# On the restriction of the moduli part to a reduced divisor

Enrica Floris  Université de Poitiers, Laboratoire de Mathématiques et
Applications,UMR CNRS 7348, Téléport 2, Boulevard Marie et Pierre Curie, BP
30179, 86962 Futuroscope Chasseneuil Cedex, France  enrica.floris@univ-
poitiers.fr

###### Abstract.

Let $f\colon(X,\Delta)\to Y$ be a fibration such that $K_{X}+\Delta$ is
torsion along the fibres of $f$. Assume that $Y$ has dimension 2, or that $Y$
has dimension 3 and the fibres have dimension at most 3. Then the restriction
of the moduli part to its augmented base locus is semiample.

This project started during the collaboration with V. Lazić for [FL19]. I am
very grateful to Vlad for all the fruitful preliminary discussions around
Theorem 10.1 without which this paper would not have been possible. I would
like to thank J. Kollár for pointing out some mistakes in a previous version
of this work, and A. Petracci for many useful discussions on the topic. I am
very grateful to G. Bini, Y. Brunebarbe, P. Cascini, G. Pacienza, A. Sarti, R.
Svaldi and J. Witaszek for the useful discussions. I was supported by the ANR
project ”FIBALGA” ANR-18-CE40-0003 and the PEPS JCJC BFC 210024.

###### Contents

  1. 1 Introduction
  2. 2 Preliminary results
  3. 3 Semiample line bundles on simple normal crossings surfaces
  4. 4 Profinite equivalence relations
  5. 5 Gluing bases of fibrations
  6. 6 Graph theory
  7. 7 Trivial line bundles on simple normal crossings varieties
  8. 8 Restriction of the moduli part to log canonical centres
  9. 9 Finiteness of the equivalence relation for the moduli part
  10. 10 Triviality of the moduli part on pseudo-fibres
  11. 11 Proof of the main results

##  1\. Introduction

In this paper we study fibrations $f\colon(X,\Delta)\to Y$ such that
$K_{X}+\Delta$ is the pullback of a $\mathbb{Q}$-Cartier divisor $D$ on $Y$.
Those arise naturally, as the abundance conjecture predicts that every log
canonical pair is birational to either a Mori fibre space or a pair
$(X,\Delta)$ with $K_{X}+\Delta$ semiample. The induced fibration $f\colon
X\to Y$ is such that $K_{X}+\Delta\sim_{\mathbb{Q}}f^{*}D$ for an ample
$\mathbb{Q}$-Cartier divisor $D$ on $Y$. The canonical bundle formula is a way
of writing $D$ as the sum of three divisors: the canonical divisor of $Y$, a
divisor $B_{Y}$ called discriminant defined in terms of the singularities of
the fibration, and a nef (on a birational model of $Y$) divisor $M_{Y}$ called
moduli part or moduli divisor, describing the variation in moduli of the
fibres. For example, by [Amb05, Theorem 3.3, 3.5] if the moduli part is
numerically zero and $(X,\Delta)$ is klt, then the fibration is essentially a
product.

The theory of the canonical bundle formula has its roots in the work by
Kodaira and Ueno on elliptic surfaces. It has been developed and generalised
in [Kaw81, Amb04, Amb05, FM00, Kol07a].

The idea of considering divisors of the form $K_{Y}+B+M$ where $K_{Y}$ is the
canonical divisor, $(Y,B)$ satisfies certain regularity conditions and $M$ is
nef on a higher model of $Y$ is central in the works by Birkar–Zhang, Birkar
who consider generalised polarised pairs instead of pairs.

The most important conjecture on the canonical bundle formula has been
formulated in [PS09, Conjecture 7.13]:

######  B-Semiampleness Conjecture.

Let $(X,\Delta)$ be a pair and let $f\colon(X,\Delta)\to Y$ be an lc-trivial
fibration to an $n$-dimensional variety $Y$, where the divisor $\Delta$ is
effective over the generic point of $Y$. If $Y$ is an Ambro model of $f$, then
the moduli divisor $M_{Y}$ is semiample.

