-2]*Corollaire 6.4.2 that $X^{\nu}$ is $S_{2}$. Moreover, since $X^{\nu}\to
B$ has reduced fibers of constant dimension as well as a normal generic fiber,
we see that the singular locus of $X^{\nu}$ has codimension at least two.
These together imply that $X^{\nu}$ is normal. Thus
$(X^{\nu},\Gamma+D^{\nu})\to B$ is the normalization of $(X,D)\to B$ and it is
stable by construction. ∎

By Kollár’s gluing theory [Kol13]*Chapter 5, a stable family is determined by
its normalization together with the induced involution on
$(\Gamma^{\nu},\operatorname{Diff}_{\Gamma^{\nu}}(D^{\nu}))$, where
$\Gamma^{\nu}$ is the normalization of $\Gamma$ and
$\operatorname{Diff}_{\Gamma^{\nu}}(D^{\nu})$ denotes the different (see e.g.
[Kol13]*Definition 4.2).

######  Definition 4.14.

Let $(X,D)\to B$ be a locally stable family over a normal variety and let
$0\leq\Delta\leq D$ be an ${\mathbb{R}}$-divisor such that $K_{X/B}+\Delta$ is
big on every irreducible component of the generic fiber. A stable family
$(Y,\Delta_{Y})\to B$ is called a semi-log-canonical (slc) model of
$(X,\Delta)\to B$ if there exists a birational map $\varphi\colon
X\dashrightarrow Y$ over $B$ such that the normalization of $(Y,\Delta_{Y})$
is the log canonical model over $B$ of the normalization of $(X,\Delta)$.

Since $X$ and $Y$ have multiple irreducible components here, birational maps
between them are defined as in [Kol13]*Definition 1.11. In particular, every
irreducible component of $X$ dominates an irreducible component of $Y$, and
the map $\varphi$ is an isomorphism over the generic points of $Y$. The
definition also implies that the conductor on $Y$ is the strict transform of
the conductor on $X$.

###### Proof of Theorem 4.10.

Let $(X^{\nu},\Gamma+D^{\nu})\to B$ be the normalization of $(X,D)\to B$ as
before and let $\tau$ be the induced involution on
$(\Gamma^{\nu},\operatorname{Diff}_{\Gamma^{\nu}}(D^{\nu}))$ where
$\Gamma^{\nu}$ is the normalization of $\Gamma$. Let $\Delta^{\nu}$ be the
strict transform of $\Delta$. By assumption, the generic fiber of the family
$(X^{\nu},\Gamma+\Delta^{\nu})\to B$ has a log canonical model, hence the
family $(X^{\nu},\Gamma+\Delta^{\nu})\to B$ also has a log canonical model
$(Y^{\nu},\Gamma_{Y}+\Delta^{\nu}_{Y})\to B$ by Remark 4.12. To construct the
slc model, we may now assume that $B$ is smooth. Indeed, if the slc model
exists after base change via a resolution of singularities $\widetilde{B}\to
B$, then as a stable family, the slc model we obtain over $\widetilde{B}$ is
automatically a trivial family along the exceptional fibers of
$\widetilde{B}\to B$ (this is because these fibers are connected, and there
are only finitely many stable pairs with a given partial normalization; c.f.
the proof of Lemma 4.8), hence by Lemma 2.20 it descends to give the slc model
over $B$.

Thus we shall assume that $B$ is smooth. To proceed, we need to show that the
induced birational involution

$\tau_{Y}\colon(\Gamma_{Y}^{\nu},\operatorname{Diff}_{\Gamma_{Y}^{\nu}}(\Delta^{\nu}_{Y}))\dashrightarrow(\Gamma_{Y}^{\nu},\operatorname{Diff}_{\Gamma_{Y}^{\nu}}(\Delta^{\nu}_{Y}))$

on the normalization $\Gamma_{Y}^{\nu}$ of $\Gamma_{Y}$ is a regular
involution. This holds over the generic point of $B$, since the generic fiber
of $(X,\Delta)\to B$ has an slc model by assumption. Note that
$(Y^{\nu},\Gamma_{Y}+\Delta^{\nu}_{Y})\to B$ is a stable family by
construction and hence
$(\Gamma_{Y}^{\nu},\operatorname{Diff}_{\Gamma_{Y}^{\nu}}(\Delta^{\nu}_{Y}))\to
B$ is a stable family by Theorem 2.17 and adjunction. By the same argument as
in the proof of [HX13]*Lemma 7.2, we then deduce that $\tau_{Y}$ is regular.

We conclude by using Kollár’s gluing theory. Note that
$(X^{\nu},\Gamma+\Delta^{\nu})$ is log canonical by Theorem 2.17, hence
$(Y^{\nu},\Gamma_{Y}+\Delta^{\nu}_{Y})$ is also log canonical. By
[Kol13]*Corollary 5.33 and Theorem 5.38, there exists a stable family
$(Y,\Delta_{Y})\to B$ whose normalization is
$(Y^{\nu},\Gamma_{Y}+\Delta^{\nu}_{Y})\to B$. This completes the proof. ∎

From the above proof, we see that the slc model is unique if it exists. By the
same argument, we also get the following corollary.

######  Corollary 4.15.

Let $B$ be an open subset of a normal variety $\overline{B}$, and let
$(X,D)\to B$ be a stable family. Assume that its normalization
$(X^{\nu},\Gamma+D^{\nu})\to B$ extends to a stable family
$(\overline{X}^{\nu},\overline{\Gamma}+\overline{D}^{\nu})\to\overline{B}$.
Then $(X,D)\to B$ extends uniquely to a stable family
$(\overline{X},\overline{D})\to\overline{B}$ whose normalization is
$(\overline{X}^{\nu},\overline{\Gamma}+\overline{D}^{\nu})\to\overline{B}$.

##  5\. Finiteness of ample models

In [BCHM10]*Corollary 1.1.5, results on the finiteness of ample models are
presented. To study the wall crossing for moduli of stable pairs in the log
canonical setting, we need to establish the finiteness of ample models in a
more general setting. This is the main task of this section.

Let us fix some notation. Let $f\colon X\to B$ be a projective morphism of
normal quasi-projective varieties. Let $D_{1},\ldots,D_{n}$ be prime divisors
on $X$. Consider the vector space

$V:=\bigoplus^{n}_{i=1}{\mathbb{R}}D_{i}\cong{\mathbb{R}}^{n}.$

For any $D=\sum^{n}_{i=1}a_{i}D_{i}$, we define

$\left\lVert D\right\rVert:=\max_{i}\\{|a_{i}|\\}$

which gives a norm on $V$. For a log canonical pair $(X,D)$ over $B$, we
define its ample model over $B$ as the ample model of $K_{X}+D$ over $B$.

We follow [BCHM10]*Section 3.7 for definitions on convex geometry. A polytope
(resp. rational polytope) in ${\mathbb{R}}^{n}$ is the convex hull of a finite
set of points (resp. rational points). A finite (resp. rational) polyhedral
decomposition of a (resp. rational) polytope $\mathcal{P}$ is a finite
decomposition $\mathcal{P}=\cup_{i=1}^{p}\mathcal{P}_{i}$ such that each
$\mathcal{P}_{i}$ is a (resp. rational) polytope, every face of
$\mathcal{P}_{i}$ coincides with $\mathcal{P}_{j}$ for some $j$, and for all
$1\leq i,j\leq p$, the intersection of $\mathcal{P}_{i}$ and $\mathcal{P}_{j}$
is either empty or a common face of both $\mathcal{P}_{i}$ and
$\mathcal{P}_{j}$. Usually the interior $\mathcal{P}_{i}^{\circ}$ of the
polytopes $\mathcal{P}_{i}$ are called the chambers, while their faces are
called the walls. The main result of this section is the following.

######  Theorem 5.1.

Notation as above. Let $\mathcal{P}\subseteq V$ be a rational polytope such
that $(X,D)$ is a log canonical pair and has a good minimal model over $B$ for
every $D\in\mathcal{P}$.

Then there exist a finite rational polyhedral decomposition
$\mathcal{P}=\cup_{i=1}^{p}\mathcal{P}_{i}$, together with finitely many
rational maps $\psi_{j}\colon X\dashrightarrow Z_{j}$ over $B$, $1\leq j\leq
q$, such that: for any $1\leq i\leq p$, there exists some $1\leq j\leq q$ such
that $\psi_{j}\colon X\dashrightarrow Z_{j}$ is the ample model of $(X,D)$
over $B$ for any $D\in\mathcal{P}_{i}^{\circ}$.