Several special cases of the conjecture are proved, mainly when the dimension
of the fibre is at most two by the classical work of Kodaira and by [PS09,
Fuj03, Fil18] and if the moduli part is numerically zero by [Amb05, Flo14].
For the klt case, if the moduli part is torsion, then by [Amb05, Theorem 3.3]
the variation of $f$ is zero.

In this paper we consider a connected divisor $\mathcal{T}=\cup T$ and assume
the B-Semiampleness Conjecture in lower dimension. In [FL19] we proved that
the divisor $M_{Y}|_{T}$ is semiample for every $T$. In this work we study the
gluing of the global sections of $mM_{Y}|_{T}$ to obtain global sections of
$mM_{Y}|_{\mathcal{T}}$.

The main result of this paper is the following:

######  Theorem A.

Let $(X,\Delta)$ be a pair and let $f\colon(X,\Delta)\to Y$ be a klt-trivial
fibration to a surface $Y$, where the divisor $\Delta$ is effective over the
generic point of $Y$. Assume that $Y$ is an Ambro model for $f$ and that
$M_{Y}$ is big.

Then there is a birational base change $Y^{\prime}\to Y$ such that the
restriction of $M_{Y^{\prime}}$ to the augmented base locus is torsion.

The semiampleness of the moduli part turns out to be deeply related to the
variation of the fibres of $f$. The variation, introduced by Viehweg [Vie83]
is roughly speaking the dimension of the moduli space of fibres of $f$ in the
sense of birational geometry (see Definition 2.23 for a precise definition).
The Kodaira dimension of the moduli part is at most the variation of $f$, and
conjecturally they coincide. On the other hand, for a fibration of maximal
variation there should be only a finite number of fibres birational to a given
one:

######  Conjecture 1.1.

Let $X$ be a $\mathbb{Q}$-factorial variety. Let $f\colon(X,\Delta)\to Y$ be a
klt-trivial fibration of maximal variation. Then there is an open set
$U\subseteq Y$ such that for every $y\in U$ the set

$\\{z\in U|\;(f^{-1}y,\Delta^{h}|_{f^{-1}y})\;\text{is crepant birational
to}\;(f^{-1}z,\Delta^{h}|_{f^{-1}z})\\}$

is finite, where $\Delta^{h}$ denotes the horizontal part of $\Delta$.

Conjecture 1.1 is true for fibrations of relative dimension at most 2. Using
this fact we are able to prove

######  Theorem B.

Let $(X,\Delta)$ be a pair and let $f\colon(X,\Delta)\to Y$ be a klt-trivial
fibration to a variety $Y$ of dimension 3 and $\dim X\leq\dim Y+3$, where the
divisor $\Delta$ is effective over the generic point of $Y$. Assume that $Y$
is an Ambro model for $f$ and that $M_{Y}$ is big.

Then there is a birational base change $Y^{\prime}\to Y$ such that the
restriction of $M_{Y^{\prime}}$ to the augmented base locus is semiample.

For the proof of Theorem A and Theorem B, we embrace the approach developed in
[Kol13] and successfully applied in [HX13] to the study of the semiampleness
of the log canonical divisor of a slc pair (roughly speaking a simple normal
crossings divisor in a smooth variety).

By [FL19] we are in the following setting: we have a line bundle $\mathcal{L}$
on a reduced, non irreducible variety $\mathcal{T}$ which is semiample on
every irreducible component of $\mathcal{T}$. We want to prove that
$\mathcal{L}$ is semiample on $\mathcal{T}$. The approach consists in
translating the semiampleness of a line bundle into the finiteness of a
certain equivalence relation. For the sake of simplicity, assume that
$\mathcal{T}=T_{1}\cup T_{2}$. Let $\phi_{i}\colon T_{i}\to V_{i}$ be the
fibration induced by $\mathcal{L}$ for $i=1,2$. We say that $x_{1}\in V_{1}$
is equivalent to $x_{2}\in V_{2}$ if
$\phi_{1}^{-1}(x_{1})\cap\phi_{2}^{-1}(x_{2})\neq\emptyset$ and we take the
closure of this equivalence relation. This is the natural relation to
consider. Indeed, if $\mathcal{L}|_{T_{1}\cup T_{2}}$ is semiample and
$\phi\colon T_{1}\cup T_{2}\to V$ is the induced fibration, then
$\phi_{1}^{-1}(x_{1})$ and $\phi_{2}^{-1}(x_{2})$ are sent to the same point
by $\phi$.