The proof largely follows the general inductive strategy of [BCHM10]. The key
step is Proposition 5.5, which essentially implies that the finite rational
polyhedral decomposition locally exists. Before the proof, we need to first
prepare a few auxiliary lemmas.

######  Lemma 5.2.

Let $\mathcal{P}\subseteq V$ be a polytope such that $(X,\Delta)$ is log
canonical for every $\Delta\in\mathcal{P}$, and let $D\in\mathcal{P}$ be such
that $K_{X}+D$ is nef over $B$. Assume that $X$ is ${\mathbb{Q}}$-factorial.
Then there exists a positive real number $\delta$ such that for any
$G\in\mathcal{P}$ satisfying $\left\lVert D-G\right\rVert<\delta$, any
$(K_{X}+G)$-MMP over $B$ is $(K_{X}+D)$-trivial.

###### Proof.

This is a well known consequence of the lengths of extremal rays. See e.g.
[Bir11]*Proposition 3.2(5) or [LT22]*Proposition 2.12. ∎

######  Lemma 5.3.

In the setting of Theorem 5.1, assume that $X$ is ${\mathbb{Q}}$-factorial and
let $D\in\mathcal{P}$. Then there exists a positive real number $\delta$ such
that:

for any $G\in\mathcal{P}$ satisfying $\left\lVert D-G\right\rVert<\delta$,
there exists a minimal model $X\dashrightarrow Y$ of $(X,G)$ over $B$ that is
also a weak log canonical model of $(X,D)$ over $B$.

###### Proof.

Since $(X,D)$ has a good minimal model over $B$, we can run a $(K_{X}+D)$-MMP
$\varphi\colon X\dashrightarrow W$ over $B$ which terminates with a good
minimal model $(W,D_{W})$ of $(X,D)$ over $B$ by [HH20]*Theorem 1.7. Since $X$
is ${\mathbb{Q}}$-factorial, $W$ is also ${\mathbb{Q}}$-factorial. Since
$\varphi$ consists of only finitely many $(K_{X}+D)$-MMP steps over $B$, there
exists a positive real number $\delta$ such that for any $G\in\mathcal{P}$
satisfying $\left\lVert D-G\right\rVert<\delta$, $\varphi$ is also a
$(K_{X}+G)$-MMP over $B$. Since $(X,G)$ has a good minimal model over $B$,
$(W,G_{W})$ has a good minimal model over $B$ by Lemma 2.9. By choosing
$\delta$ sufficiently small, we can assume that for any $G\in\mathcal{P}$
satisfying $\left\lVert D-G\right\rVert<\delta$, any $(K_{W}+G_{W})$-MMP over
$B$ is $(K_{W}+D_{W})$-trivial by Lemma 5.2 since $K_{W}+D_{W}$ is nef over
$B$. We can run a $(K_{W}+G_{W})$-MMP over $B$ which terminates with a good
minimal model $(Y,G_{Y})$ of $(W,G_{W})$ over $B$ by [HH20]*Theorem 1.7. Since
this $(K_{W}+G_{W})$-MMP over $B$ is $(K_{W}+D_{W})$-trivial, we see that
$(Y,D_{Y})$ is a weak log canonical model of $(X,D)$ over $B$. ∎

We also need a slight improvement of the above lemma that allows perturbation
of the divisor $D$ to some ${\mathbb{Q}}$-divisor in the same neighbourhood.
This is crucial in order to show that the polyhedral decomposition in Theorem
5.1 can be made rational.

######  Lemma 5.4.

In the setting of Theorem 5.1, assume that $X$ is ${\mathbb{Q}}$-factorial and
let $D\in\mathcal{P}$. Then there exists a positive real number $\delta$ such
that:

for any $\Delta\in\mathcal{P}$ in the rational envelope of $D$ satisfying
$\left\lVert D-\Delta\right\rVert<\delta$, and any $G\in\mathcal{P}$ with
$\left\lVert D-G\right\rVert<\delta$, there exists a minimal model
$X\dashrightarrow Y$ of $(X,G)$ over $B$ that is also a weak log canonical
model of $(X,\Delta)$ over $B$.

Recall that the rational envelope of $D\in V$ is defined as the smallest
rational affine subspace of $V$ that contains $D$.

###### Proof.

Let $\delta$ be the positive real number given by Lemma 5.3. We analyze the
maps in the proof of Lemma 5.3:

${(X,D)}$${(W,D_{W})}$${(Y,D_{Y}).}$$\scriptstyle{\varphi}$$\scriptstyle{\psi}$

Recall that $\varphi$ is a $(K_{X}+G)$-MMP over $B$ for all $G\in\mathcal{P}$
with $\left\lVert D-G\right\rVert<\delta$ and $(Y,G_{Y})$ is a minimal model
of $(X,G)$ over $B$. Moreover, any $(K_{W}+G_{W})$-MMP over $B$ is
$(K_{W}+D_{W})$-trivial. For any curve $C\subseteq W$, the condition
$(K_{W}+D_{W})\cdot C=0$ on $D$ defines a hyperplane in $V$ with rational
coefficients, hence it contains the rational envelope of $D$. In particular,
if $\Delta\in\mathcal{P}$ lies in the rational envelope of $D$, then
$(K_{W}+D_{W})\cdot C=0$ implies $(K_{W}+\Delta_{W})\cdot C=0$. Thus the
$(K_{W}+G_{W})$-MMP over $B$ is also $(K_{W}+\Delta_{W})$-trivial. If in
addition $\left\lVert D-\Delta\right\rVert<\delta$, then we also know that
$K_{W}+\Delta_{W}$ is nef over $B$: otherwise, the $(K_{W}+\Delta_{W})$-MMP
over $B$ contracts some curve $C\subseteq W$ with $(K_{W}+\Delta_{W})\cdot
C<0$, but since $\left\lVert D-\Delta\right\rVert<\delta$, we also have
$(K_{W}+D_{W})\cdot C=0$, which implies $(K_{W}+\Delta_{W})\cdot C=0$, a
contradiction. Again because $\left\lVert D-\Delta\right\rVert<\delta$, we
have that $\varphi$ is a $(K_{X}+\Delta)$-MMP over $B$, thus $(W,\Delta_{W})$
is also a minimal model of $(X,\Delta)$ over $B$. Since $\psi$ is
$(K_{W}+\Delta_{W})$-trivial, we get that $Y$ is a weak log canonical model of
$(X,\Delta)$ over $B$. The proof is now complete. ∎

######  Proposition 5.5.

In the setting of Theorem 5.1, let $D\in\mathcal{P}$. Then there exist a
positive real number $\delta$ such that:

for any $\Delta\in\mathcal{P}$ in the rational envelope of $D$ satisfying
$\left\lVert D-\Delta\right\rVert<\delta$, and any $G\in\mathcal{P}$ with
$\left\lVert D-G\right\rVert<\delta$, the ample model of $(X,tG+(1-t)\Delta)$
over $B$ is independent of $t\in(0,1]$.

###### Proof.

First, we claim that we only need to show that there exists a positive real
number $\delta$ such that for any $\Delta,G\in\mathcal{P}$ as in the
proposition, $(X,tG+(1-t)\Delta)$ has the same ample model over $B$ for every
$t\in(0,1)$. Assume the existence of such a $\delta$. Since $\mathcal{P}$ is a
rational polytope and $\Delta\in\mathcal{P}$ is in the rational envelope of
$D$, possibly after replacing $\delta$ by some smaller $\delta^{\prime}>0$, we
may further assume that for any $G\in\mathcal{P}$ satisfying $\left\lVert
D-G\right\rVert<\frac{\delta}{3}$ and any $\Delta\in\mathcal{P}$ satisfying
$\left\lVert D-\Delta\right\rVert<\frac{\delta}{3}$, we have
$G^{\prime}=2G-\Delta\in\mathcal{P}$ (in other words,
$G=\frac{1}{2}(G^{\prime}+\Delta)$) and $\left\lVert
D-G^{\prime}\right\rVert<\delta$. Any convex combination $tG+(1-t)\Delta$
where $t\in(0,1]$ is also a convex combination $sG^{\prime}+(1-s)\Delta$ for
some $s\in(0,1)$ (in fact $s=\frac{t}{2}$). By the choice of $\delta$, we know
that the ample model of $(X,sG^{\prime}+(1-s)\Delta)$ over $B$ is independent
of $s\in(0,1)$, hence the ample model of $(X,tG+(1-t)\Delta)$ over $B$ is
independent of $t\in(0,1]$ whenever $\left\lVert
D-\Delta\right\rVert,\left\lVert D-G\right\rVert<\frac{\delta}{3}$.