By considering the union of the fibres of $\phi_{1}$ and $\phi_{2}$ which
intersect, we construct subsets of $T_{1}\cup T_{2}$ called pseudofibres.

The reason why we cannot fully apply Kollár’s gluing theory is that many of
the required regularity hypotheses are not satisfied in our setup.

We now describe the structure of the paper as well as the techniques used in
every section. Section 2 contains some preliminary results as well as some
refinements of results on the canonical bundle formula. Section 3 is a
semiampleness criterion for a line bundle on a simple normal crossings
surface. In section 4 we recall the basic notions on equivalence relations and
prove some technical lemmas necessary for the study of the equivalence
relation $R_{\mathcal{L}}$, which is done in section 5. In section 6 we gather
some results from [Sta83] and we apply them in section 7 where we develop a
criterion for the triviality a line bundle on a simple normal crossing
variety. Section 8 uses techniques from the minimal model program and is a
study of the restriction of the moduli part to higher codimensional log
canonical centres.

In section 9 we prove that, assuming the B-Semiampleness Conjecture in
dimension $n-1$ and Conjecture 1.1 in dimension $d-1$, the equivalence
relation is finite for $\mathcal{L}=\mathcal{O}(mM_{Y})$ for $Y$ of dimension
$n$ and $X$ of dimension $d+n$. In section 10 we prove that the restriction of
$\mathcal{L}$ to a simple normal crossings pseudofibre is torsion.

The last section contains the proofs of Theorems A and B.

##  2\. Preliminary results

We work over the complex numbers. For the notions on the minimal model program
and singularities of pairs we refer to [KM92]. We will use without defining
them the notions of log canonical, klt and dlt singularities, as well as of
centre of a log canonical singularity. We refer to [Kol97] and [KM92] for a
presentation of these concepts and to [FL19, Definition 2.5] for a summary of
all the required notions in our setup.

We recall that a pair $(X,\Delta)$ is the data of a normal projective variety
$X$ and a $\mathbb{Q}$-Weil divisor $\Delta$ such that $K_{X}+\Delta$ is
$\mathbb{Q}$-Cartier. In this paper we do not require $\Delta$ to be an
effective divisor.

We say that a closed subvariety $S$ of $X$ is a minimal log canonical centre
of $(X,\Delta)$ over $Z$ if $S$ is a minimal log canonical centre of
$(X,\Delta)$ (with respect to inclusion) which dominates $Z$.

###  2.1. Semistable morphisms

In this paragraph we recall the definition of semistable morphisms and the
statement of the semistable reduction theorem, proved in [ALT18], which will
be crucial in the proof of our main results, Theorem 9.2 and 10.1. We refer to
[Ogu18, section II.1.1] for the definition of log scheme and morphism of log
schemes, and to [Stack, Definition 15.52.1, Proposition 15.52.3] for the
definition of quasi excellent rings and the first properties.

######  Definition 2.1 (4.2.1 [ALT18]).

A morphism of log schemes $f\colon X\to B$, $f^{\sharp}\colon
f^{-1}\mathcal{O}_{Y}\to\mathcal{O}_{X}$ is semistable if the following
conditions hold:

  1. (1)

$X$ and $B$ are regular and the log structures are given by normal crossings
divisors $Z\subseteq X$ and $W\subseteq B$.