We may also assume that $X$ is smooth. In fact, if $\pi\colon X^{\prime}\to X$
is a log resolution of $(X,D_{1}+\dots+D_{n})$ and let $E$ be the sum of the
$\pi$-exceptional divisors, we may consider the rational polytope

$\mathcal{P}^{\prime}:=\\{\Delta^{\prime}=\Delta_{X^{\prime}}+E\,|\,\Delta\in\mathcal{P}\\}.$

For any $\Delta\in\mathcal{P}$, since $(X,\Delta)$ is log canonical, we have
$(X^{\prime},\Delta^{\prime})$ is also log canonical where
$\Delta^{\prime}=\Delta_{X^{\prime}}+E$ is as above and
$K_{X^{\prime}}+\Delta^{\prime}\geq\pi^{*}(K_{X}+\Delta)$. Hence by Lemma 2.9,
$(X^{\prime},\Delta^{\prime})$ also has a good minimal model over $B$ and its
ample model over $B$ is the same as the ample model of $(X,\Delta)$ over $B$.
Thus in order to prove the proposition, we may replace $X$ (resp. $D$, resp.
$\mathcal{P}$) by $X^{\prime}$ (resp. $D^{\prime}$, resp.
$\mathcal{P}^{\prime}$) and assume that $X$ is smooth.

We now proceed to prove that there exists a positive real number $\delta$ such
that for any $\Delta,G\in\mathcal{P}$ as in the proposition,
$(X,tG+(1-t)\Delta)$ has the same ample model over $B$ for every $t\in(0,1)$.
Let $\delta$ be the positive real number given by Lemma 5.4. Then as long as
$\left\lVert D-G\right\rVert,\left\lVert D-\Delta\right\rVert<\delta$ and
$\Delta$ is in the rational envelope of $D$, there exists a $(K_{X}+G)$-MMP
$X\dashrightarrow Y$ over $B$ such that $Y$ is also a weak log canonical model
of $(X,\Delta)$ over $B$.

${Z}$${(X,\Delta)}$${(Y,\Delta_{Y})}$${Z_{0}}$$\scriptstyle{i}$$\scriptstyle{g}$$\scriptstyle{h}$

Since $(X,\Delta)$ has a good minimal model over $B$, we can take the ample
model $g\colon Y\to Z_{0}$ of $(Y,\Delta_{Y})$ over $B$ such that

$K_{Y}+\Delta_{Y}\sim_{{\mathbb{R}}}g^{*}H$

where $H$ is an ${\mathbb{R}}$-Cartier ${\mathbb{R}}$-divisor on $Z_{0}$ that
is ample over $B$. Similarly, we know that $K_{Y}+G_{Y}$ is semiample over
$B$, hence it is also semiample over $Z_{0}$ and we take an ample model
$h\colon Y\to Z$ of $(Y,G_{Y})$ over $Z_{0}$ such that

$K_{Y}+G_{Y}\sim_{{\mathbb{R}}}h^{*}H_{Z}$

where $H_{Z}$ is an ${\mathbb{R}}$-Cartier ${\mathbb{R}}$-divisor on $Z$ that
is ample over $Z_{0}$. Since $Y$ is a weak log canonical model of both
$(X,\Delta)$ and $(X,G)$ over $B$, it is also a weak log canonical model of
$(X,tG+(1-t)\Delta)$ over $B$ for any $t\in[0,1]$ by the definition of weak
log canonical models. Since $H$ is ample over $B$ and $H_{Z}$ is ample over
$Z_{0}$, for any $t\in(0,1)$ we may choose some sufficiently small
$\varepsilon\in(0,t)$ such that $\varepsilon H_{Z}+(1-t)i^{*}H$ is ample over
$B$. Note also that $H_{Z}$ is nef over $B$ since $K_{Y}+G_{Y}$ is nef over
$B$. It follows that

$\displaystyle K_{Y}+tG_{Y}+(1-t)\Delta_{Y}$
$\displaystyle=t(K_{Y}+G_{Y})+(1-t)(K_{Y}+\Delta_{Y})$
$\displaystyle\sim_{{\mathbb{R}}}h^{*}(tH_{Z}+(1-t)i^{*}H)$
$\displaystyle=h^{*}((t-\varepsilon)H_{Z}+(\varepsilon H_{Z}+(1-t)i^{*}H))$
$\displaystyle=h^{*}(\mathrm{nef}/B+\mathrm{ample}/B)=h^{*}(\mathrm{ample}/B).$

Therefore, $h\colon Y\to Z$ is the common ample model of
$(Y,tG_{Y}+(1-t)\Delta_{Y})$ over $B$ for all $t\in(0,1)$ and we are done. ∎

###### Proof of Theorem 5.1.

Since $\mathcal{P}$ is compact, we only need to prove the result locally
around any $D\in\mathcal{P}$. We proceed by induction on $\dim\mathcal{P}$.
The base case when $\dim\mathcal{P}=0$ is clear. In the general case, by
Proposition 5.5, there exist a positive real number $\delta$ and a
${\mathbb{Q}}$-divisor $\Delta\in\mathcal{P}$ with $\left\lVert
D-\Delta\right\rVert<\delta$ such that for any $G\in\mathcal{P}$ satisfying
$\left\lVert D-G\right\rVert<\delta$, $(X,tG+(1-t)\Delta)$ has the same ample
model over $B$ for every $t\in(0,1]$. Since we are allowed to work locally
around $D\in\mathcal{P}$, we only need to prove the result for any rational
polytope $\mathcal{Q}\subseteq\mathcal{P}$ containing $D$ and $\Delta$ such
that for any $Q\in\mathcal{Q}$, $\left\lVert D-Q\right\rVert<\delta$. For any
$Q\in\mathcal{Q}$ such that $Q\neq\Delta$, there exists
$Q^{\prime}\in\mathcal{Q}$ such that $Q^{\prime}$ is on the boundary of
$\mathcal{Q}$ and $Q$ is in the line segment connecting $Q^{\prime}$ and
$\Delta$. Since $\left\lVert D-Q^{\prime}\right\rVert<\delta$, $(X,Q)$ and
$(X,Q^{\prime})$ have the same ample model over $B$. Since the boundary
$\partial\mathcal{Q}$ of $\mathcal{Q}$ consists of finitely many rational
polytopes of lower dimension, we deduce from the induction hypothesis that
there is a finite rational polyhedral decomposition of $\partial\mathcal{Q}$
corresponding to the finitely many ample models of $(X,Q^{\prime})$ over $B$
(as $Q^{\prime}$ varies in $\partial\mathcal{Q}$). As $\Delta$ is a
${\mathbb{Q}}$-divisor, we get an induced finite rational polyhedral
decomposition of $\mathcal{Q}$ by taking cones with vertex $\Delta$. By
construction, the ample model over $B$ stays the same in each chamber. ∎

By the same argument as in this section, we can deduce a variant of Theorem
5.1. We omit the proof since it is almost the same as the proof of Theorem
5.1.

######  Theorem 5.6.

Notation as above. Let $\mathcal{P}\subseteq V$ be a polytope such that
$(X,D)$ is a log canonical pair and has a good minimal model over $B$ for
every $D\in\mathcal{P}$.

Then there exist a finite polyhedral decomposition
$\mathcal{P}=\cup_{i=1}^{p}\mathcal{P}_{i}$, together with finitely many
rational maps $\psi_{j}\colon X\dashrightarrow Z_{j}$ over $B$, $1\leq j\leq
q$, such that: for any $1\leq i\leq p$, there exists some $1\leq j\leq q$ such
that $\psi_{j}\colon X\dashrightarrow Z_{j}$ is the ample model of $(X,D)$
over $B$ for any $D\in\mathcal{P}_{i}^{\circ}$.