  2. (2)

Étale-locally at any $x\in X$ with $b=f(x)$ there exist regular parameters
$t_{1},\ldots,t_{n},t^{\prime}_{1},\ldots,t^{\prime}_{n^{\prime}}\in\mathcal{O}_{X,x}$
and
$\pi_{1},\ldots,\pi_{l},\pi^{\prime}_{1},\ldots,\pi^{\prime}_{l^{\prime}}\in\mathcal{O}_{B,b}$
such that $Z=V(t_{1}\cdot\ldots\cdot t_{n})$ at $x$,
$W=V(\pi_{1}\cdot\ldots\cdot\pi_{l})$ at $b$,
$f^{\sharp}(\pi_{i})=t_{n_{i}+1}\ldots t_{n_{i+1}}$ for
$0=n_{1}<n_{2}<\ldots<n_{l+1}\leq n$.

  3. (3)

$f$ is log smooth.

In characteristic zero, the third condition can be replaced by the condition
that $f^{\sharp}(\pi^{\prime}_{j})=t^{\prime}_{j}$ for $1\leq j\leq
l^{\prime}$.

The following semistable reduction theorem is proved in [ALT18] and uses a
finer toroidalization proved in [ATW20].

######  Theorem 2.2 (Theorem 4.7 [ALT18]).

Assume that $X\to B$ is a dominant morphism of finite type between quasi
excellent integral schemes of characteristic zero and $Z\subseteq X$ is a
closed subset. Then there exists a stack-theoretic modification $b\colon
B^{\prime}\to B$, a projective modification $a\colon
X^{\prime}\to(X\times_{B}B^{\prime})^{pr}$, and divisors $W^{\prime}\subseteq
B^{\prime}$, $Z^{\prime}\subseteq X^{\prime}$such that:

  1. (1)

$a^{-1}Z\cup f^{\prime-1}W^{\prime}\subseteq Z^{\prime}$ and the morphism
$f^{\prime}\colon(X^{\prime},Z^{\prime})\to(B^{\prime},W^{\prime})$ is
semistable. In particular, $X^{\prime},B^{\prime}$ are regular and
$Z^{\prime},W^{\prime}$ are snc.

  2. (2)

If a regular open $B_{0}\subseteq B$ is such that $X_{0}=X\times_{B}B_{0}\to
B_{0}$ is smooth and $Z_{0}=Z\times_{B}B_{0}\to B_{0}$ is a relative divisor
over $B_{0}$ with normal crossings (in other words,
$f\colon(X_{0},Z_{0})\to(B_{0},W_{0})$ is semistable), then $a$ and $b$ are
isomorphisms over $X_{0}$ and $B_{0}$, respectively.

######  Remark 2.3.

Let $f\colon(X,Z)\to(B,W)$ be a semistable map, and let $S\subseteq X$ be a
stratum of $Z$. Let $C=f(S)$. It follows from the definition that
$f|_{S}\colon S\to C$ is semistable. Moreover, if $f|_{S}\colon
S\overset{h}{\longrightarrow}C^{\prime}\overset{\tau}{\longrightarrow}C$ is
the Stein factorisation, then $h$ is semistable.

###  2.2. Groups of crepant birational automorphisms

In this paragraph we state two results on the group of crepant birational
selfmaps of a pair. The first one is the finiteness of pluricanonical
representations [Gon13, Theorem 4.5] and the second one is a generalisation to
pairs of the finiteness of the group of selfmaps of a manifold of general
type.

######  Definition 2.4.

Let $f_{1}\colon(X_{1},\Delta_{1})\to Y$ and $f_{2}\colon(X_{2},\Delta_{2})\to
Y$ be two fibrations of pairs to the same base $Y$. A birational map
$\theta\colon X_{1}\dasharrow X_{2}$ is _crepant birational over $Y$_ if
$a(E,X_{1},\Delta_{1})=a(E,X_{2},\Delta_{2})$ for every geometric valuation
$E$ over $X_{1}$ and $X_{2}$ and we have the commutative diagram

$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\scriptstyle{\theta}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\textstyle{Y.}$

The map $\theta$ is _crepant birational_ if $Y$ is a point.

The set of all crepant birational maps of a pair $(X,\Delta)$ to itself is a
group, denoted by $\operatorname{Bir}^{\mathrm{c}}(X,\Delta)$. For a positive
integer $m$ such that $m(K_{X}+\Delta)$ is Cartier, every
$\sigma\in\operatorname{Bir}^{\mathrm{c}}(X,\Delta)$ defines an automorphism
of $H^{0}(X,m(K_{X}+\Delta))$, and hence the _pluricanonical representation_

$\rho_{m}\colon\operatorname{Bir}^{\mathrm{c}}(X,\Delta)\to\operatorname{GL}\big{(}H^{0}(X,m(K_{X}+\Delta))\big{)}.$

######  Remark 2.5.

If the condition $p^{*}(K_{X_{1}}+\Delta_{1})=q^{*}(K_{X_{2}}+\Delta_{2})$ is
true for one resolution of the indeterminacy, then it is true for every
resolution of indeterminacy. Indeed, let $(p^{\prime},q^{\prime})\colon
W^{\prime}\to X_{1}\times X_{2}$ be another resolution of the indeterminacy.
Let $(\nu,\mu)\colon\widehat{W}\to W\times W^{\prime}$ be a dominating
birational model. Then
$\nu^{*}p^{*}(K_{X_{1}}+\Delta_{1})=\nu^{*}q^{*}(K_{X_{2}}+\Delta_{2})$. By
commutativity,
$\nu^{*}p^{*}(K_{X_{1}}+\Delta_{1})=\mu^{*}p^{\prime*}(K_{X_{1}}+\Delta_{1})$
and
$\nu^{*}q^{*}(K_{X_{1}}+\Delta_{1})=\mu^{*}q^{\prime*}(K_{X_{1}}+\Delta_{1})$.
We conclude by pushing forward with $\nu$.

######  Theorem 2.6.

Let $(X,\Delta)$ be a klt pair such that $K_{X}+\Delta\sim_{\mathbb{Q}}0$.
Then for every $m$, the image of the pluricanonical representation $\rho_{m}$
is finite. In particular, there is a positive integer $\ell$ such that the
image of $\rho_{\ell}$ is trivial.

###### Proof.

The first statement is [Gon13, Theorem 4.5], and then the second statement is
straightforward. ∎

###  2.3. Canonical bundle formula

In this subsection we define lc-trivial fibration and recall several
fundamental results. We refer the reader to [FL20] for a survey of the general
results on the canonical bundle formula.

######  Definition 2.7.

Let $(X,\Delta)$ be a pair and let $\pi\colon X^{\prime}\rightarrow X$ be a
log resolution of the pair. A morphism $f\colon(X,\Delta)\rightarrow Y$ to a
normal projective variety $Y$ is a _klt-trivial_ , respectively _lc-trivial_ ,
fibration if $f$ is a surjective morphism with connected fibres, $(X,\Delta)$
has klt, respectively log canonical, singularities over the generic point of
$Y$, there exists a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $D$ on $Y$ such
that

$K_{X}+\Delta\sim_{\mathbb{Q}}f^{*}D,$

and if $f^{\prime}=f\circ\pi$, then

$\operatorname{rk}f^{\prime}_{*}\mathcal{O}_{X}(\lceil
K_{X^{\prime}}-\pi^{*}(K_{X}+\Delta)\rceil)=1,$

respectively

$\textstyle\operatorname{rk}f^{\prime}_{*}\mathcal{O}_{X}\big{(}\lceil
K_{X^{\prime}}-\pi^{*}(K_{X}+\Delta)+\sum_{a(E,X,\Delta)=-1}E\rceil\big{)}=1.$

######  Remark 2.8.

This last condition in the previous definition is verified, for instance, if
$\Delta$ is effective on the generic fibre, which is mostly the case in this
paper.