######  Remark 5.7.

We remark that Theorems 5.1 and 5.6 also work when $X$ and $B$ are normal
varieties but not necessarily quasi-projective. We explain this briefly. We
cover $B$ by finitely many quasi-projective open subsets $B_{i}$ and restrict
the log canonical pairs given by the (rational) polytope $\mathcal{P}$ to
$X_{i}:=f^{-1}(B_{i})$ for each $i$ where $X_{i}$ is quasi-projective (since
$f$ is projective). We can apply Theorems 5.1 and 5.6 to these pairs over
$B_{i}$ and get a finite (rational) polyhedral decomposition of $\mathcal{P}$
for each $i$. A common refinement of these decompositions gives the desired
decomposition of $\mathcal{P}$ regarding the log canonical pairs over $B$.

##  6\. Wall-crossing morphisms

In this section, we study the wall-crossing morphisms between (the reduced
closure of) the images of the reduction morphisms and prove our main theorems.
The results of this section rely on, among many others things, the MMP for
locally stable families and the finiteness of ample models, see Sections 3, 4
and 5.

We fix the following notation. Let $f\colon(X;D_{1},\dots,D_{n})\to B$ be an
$n$-marked family over a reduced scheme. For any coefficient vector
$\mathbf{v}=(v_{1},\dots,v_{n})$, we define
$\mathbf{v}D:=\sum^{n}_{i=1}v_{i}D_{i}$. If $B$ is a normal variety and we let
$\mathcal{P}\subseteq{\mathbb{R}}_{\geq 0}^{n}$ be a polytope such that for
every $\mathbf{b}\in\mathcal{P}$, the family $(X,\mathbf{b}D)\to B$ is locally
stable and its generic fiber has a semi-log-canonical model (see Definition
4.14), then by Theorem 4.10, for every $\mathbf{b}\in\mathcal{P}$, the family
$(X,\mathbf{b}D)\to B$ has an slc model
$f_{\mathbf{b}}\colon(Z_{\mathbf{b}},\mathbf{b}\Delta_{\mathbf{b}})\to B$
which is a stable family and thus induces a unique morphism
$\Phi_{\mathbf{b}}\colon B\to\mathcal{K}_{\mathbf{b}}$. We define
$\mathcal{M}_{\mathbf{b}}=\overline{\Phi_{\mathbf{b}}(B)}$ as the reduced
closure of the image of $\Phi_{\mathbf{b}}$ and $\mathcal{N}_{\mathbf{b}}$ as
the normalization of $\mathcal{M}_{\mathbf{b}}$.

We shall analyze the variation of $\mathcal{M}_{\mathbf{b}}$ as the
coefficient vector $\mathbf{b}$ varies in the polytope $\mathcal{P}$. First,
we prove an auxiliary lemma.

######  Lemma 6.1.

Let $(X;D_{1},\dots,D_{n})\to B$ be an $n$-marked family over a reduced proper
scheme. Assume that for some
$\mathbf{b},\mathbf{b}^{\prime}\in{\mathbb{R}}_{\geq 0}^{n}$,
$(X,\mathbf{b}D)\to B$ and $(X,\mathbf{b}^{\prime}D)\to B$ are stable
families. Then
$\mathcal{M}_{\mathbf{b}}\cong\mathcal{M}_{\mathbf{b}^{\prime}}$.

###### Proof.

We denote by
$(\mathcal{X}_{\mathbf{b}},\mathbf{b}\mathcal{D}_{\mathbf{b}})\to\mathcal{M}_{\mathbf{b}}$
the universal family over $\mathcal{M}_{\mathbf{b}}$. Since $B$ is proper and
$\mathcal{M}_{\mathbf{b}}$ is separated, $B\to\mathcal{M}_{\mathbf{b}}$ is
proper and surjective. Since $(X,\mathbf{b}^{\prime}D)\to B$ is a stable
family and $B\to\mathcal{M}_{\mathbf{b}}$ is proper and surjective, it is not
hard to check that
$(\mathcal{X}_{\mathbf{b}},\mathbf{b}^{\prime}\mathcal{D}_{\mathbf{b}})\to\mathcal{M}_{\mathbf{b}}$
is a stable family. See the argument in [ABIP23]*Corollary 4.10. Thus it
induces a morphism
$\mathcal{M}_{\mathbf{b}}\to\mathcal{M}_{\mathbf{b}^{\prime}}$. Similarly,
there also exists a morphism
$\mathcal{M}_{\mathbf{b}^{\prime}}\to\mathcal{M}_{\mathbf{b}}$. They are
inverse to each other by construction. ∎

The following is the main result of this section.

######  Theorem 6.2.

Let $f\colon(X;D_{1},\dots,D_{n})\to B$ be an $n$-marked family over a normal
variety and let $\mathcal{P}\subseteq{\mathbb{R}}_{\geq 0}^{n}$ be a polytope
such that for every $\mathbf{b}\in\mathcal{P}$, the family $(X,\mathbf{b}D)\to
B$ is locally stable and its generic fiber has a semi-log-canonical model.
Then

  1. (1)

For any $\mathbf{a},\mathbf{b}\in\mathcal{P}$ such that the generic fiber of
$(X,\mathbf{a}D)\to B$ is stable and $\mathbf{a}\geq\mathbf{b}$, there exists
a unique morphism
$r_{\mathbf{a},\mathbf{b}}\colon\mathcal{N}_{\mathbf{a}}\to\mathcal{N}_{\mathbf{b}}$
making the following diagram commute.

$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{N}_{\mathbf{a}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r_{\mathbf{a},\mathbf{b}}}$$\textstyle{\mathcal{N}_{\mathbf{b}}}$

  2. (2)

There exists a finite polyhedral decomposition
$\mathcal{P}=\cup_{i=1}^{p}\mathcal{P}_{i}$ such that for any $1\leq i\leq p$,
the birational map $X\dashrightarrow Z_{\mathbf{b}}$ and the underlying
$n$-marked family
$(Z_{\mathbf{b}};\Delta_{\mathbf{b},1},\ldots,\Delta_{\mathbf{b},n})$ are
independent of $\mathbf{b}\in\mathcal{P}_{i}^{\circ}$, and
$\mathcal{M}_{\mathbf{b}}\cong\mathcal{M}_{\mathbf{b}^{\prime}}$ for any
$\mathbf{b},\mathbf{b}^{\prime}\in\mathcal{P}_{i}^{\circ}$. Moreover, we can
choose $\mathcal{P}_{i}$ to be rational if $\mathcal{P}$ is a rational
polytope.

  3. (3)

For any $1\leq i\leq p$, we denote $\mathcal{M}_{\mathbf{b}}$ by
$\mathcal{M}_{i}$ for any $\mathbf{b}\in\mathcal{P}_{i}^{\circ}$. Assume that
$f$ has a normal generic fiber. Then for any $1\leq i,j\leq p$ such that
$\mathcal{P}_{j}$ is a face of $\mathcal{P}_{i}$, there exists a unique
morphism $\alpha_{ij}\colon\mathcal{M}_{i}\to\mathcal{M}_{j}$ making the
following diagram commute. Moreover, for any $1\leq i,j,k\leq p$ such that
$\mathcal{P}_{j}$ is a face of $\mathcal{P}_{i}$ and $\mathcal{P}_{k}$ is a
face of $\mathcal{P}_{j}$, $\alpha_{jk}\circ\alpha_{ij}=\alpha_{ik}$. The same
statement holds for the normalization $\mathcal{N}_{i}$ of $\mathcal{M}_{i}$
without assuming that $f$ has a normal generic fiber.

(6.1)
$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{M}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{ij}}$$\textstyle{\mathcal{M}_{j}}$

We remark that when the base $B$ is proper, the statement is essentially a
formal consequence of the results in the previous sections. The main missing
part is the finite wall-and-chamber decomposition corresponding to
$\mathcal{M}_{\mathbf{b}}$ when $B$ is not proper. We will reduce this to the
proper case by applying weak semistable reduction.

###### Proof.

Since $B$ is normal, the morphism $B\to\mathcal{M}_{\mathbf{b}}$ lifts to a
unique morphism $B\to\mathcal{N}_{\mathbf{b}}$ for every
$\mathbf{b}\in\mathcal{P}$. Note that $B$ dominates
$\mathcal{N}_{\mathbf{b}}$. Since we also have that $\mathcal{N}_{\mathbf{a}}$
and $\mathcal{N}_{\mathbf{b}}$ are normal and separated, the morphism
$r_{\mathbf{a},\mathbf{b}}$ is unique. Let
$(\mathcal{X}_{\mathcal{N}_{\mathbf{a}}},\mathbf{a}\mathcal{D}_{\mathcal{N}_{\mathbf{a}}})\to\mathcal{N}_{\mathbf{a}}$
be the universal family over $\mathcal{N}_{\mathbf{a}}$. Since the generic
fiber of $(X,\mathbf{a}D)\to B$ is stable, the pullback of
$(\mathcal{X}_{\mathcal{N}_{\mathbf{a}}},\mathbf{a}\mathcal{D}_{\mathcal{N}_{\mathbf{a}}})\to\mathcal{N}_{\mathbf{a}}$
via $B\to\mathcal{N}_{\mathbf{a}}$ is isomorphic to $(X,\mathbf{a}D)\to B$
over the generic point of $B$. Since $\mathcal{N}_{\mathbf{a}}$ is normal and
the generic fiber of $(X,\mathbf{b}D)\to B$ has an slc model,
$(\mathcal{X}_{\mathcal{N}_{\mathbf{a}}},\mathbf{b}\mathcal{D}_{\mathcal{N}_{\mathbf{a}}})\to\mathcal{N}_{\mathbf{a}}$
has an slc model by Theorem 4.10 which induces a unique morphism
$r_{\mathbf{a},\mathbf{b}}\colon\mathcal{N}_{\mathbf{a}}\to\mathcal{N}_{\mathbf{b}}$.
The composition of $r_{\mathbf{a},\mathbf{b}}$ and
$B\to\mathcal{N}_{\mathbf{a}}$ is clearly $B\to\mathcal{N}_{\mathbf{b}}$. This
proves statement (1).

For the statement (2), we divide its proof in two parts. First we construct a
polyhedral decomposition corresponding to the birational map $X\dashrightarrow
Z_{\mathbf{b}}$ to the slc model. Observe that we may assume that $B$ is
smooth. To see this, let $\widetilde{B}\to B$ be a resolution of
singularities. For every $\mathbf{b}\in\mathcal{P}$, let
$(\widetilde{X},\mathbf{b}\widetilde{D})\to\widetilde{B}$ be the locally
stable family obtained by base change. It has an slc model
$\widetilde{f}_{\mathbf{b}}\colon(\widetilde{Z}_{\mathbf{b}},\mathbf{b}\widetilde{\Delta}_{\mathbf{b}})\to\widetilde{B}$
by Theorem 4.10 and the assumptions. As in the proof of Theorem 4.10,
$\widetilde{f}_{\mathbf{b}}\colon(\widetilde{Z}_{\mathbf{b}},\mathbf{b}\widetilde{\Delta}_{\mathbf{b}})\to\widetilde{B}$
descends to
$f_{\mathbf{b}}\colon(Z_{\mathbf{b}},\mathbf{b}\Delta_{\mathbf{b}})\to B$ for
every $\mathbf{b}\in\mathcal{P}$. If there exists a finite (rational)
polyhedral decomposition $\mathcal{P}=\cup_{i=1}^{p}\mathcal{P}_{i}$ such that
for any $1\leq i\leq p$, the birational map
$\widetilde{X}\dashrightarrow\widetilde{Z}_{\mathbf{b}}$ is independent of
$\mathbf{b}\in\mathcal{P}_{i}^{\circ}$, then the birational map
$X\dashrightarrow Z_{\mathbf{b}}$ is also independent of
$\mathbf{b}\in\mathcal{P}_{i}^{\circ}$. Thus we may assume $B$ is smooth.

Since $B$ is smooth, we deduce that $(X,\mathbf{b}D)$ is slc for every
$\mathbf{b}\in\mathcal{P}$ by Theorem 2.17. Let
$(X^{\nu},\Gamma+\mathbf{b}D^{\nu})\to B$ be the normalization of
$(X,\mathbf{b}D)\to B$ where $\Gamma$ is the conductor and $D^{\nu}$ is the
strict transform of $D$. Note that the induced involution $\tau$ on
$\Gamma^{\nu}$ is determined by the normalization $X^{\nu}\to X$ and is
independent of $\mathbf{b}\in\mathcal{P}$. By definition, the normalization
$(Z^{\nu}_{\mathbf{b}},\Gamma_{\mathbf{b}}+\mathbf{b}\Delta^{\nu}_{\mathbf{b}})$
of $(Z_{\mathbf{b}},\mathbf{b}\Delta_{\mathbf{b}})$ is the log canonical model
of $(X^{\nu},\Gamma+\mathbf{b}D^{\nu})$ over $B$. By applying Theorems 5.1 and
5.6 to the log canonical pairs $(X^{\nu},\Gamma+\mathbf{b}D^{\nu})$ over $B$
where $\mathbf{b}$ varies in $\mathcal{P}$, we get a finite (rational)
polyhedral decomposition $\mathcal{P}=\cup_{i=1}^{p}\mathcal{P}_{i}$ such that
for any $1\leq i\leq p$, the birational contraction $X^{\nu}\dashrightarrow
Z^{\nu}_{\mathbf{b}}$ is independent of
$\mathbf{b}\in\mathcal{P}_{i}^{\circ}$. Thus the induced regular involution
$\tau_{\mathbf{b}}$ on the normalization $\Gamma^{\nu}_{\mathbf{b}}$ of the
conductor $\Gamma_{\mathbf{b}}$ is also independent of
$\mathbf{b}\in\mathcal{P}_{i}^{\circ}$. By [Kol13]*Proposition 5.3, we
conclude that for any $1\leq i\leq p$, the birational map $X\dashrightarrow
Z_{\mathbf{b}}$ is independent of $\mathbf{b}\in\mathcal{P}_{i}^{\circ}$. Note
that this also implies that the underlying $n$-marked family
$(Z_{\mathbf{b}};\Delta_{\mathbf{b},1},\ldots,\Delta_{\mathbf{b},n})$ is
independent of $\mathbf{b}\in\mathcal{P}_{i}^{\circ}$. In particular, if $B$
is also proper, then by Lemma 6.1, we obtain
$\mathcal{M}_{\mathbf{b}}\cong\mathcal{M}_{\mathbf{b}^{\prime}}$ for any
$\mathbf{b},\mathbf{b}^{\prime}\in\mathcal{P}_{i}^{\circ}$ and therefore
statement (2) holds in this case.

In general, we prove that there exists a finite (rational) polyhedral
decomposition $\mathcal{P}=\cup_{i=1}^{q}\mathcal{Q}_{i}$ such that for any
$1\leq i\leq q$,
$\mathcal{M}_{\mathbf{b}}\cong\mathcal{M}_{\mathbf{b}^{\prime}}$ for any
$\mathbf{b},\mathbf{b}^{\prime}\in\mathcal{Q}_{i}^{\circ}$. We will reduce it
to the case when the base is proper. We may freely shrink $B$ as needed since
it will not change $\mathcal{M}_{\mathbf{b}}$. In particular, we may assume
that $B$ is smooth. We continue to use the same notation as in the paragraph
above. Let $T:=(\sum^{n}_{i=1}D_{i})_{\operatorname{red}}$. Let $Y\to X^{\nu}$
be a log resolution of $(X^{\nu},\Gamma+T^{\nu})$ and $E_{Y}$ the sum of its
exceptional divisors. Then $(Y,\Gamma_{Y}+T^{\nu}_{Y}+E_{Y})$ is log smooth.
By shrinking $B$, we can assume that $(Y,\Gamma_{Y}+T^{\nu}_{Y}+E_{Y})$ is log
smooth over $B$, and thus $(Y,\Gamma_{Y}+T^{\nu}_{Y}+E_{Y})\to B$ is locally
stable (note that $\Gamma_{Y}+T^{\nu}_{Y}+E_{Y}$ is a reduced divisor). By
Abramovich–Karu’s weak semistable reduction theorem [AK00] as used in the
proof of [Kol23]*Theorem 4.59, there exist a projective generically finite
surjective morphism $V\to B$ and a compactification $V\to\overline{V}$ such
that the pullback of $(Y,\Gamma_{Y}+T^{\nu}_{Y}+E_{Y})\to B$ via $V\to B$
extends to a locally stable family over $\overline{V}$ (in fact one can make
$\overline{V}$ smooth and the family toroidal, but we will not need this fact
here). We can assume that $V\to B$ is étale by shrinking $B$ further. Then the
formation of the slc model of $(X,\mathbf{b}D)\to B$ commutes with the base
change via $V\to B$ for every $\mathbf{b}\in\mathcal{P}$. Thus
$\mathcal{M}_{\mathbf{b}}$ will not change if we replace $B$ by $V$ and pull
back everything to $V$. Normalize $\overline{V}$ if necessary, we may assume
that $B$ has a normal compactification $\overline{B}$ such that
$(Y,\Gamma_{Y}+T^{\nu}_{Y}+E_{Y})\to B$ extends to a locally stable family
$(\overline{Y},\Gamma_{\overline{Y}}+T^{\nu}_{\overline{Y}}+E_{\overline{Y}})\to\overline{B}$
with normal generic fiber. By applying Lemma 4.6 finitely many times, we can
assume that
$K_{\overline{Y}/\overline{B}}+\Gamma_{\overline{Y}}+\mathbf{b}D^{\nu}_{\overline{Y}}+E_{\overline{Y}}$
is ${\mathbb{R}}$-Cartier for every $\mathbf{b}\in\mathcal{P}$. Since
$(X,\mathbf{b}D)\to B$ is locally stable, we have $\mathbf{b}D\leq T$ and thus
$\mathbf{b}D^{\nu}_{\overline{Y}}\leq T^{\nu}_{\overline{Y}}$. Thus
$(\overline{Y},\Gamma_{\overline{Y}}+\mathbf{b}D^{\nu}_{\overline{Y}}+E_{\overline{Y}})\to\overline{B}$
is locally stable for every $\mathbf{b}\in\mathcal{P}$. The situation is
partly summarized in the following diagram.