######  Definition 2.9.

Let $f\colon(X,\Delta)\to Y$ be an lc-trivial fibration, and let $P\subseteq
Y$ be a prime divisor with the generic point $\eta_{P}$. The _log canonical
threshold_ of $f^{*}P$ with respect to $(X,\Delta)$ is

$\gamma_{P}=\sup\\{t\in\mathbb{R}\mid(X,\Delta+tf^{*}P)\textrm{ is log
canonical over }\eta_{P}\\}.$

The _discriminant_ of $f$ is

(1) $B_{f}=\sum_{P}(1-\gamma_{P})P.$

This is a Weil $\mathbb{Q}$-divisor on $Y$, and it is effective if $\Delta$ is
effective. Fix $\varphi\in\mathbb{C}(X)$ and the smallest positive integer $r$
such that $K_{X}+\Delta+\frac{1}{r}\operatorname{div}\varphi=f^{*}D$. Then
there exists a unique Weil $\mathbb{Q}$-divisor $M_{f}$, the _moduli part_ of
$f$, such that

(2)
$K_{X}+\Delta+\frac{1}{r}\operatorname{div}\varphi=f^{*}(K_{Y}+B_{f}+M_{f}).$

The formula (2) is the _canonical bundle formula_ associated to $f$.

######  Remark 2.10.

As in [FL19], we adopt here the notation $B_{f},M_{f}$ for the discriminant
and moduli part of $f$ instead of the usual one $B_{Y},M_{Y}$. We will
occasionally write $B_{Y},M_{Y}$ when the fibration is clear from the contest.

######  Remark 2.11.

If $f_{1}\colon(X_{1},\Delta_{1})\to Y$ and $f_{2}\colon(X_{2},\Delta_{2})\to
Y$ are two lc-trivial fibrations over the same base which are crepant
birational over $Y$, then $f_{1}$ and $f_{2}$ have the same discriminant and
moduli part.

The canonical bundle formula satisfies several desirable properties. The first
is the _base change property_ , [Amb04, Theorem 0.2] and [Kaw98, Theorem 2].

######  Theorem 2.12.

Let $f\colon(X,\Delta)\to Y$ be a klt-trivial fibration. Then there exists a
proper birational morphism $Y^{\prime}\to Y$ such that for every proper
birational morphism $\pi\colon Y^{\prime\prime}\to Y^{\prime}$ we have:

  1. (i)

$K_{Y^{\prime}}+B_{Y^{\prime}}$ is a $\mathbb{Q}$-Cartier divisor and
$K_{Y^{\prime\prime}}+B_{Y^{\prime\prime}}=\pi^{*}(K_{Y^{\prime}}+B_{Y^{\prime}})$,

  2. (ii)

$M_{Y^{\prime}}$ is a nef $\mathbb{Q}$-Cartier divisor and
$M_{Y^{\prime\prime}}=\pi^{*}M_{Y^{\prime}}$.

In the context of the previous theorem, we say that $M_{Y}$ _descends_ to
$Y^{\prime}$, and we call $Y^{\prime}$ an _Ambro model_ for $f$. One of the
reasons why base change property is important is the following _inversion of
adjunction_ [Amb04, Theorem 3.1].

Moreover, by [Kol07a, Proposition 8.4.9, Definition 8.3.6, Theorem 8.5.1] if
$f\colon(X,\Delta)\to Y$ is an lc-trivial fibration such that the non-smooth
locus $\Sigma$ of the fibration is a simple normal crossings divisor and
$f^{-1}\Sigma+\Delta$ is simple normal crossings, then $Y$ is an Ambro model.

######  Remark 2.13.

Theorem 2.12 implies in particular that the moduli part is always
pseudoeffective, even when it is not nef, as it is the push-forward of a nef
divisor by a birational model.

We prove now that if the moduli part descends on $Y$, then it descends on
$Y^{\prime}$ with $Y^{\prime}\to Y$ generically finite.

######  Lemma 2.14.

Let $f\colon(X,\Delta)\to Y$ be an lc-trivial fibration and $\tau\colon
Y^{\prime}\to Y$ be a genrically finite map. If $Y$ is an Ambro model, then
$Y^{\prime}$ is an Ambro model for the fibration obtained by base change.