$\textstyle{(Y,\Gamma_{Y}+\mathbf{b}D^{\nu}_{Y}+E_{Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
log resolution
$\textstyle{(X^{\nu},\Gamma+\mathbf{b}D^{\nu})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
normalization $\textstyle{(X,\mathbf{b}D)}$

Since $(X^{\nu},\Gamma+\mathbf{b}D^{\nu})$ is log canonical by Theorem 2.17,
the log canonical model of $(Y,\Gamma_{Y}+\mathbf{b}D^{\nu}_{Y}+E_{Y})$ over
$B$ is the same as that of $(X^{\nu},\Gamma+\mathbf{b}D^{\nu})$ by Lemma 2.9,
the latter being
$(Z^{\nu}_{\mathbf{b}},\Gamma_{\mathbf{b}}+\mathbf{b}\Delta^{\nu}_{\mathbf{b}})$.
Thus
$(\overline{Y},\Gamma_{\overline{Y}}+\mathbf{b}D^{\nu}_{\overline{Y}}+E_{\overline{Y}})$
also has a log canonical model over $\overline{B}$ by Theorem 4.10, and it is
a stable family over $\overline{B}$ (by definition) that extends the stable
family
$(Z^{\nu}_{\mathbf{b}},\Gamma_{\mathbf{b}}+\mathbf{b}\Delta^{\nu}_{\mathbf{b}})\to
B$. Denote by
$(\overline{Z}^{\nu}_{\mathbf{b}},\overline{\Gamma}_{\mathbf{b}}+\mathbf{b}\overline{\Delta}^{\nu}_{\mathbf{b}})\to\overline{B}$
the extended family. By Corollary 4.15, the stable family
$(Z_{\mathbf{b}},\mathbf{b}\Delta_{\mathbf{b}})\to B$ also extends to some
stable family
$(\overline{Z}_{\mathbf{b}},\mathbf{b}\overline{\Delta}_{\mathbf{b}})\to\overline{B}$,
and the normalization of $\overline{Z}_{\mathbf{b}}$ is exactly
$\overline{Z}^{\nu}_{\mathbf{b}}$. Since $\overline{B}$ is proper, from the
already established case we see that there is a finite (rational) polyhedral
decomposition $\mathcal{P}=\cup_{i=1}^{q}\mathcal{Q}_{i}$ such that for any
$1\leq i\leq q$, the birational contraction
$\overline{Y}\dashrightarrow\overline{Z}^{\nu}_{\mathbf{b}}$ is independent of
$\mathbf{b}\in\mathcal{Q}_{i}^{\circ}$. In particular, the birational
involution $\tau_{\mathbf{b}}$ on the normalization
$\overline{\Gamma}^{\nu}_{\mathbf{b}}$ of the conductor
$\overline{\Gamma}_{\mathbf{b}}$ is also independent of
$\mathbf{b}\in\mathcal{Q}_{i}^{\circ}$. By [Kol13]*Proposition 5.3, this
implies that the birational map
$\overline{Y}\dashrightarrow\overline{Z}_{\mathbf{b}}$ is independent of
$\mathbf{b}\in\mathcal{Q}_{i}^{\circ}$ as well. Thus the underlying $n$-marked
family
$(\overline{Z}_{\mathbf{b}};\overline{\Delta}_{\mathbf{b},1},\ldots,\overline{\Delta}_{\mathbf{b},n})$
is independent of $\mathbf{b}\in\mathcal{Q}_{i}^{\circ}$ for any $1\leq i\leq
q$. On the other hand, the stable family
$(\overline{Z}_{\mathbf{b}},\mathbf{b}\overline{\Delta}_{\mathbf{b}})\to\overline{B}$
induces a unique morphism
$\overline{\Phi}_{\mathbf{b}}\colon\overline{B}\to\mathcal{K}_{\mathbf{b}}$
that extends $\Phi_{\mathbf{b}}\colon B\to\mathcal{K}_{\mathbf{b}}$. Thus
$\mathcal{M}_{\mathbf{b}}$ is the reduced image of
$\overline{\Phi}_{\mathbf{b}}$ since $\overline{B}$ is proper. By Lemma 6.1,
we get $\mathcal{M}_{\mathbf{b}}\cong\mathcal{M}_{\mathbf{b}^{\prime}}$ for
any $1\leq i\leq q$ and any
$\mathbf{b},\mathbf{b}^{\prime}\in\mathcal{Q}_{i}^{\circ}$. A common
refinement of the decompositions $\mathcal{P}=\cup_{i=1}^{p}\mathcal{P}_{i}$
and $\mathcal{P}=\cup_{i=1}^{q}\mathcal{Q}_{i}$ gives the desired finite
(rational) polyhedral decomposition of $\mathcal{P}$ in statement (2).

Finally, we prove statement (3). For any $1\leq i\leq p$, we denote the
underlying $n$-marked families
$(\mathcal{X}_{\mathbf{b}},\mathcal{D}_{\mathbf{b}})\to\mathcal{M}_{\mathbf{b}}$
by $(\mathcal{X}_{i},\mathcal{D}_{i})\to\mathcal{M}_{i}$,
$(\mathcal{X}_{\mathcal{N}_{\mathbf{b}}},\mathcal{D}_{\mathcal{N}_{\mathbf{b}}})\to\mathcal{N}_{\mathbf{b}}$
by
$(\mathcal{X}_{\mathcal{N}_{i}},\mathcal{D}_{\mathcal{N}_{i}})\to\mathcal{N}_{i}$
and $(Z_{\mathbf{b}},\Delta_{\mathbf{b}})\to B$ by $(Z_{i},\Delta_{i})\to B$
for any $\mathbf{b}\in\mathcal{P}_{i}^{\circ}$. This notation is well-defined
by statement (2). Since $\mathcal{M}_{i}$ and $\mathcal{M}_{j}$ are reduced
and separated and $B$ dominates $\mathcal{M}_{i}$, we see that $\alpha_{ij}$
is unique and the diagram (6.1) commutes if it commutes over an open set of
$B$. Thus to construct $\alpha_{ij}$, we may shrink $B$ and assume that $B$ is
smooth. We first prove statement (3) for $\mathcal{M}_{i}$ assuming that the
generic fiber of $f$ is normal. We deduce that $X$ is normal. Since
$(Z_{i},\mathbf{b}\Delta_{i})$ is the log canonical model of $(X,\mathbf{b}D)$
over $B$ for every $\mathbf{b}\in\mathcal{P}_{i}^{\circ}$,
$(Z_{i},\mathbf{b}\Delta_{i})$ is a weak log canonical model of
$(X,\mathbf{b}D)$ over $B$ for every $\mathbf{b}\in\mathcal{P}_{i}$ by
definition. Since $(X,\mathbf{b}D)$ has a good minimal model over $B$, we
deduce that $K_{Z_{i}/B}+\mathbf{b}\Delta_{i}$ is semiample over $B$ for every
$\mathbf{b}\in\mathcal{P}_{i}$, hence
$K_{\mathcal{X}_{i}/\mathcal{M}_{i}}+\mathbf{b}\mathcal{D}_{i}$ is semiample
over the generic point of $\mathcal{M}_{i}$. Since
$(\mathcal{X}_{i},\mathbf{b}\mathcal{D}_{i})\to\mathcal{M}_{i}$ is stable for
every $\mathbf{b}\in\mathcal{P}_{i}^{\circ}$, we see that
$(\mathcal{X}_{i},\mathbf{b}\mathcal{D}_{i})\to\mathcal{M}_{i}$ is locally
stable, and $K_{\mathcal{X}_{i}/\mathcal{M}_{i}}+\mathbf{b}\mathcal{D}_{i}$ is
nef over $\mathcal{M}_{i}$ for every $\mathbf{b}\in\mathcal{P}_{i}$. By Lemma
4.4, this further implies that
$K_{\mathcal{X}_{i}/\mathcal{M}_{i}}+\mathbf{b}\mathcal{D}_{i}$ is semiample
over $\mathcal{M}_{i}$. Let $\mathbf{b}_{j}$ be any point in
$\mathcal{P}_{j}^{\circ}\subseteq\mathcal{P}_{i}$. By Lemma 4.9, the log
canonical model of
$(\mathcal{X}_{i},\mathbf{b}_{j}\mathcal{D}_{i})\to\mathcal{M}_{i}$ is a
stable family and induces a morphism
$\alpha_{ij}\colon\mathcal{M}_{i}\to\mathcal{M}_{j}$. By construction and
Lemma 4.4, the composition of $\alpha_{ij}$ and $B\to\mathcal{M}_{i}$ is
$B\to\mathcal{M}_{j}$. For any $1\leq i,j,k\leq p$ such that $\mathcal{P}_{j}$
is a face of $\mathcal{P}_{i}$ and $\mathcal{P}_{k}$ is a face of
$\mathcal{P}_{j}$, the compositions of $\alpha_{ik}$ or
$\alpha_{jk}\circ\alpha_{ij}$ with $B\to\mathcal{M}_{i}$ both equal
$B\to\mathcal{M}_{k}$, thus the uniqueness of the diagram (6.1) implies that
$\alpha_{jk}\circ\alpha_{ij}=\alpha_{ik}$.

Next, without assuming that the generic fiber of $f$ is normal, we prove
statement (3) for $\mathcal{N}_{i}$. We only need to prove that for any $1\leq
i,j\leq p$ such that $\mathcal{P}_{j}$ is a face of $\mathcal{P}_{i}$, there
exists a morphism $\beta_{ij}\colon\mathcal{N}_{i}\to\mathcal{N}_{j}$ whose
composition with $B\to\mathcal{N}_{i}$ is $B\to\mathcal{N}_{j}$ since the rest
of the statement follows from the same argument. Similarly as above, we can
assume that $B$ is smooth. For every $\mathbf{b}\in\mathcal{P}_{i}^{\circ}$,
$(Z_{i},\mathbf{b}\Delta_{i})\to B$ is the slc model of $(X,\mathbf{b}D)\to B$
and thus its normalization
$(Z^{\nu}_{i},\Gamma_{i}+\mathbf{b}\Delta^{\nu}_{i})$ is the log canonical
model of $(X^{\nu},\Gamma+\mathbf{b}D^{\nu})$ over $B$. By the same argument
as above, for every $\mathbf{b}\in\mathcal{P}_{i}$, the log canonical model of
$(Z^{\nu}_{i},\Gamma_{i}+\mathbf{b}\Delta^{\nu}_{i})$ over $B$ exists and it
is the same as that of $(X^{\nu},\Gamma+\mathbf{b}D^{\nu})$. Let
$\mathbf{b}_{j}$ be any point in
$\mathcal{P}_{j}^{\circ}\subseteq\mathcal{P}_{i}$. We deduce that the slc
model $(Z_{j},\mathbf{b}_{j}\Delta_{j})\to B$ of $(X,\mathbf{b}_{j}D)\to B$ is
the same as that of $(Z_{i},\mathbf{b}_{j}\Delta_{i})\to B$ (see Definition
4.14 and Theorem 4.10). The pullback of
$(\mathcal{X}_{\mathcal{N}_{i}},\mathbf{b}_{j}\mathcal{D}_{\mathcal{N}_{i}})\to\mathcal{N}_{i}$
via $B\to\mathcal{N}_{i}$ is isomorphic to
$(Z_{i},\mathbf{b}_{j}\Delta_{i})\to B$. Since $\mathcal{N}_{i}$ is normal and
the generic fiber of $(Z_{i},\mathbf{b}_{j}\Delta_{i})\to B$ has an slc model,
$(\mathcal{X}_{\mathcal{N}_{i}},\mathbf{b}_{j}\mathcal{D}_{\mathcal{N}_{i}})\to\mathcal{N}_{i}$
has an slc model by Theorem 4.10 which induces a unique morphism
$\beta_{ij}\colon\mathcal{N}_{i}\to\mathcal{N}_{j}$. The composition of
$\beta_{ij}$ and $B\to\mathcal{N}_{i}$ is $B\to\mathcal{N}_{j}$ by the
discussion above. ∎

We remark that the morphism $\alpha_{ij}$ in the theorem above is given by
taking fiberwise ample model by the construction of $\alpha_{ij}$ and Lemma
4.4.

###  6.1. Proof of main theorems

We are now in a position to prove our main theorems.

###### Proof of Theorem 1.1.

The generic fiber of $(X,\mathbf{a}D)\to B$ has a good minimal model by Lemma
2.9 since it has a log canonical model. Since each irreducible component of
$D$ dominates $B$ and $\mathbf{0}<\mathbf{b}\leq\mathbf{a}$, the restrictions
of $\mathbf{a}D$ and $\mathbf{b}D$ to the generic fiber of $X\to B$ have the
same support. By Theorem 3.9, the generic fiber of $(X,\mathbf{b}D)\to B$ has
a log canonical model since $K_{X/B}+\mathbf{b}D$ is big over $B$. By Theorem
4.10, we finish the proof. ∎

###### Proof of Theorem 1.2.

By applying Lemma 4.6 finitely many times, we get a small birational
projective morphism $\pi\colon Y\to X$ such that $K_{Y/B}$ and each
irreducible component of $D_{Y}$ are ${\mathbb{Q}}$-Cartier and
$(Y,\mathbf{a}D_{Y})\to B$ is locally stable with normal generic fiber. Thus
$(Y,\mathbf{b}D_{Y})\to B$ is locally stable for every
$\mathbf{b}\in\mathcal{P}$ since $\mathbf{b}\leq\mathbf{a}$ and
$K_{Y/B}+\mathbf{b}D_{Y}$ is ${\mathbb{R}}$-Cartier. The log canonical model
of $(Y,\mathbf{b}D_{Y})\to B$ exists by Theorem 1.1 and coincides with that of
$(X,\mathbf{b}D)\to B$ by Remark 4.12. By applying Theorem 6.2(2)(3) to the
locally stable families $(Y,\mathbf{b}D_{Y})\to B$ where $\mathbf{b}$ varies
in $\mathcal{P}$, we finish the proof. ∎

If the family $(X,\mathbf{a}D)\to B$ is stable, its generic fiber has a log
canonical model by definition. Thus one can apply Theorems 1.1 and 1.2 to this
special case.