###### Proof.

By taking the Stein factorisation, it is enough to consider $\tau$ finite. Let
$\tilde{\nu}\colon\widetilde{Y}\to Y^{\prime}$ be a birational map. By [FL19,
Lemma 2.4] there is a diagram

$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{\widetilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\nu}}$$\textstyle{W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu^{\prime}}$$\scriptstyle{\sigma}$$\textstyle{Y}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$

such that $\nu$ and $\nu^{\prime}$ are birational and $(\nu^{\prime})^{-1}$ is
an isomorphism along the generic point of every $\tilde{\nu}$-exceptional
divisor.

Let $(p,q)\colon\widehat{W}\to W^{\prime}\times\widetilde{Y}$ be a resolution
of the indeterminacies. Then we have

$M_{\widehat{W}}=p^{*}\sigma^{*}\nu^{*}M_{Y}=q^{*}\tilde{\nu}^{*}\tau^{*}M_{Y}=q^{*}\tilde{\nu}^{*}M_{Y^{\prime}}$

which implies
$M_{\widetilde{Y}}=q_{*}M_{\widehat{W}}=\tilde{\nu}^{*}M_{Y^{\prime}}$.

∎

######  Theorem 2.15.

Let $f\colon(X,\Delta)\rightarrow Y$ be an lc-trivial fibration, and assume
that $Y$ is an Ambro model for $f$. Then $(Y,B_{Y})$ has klt, respectively log
canonical, singularities in a neighbourhood of a point $y\in Y$ if and only if
$(X,\Delta)$ has klt, rerspectively log canonical, singularities in a
neighbourhood of $f^{-1}(y)$.

The following is [Amb05, Theorem 3.3]. It will be a key result in the proof of
both Theorem 9.2 and 10.1.

######  Theorem 2.16.

Let $f\colon(X,\Delta)\to Y$ be a klt-trivial fibration between normal
projective varieties such that $\Delta$ is effective over the generic point of
$Y$. Then there exists a diagram

$\textstyle{(X,\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{(X^{+},\Delta^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{+}}$$\textstyle{Y}$$\textstyle{\widetilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta}$$\scriptstyle{\chi}$$\textstyle{Y^{+}}$

such that:

  1. (i)

$f^{+}\colon(X^{+},\Delta^{+})\rightarrow Y^{+}$ is a klt-trivial fibration,

  2. (ii)

$\vartheta$ is generically finite and surjective, and $\chi$ is surjective,

  3. (iii)

there exists a non-empty open set $U\subseteq\widetilde{Y}$ and an isomorphism

$\textstyle{(X,\Delta)\times_{Y}U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{(X^{+},\Delta^{+})\times_{Y^{+}}U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U,}$

  4. (iv)

the moduli part $M_{f^{+}}$ is big and, after possibly a birational base
change, we have $\vartheta^{*}M_{f}=\chi^{*}M_{f^{+}}$.

The following remark will be useful at the end of this section.

######  Lemma 2.17.

Notation as in Theorem 2.16. Assume that $Y$ is an Ambro model and $M_{f}$ is
semiample, let $\phi\colon Y\to V$ be the fibration induced by $M_{f}$. Then
there is $\vartheta$ such that $\operatorname{Exc}(\vartheta)$ is vertical
with respect to $\phi\circ\vartheta$. Moreover there is a generically finite
map $\lambda\colon Y^{+}\to V$.

###### Proof.

Since $M_{f}$ is semiample, $M_{f^{+}}$ is semiample as well. Let
$\phi^{+}\colon Y^{+}\to V^{+}$ be the fibration defined by $M_{f^{+}}$. We
notice that as $M_{f^{+}}$ is big, the fibration $\phi^{+}$ is birational.
Since $\phi^{+}\circ\chi$ is a fibration, there is a finite map $V^{+}\to V$.
We set $\lambda\colon Y^{+}\to V$ the induced generically finite map.