###### Proof of Theorem 1.3.

By Theorem 6.2, the reduction morphism $r_{\mathbf{a},\mathbf{b}}$ exists for
any $\mathbf{a}\geq\mathbf{b}$ in $\mathcal{P}$, and there is a wall-and-
chamber and wall-crossing structure on $\mathcal{P}$ corresponding to
$\mathcal{N}_{\mathbf{a}}$. By the same argument as in [ABIP23]*Proposition
7.7, $r_{\mathbf{a},\mathbf{b}}$ is birational. Since $(X,\mathbf{a}D)\to B$
is stable for every $\mathbf{a}\in\mathcal{P}$,
$r_{\mathbf{b},\mathbf{c}}\circ r_{\mathbf{a},\mathbf{b}}$ and
$r_{\mathbf{a},\mathbf{c}}$ coincide on a dense open substack of
$\mathcal{N}_{\mathbf{a}}$. Thus $r_{\mathbf{b},\mathbf{c}}\circ
r_{\mathbf{a},\mathbf{b}}=r_{\mathbf{a},\mathbf{c}}$ since
$\mathcal{N}_{\mathbf{a}}$ and $\mathcal{N}_{\mathbf{c}}$ are normal and
separated. ∎

######  Proposition 6.3.

In the setting of Theorem 1.2, there exists a nonempty open subset $U\subseteq
B$ which is independent of $\mathbf{b}\in\mathcal{P}$ such that
$\Phi_{\mathbf{b}}(u)$ is the point classifying the log canonical model of
$(X_{u},\mathbf{b}D_{u})$ for every $u\in U$ and every
$\mathbf{b}\in\mathcal{P}$.

###### Proof.

Recall that $\Phi_{\mathbf{b}}\colon B\to\mathcal{K}_{\mathbf{b}}$ is the
unique morphism induced by the stable family
$f_{\mathbf{b}}\colon(Z_{\mathbf{b}},\mathbf{b}\Delta_{\mathbf{b}})\to B$. By
shrinking $B$, we can assume that $B$ is smooth and quasi-projective. By
Theorem 2.17, $(X,\mathbf{a}D)$ is log canonical. Let $g\colon Y\to X$ be a
log resolution of $(X,\mathbf{a}D)$ and $E$ the sum of the $g$-exceptional
divisors. Then $(Y,\mathbf{a}D_{Y}+E)$ is log smooth. By Remark 3.8,
$f_{\mathbf{b}}\colon(Z_{\mathbf{b}},\mathbf{b}\Delta_{\mathbf{b}})\to B$ is
also the log canonical model of $(Y,\mathbf{b}D_{Y}+E)$ over $B$. We can
choose an affine open subset $U\subseteq B$ such that $(Y,\mathbf{a}D_{Y}+E)$
is log smooth over $U$. By shrinking $U$, we can also assume that
$(X_{u},\mathbf{a}D_{u})$ is log canonical for every $u\in U$ since
$(X,\mathbf{a}D)$ is log canonical. We deduce that the log canonical model of
$(X_{u},\mathbf{b}D_{u})$ is the same as that of
$(Y_{u},\mathbf{b}(D_{Y})_{u}+E_{u})$ for every $u\in U$ and every
$\mathbf{b}\in\mathcal{P}$ by Remark 3.8. Thus we only need to prove that the
log canonical model of $(Y_{u},\mathbf{b}(D_{Y})_{u}+E_{u})$ is the same as
the fiber of the log canonical model of $(Y,\mathbf{b}D_{Y}+E)$ over $B$ at
the point $u\in U$ for every $u\in U$ and every $\mathbf{b}\in\mathcal{P}$. By
the same argument as in Lemma 4.5, we can assume that $\mathbf{b}$ has
rational coordinates. By [HMX18]*Corollary 1.4, we only need to prove that
there exists a closed point $0\in U$ such that
$(Y_{0},\mathbf{b}(D_{Y})_{0}+E_{0})$ has a good minimal model. Let $W=(f\circ
g)^{-1}(U)$. By [HMX18]*Lemma 6.2, it suffices to prove that
$(W,\mathbf{b}D_{Y}|_{W}+E|_{W})$ has a good minimal model. Since $U$ is
affine, $(W,\mathbf{b}D_{Y}|_{W}+E|_{W})$ has a good minimal model if it has a
good minimal model over $U$. Thus we only need to prove that
$(Y,\mathbf{b}D_{Y}+E)$ has a good minimal model over $B$ for every
$\mathbf{b}\in\mathcal{P}$. Since $(X,\mathbf{a}D)$ has a log canonical model
over $B$ by Theorem 4.1, $(Y,\mathbf{a}D_{Y}+E)$ also has one and thus it has
a good minimal model over $B$ by Lemma 2.9. Since
$\mathbf{0}<\mathbf{b}\leq\mathbf{a}$,
$\operatorname{Supp}(\mathbf{b}D_{Y}+E)=\operatorname{Supp}(\mathbf{a}D_{Y}+E)$.
By Theorem 3.1, $(Y,\mathbf{b}D_{Y}+E)$ has a good minimal model over $B$ for
every $\mathbf{b}\in\mathcal{P}$. ∎

######  Remark 6.4.

In addition to weight reduction morphisms, [Has03] also studies forgetting
morphisms (which is equivalent to reducing certain weights to zero) between
moduli spaces $\mathcal{M}_{g,\mathbf{a}}$ of curves with marked points. For
moduli of stable pairs, if we restrict to the components whose generic points
parameterize log canonical pairs, then the existence of forgetting morphisms
is essentially equivalent to the abundance conjecture. In fact, if abundance
holds, then the assumption of Theorem 6.2 is automatically satisfied and hence
we get the forgetting morphisms. On the other hand, if $(X,D)$ is a log
canonical pair such that $K_{X}+D$ is nef and big and $H\geq 0$ is some
general ample divisor on $X$, then $(X,D+\varepsilon H)$ is a stable pair for
any $0<\varepsilon\ll 1$. To construct the forgetting morphism when
$\varepsilon\to 0$ requires taking the ample model of $K_{X}+D$, and it is
well known that abundance for log canonical pairs of log general type is
equivalent to abundance in general.

##  7\. Examples

In this section, we give some examples to show that some of the assumptions in
our main results are optimal.

###  7.1. Reducing weights on non-normal stable pairs

Since MMP can fail for semi-log-canonical pairs, we cannot reduce coefficients
in an arbitrary way if the locally stable family has a non-normal generic
fiber. The following is an explicit example.

######  Example 7.1.

Take $X_{1}=\mathbb{P}^{1}\times\mathbb{P}^{1}$ and let $X_{2}$ be the blowup
of a point on $\mathbb{P}^{1}\times\mathbb{P}^{1}$. Let $E\subseteq X_{2}$ be
the exceptional curve. On both $X_{i}$, let $F$ be the divisor class of
$\\{\mathrm{point}\\}\times\mathbb{P}^{1}$ and let $\Gamma$ be the divisor
class of $\mathbb{P}^{1}\times\\{\mathrm{point}\\}$. Our surface $X$ is
obtained by gluing together $X_{1}$ and $X_{2}$ along some curves
$\Gamma_{i}\subseteq X_{i}$ ($i=1,2$) where $\Gamma_{1}$ is a general member
of the linear system $|\Gamma|$ and $\Gamma_{2}$ is the unique member of
$|\Gamma-E|$. We need to also specify the boundary divisor $D$. On $X_{1}$ we
take $D$ as the sum of two general members of $|\Gamma|$ together with three
general members of $|F|$. On $X_{2}$ we take $D$ as the sum of two general
members $D_{1},D_{2}$ of the (base point free) linear system $|\Gamma+F-E|$
plus $E$ plus two general members of $|\Gamma+F|$. The following picture is an
illustration of the resulting configuration.

$X_{1}=\mathbb{P}^{1}\times\mathbb{P}^{1}$$\Gamma_{1}$$p_{1}$$q_{1}$$r_{1}$$X_{2}=\mathrm{Bl}_{x}(\mathbb{P}^{1}\times\mathbb{P}^{1})$$E$$\Gamma_{2}$$r_{2}$$D_{1}$$D_{2}$$p_{2}$$q_{2}$

On each component $X_{i}$ the divisor $D$ intersects $\Gamma_{i}$ in three
points $p_{i},q_{i},r_{i}$, and the gluing between $X_{1}$ and $X_{2}$ should
identify $\\{p_{1},q_{1},r_{1}\\}$ with $\\{p_{2},q_{2},r_{2}\\}$. It is not
hard to check that $K_{X_{i}}+D|_{X_{i}}+\Gamma_{i}$ is ample on each
component, hence by Kollár’s gluing theory [Kol13]*Theorem 5.13, we know that
$(X,D)$ is a stable pair.

Note that $D_{1}$ and $D_{2}$ do not intersect the double locus of $X$, hence