By the proof of [Amb05, Theorem 2.2], we have
$\vartheta=\varepsilon\circ\sigma\circ p$ where

  * •

$p\colon Y^{\prime}\to Y$ is birational, such that $Y^{\prime}$ is smooth and
the period map extends to a fibration $q\colon Y^{\prime}\to Y_{0}$ and $p$
can be taken as a composition of blow-ups along smooth centres;

  * •

$\sigma\colon Y^{\prime\prime}\to Y^{\prime}$ is finite and such that, if
$\sigma_{0}\circ\alpha$ is the Stein factorisation of $q\circ\sigma$ then
$\alpha\colon Y^{\prime\prime}\to Y^{+}$ admits a section;

  * •

$\varepsilon$ is a desingularisation of $Y^{\prime\prime}$ and
$\chi=\alpha\circ\varepsilon$.

We prove first that $p\operatorname{Exc}(p)$ is $\phi$-vertical. Indeed, let
$C\subseteq\operatorname{Exc}(p)$ be a curve not contracted by $p$ but
contracted by $q$. Let $\widetilde{C}\subseteq Y^{\prime\prime}$ be such that
$\sigma(\widetilde{C})=C$. Since $q\circ\sigma=\sigma_{0}\circ\alpha$, the
image $\alpha(\widetilde{C})$ is a point. Then $\sigma\circ
p\circ\phi(\widetilde{C})=\lambda\circ\alpha(\widetilde{C})$ is a point.
Therefore $C$ is contracted by $\phi$. This implies that the indeterminacy
locus of $q\circ p^{-1}$ is $\phi$-vertical. Therefore the indeterminacy locus
of $q\circ p^{-1}$ is $\phi$-vertical and we can find $p,q$ such that the
exceptional locus of $p,q$ is $\phi\circ p$-vertical.

The morphism $\sigma$ is generically étale, therefore the singularities of
$Y^{\prime\prime}$ are vertical with respect to $\alpha$. Therefore we can
chose $\varepsilon$ which is an isomorphism over the generic point of $V$. ∎

The following is [Amb05, Proposition 4.4], and it allows to extend the
isomorphism from Theorem 2.16(iii) to a suitable bigger open subset.

######  Proposition 2.18.

Let $f\colon(X,\Delta)\rightarrow Y$ be a klt-trivial fibration of normal
projective varieties such that there exists an isomorphism

$\Phi\colon(X,\Delta)\times_{Y}U\to(F,\Delta_{F})\times U$

over a non-empty open subset $U\subseteq Y$. Then $\Phi$ extends to an
isomorphism over

$Y^{0}=Y\setminus\big{(}\operatorname{Supp}B_{Y}\cup\operatorname{Sing}(Y)\cup
f(\operatorname{Supp}\Delta_{v}^{<0})\big{)},$

where $\Delta_{v}^{<0}$ consists of the vertical components of $\Delta$ with
negative coefficients in $\Delta$.

The following two lemmas were written in collaboration with V. Lazić.

######  Lemma 2.19.

Let $S,T,\widetilde{T}$ be quasi-projective varieties, assume that $T$ is
smooth. Let $h\colon S\to T$ be a projective fibration and let
$\vartheta\colon\widetilde{T}\to T$ be a finite map. Let

$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{F\times\widetilde{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{h}}$$\scriptstyle{\tau}$$\textstyle{T}$$\textstyle{\widetilde{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\vartheta}$

be a base change where $\tilde{h}$ is the second projection. Let $G$ be a
reduced fibre of $h$. Let $y\in\widetilde{T}$ be such that $\tau(y)=x$. Then
$\tau\colon F\times\\{y\\}\rightarrow G$ is an isomorphism.

###### Proof.

After cutting the base with $\dim T-1$ hyperplane sections through $x$, we can
assume that $\dim T=1$.

The morphisms $\vartheta$ and $\tau$ have the same degree, set
$d=\deg\tau=\deg\vartheta$. Let $x\in\tilde{T}$ be such that $G=h^{*}x$. Write
$\tau^{*}G=\sum a_{i}F\times\\{y_{i}\\}$ and $\vartheta^{*}x=\sum e_{i}y_{i}$.
Thus $\tau^{*}h^{*}x=\tau^{*}G=\sum
a_{i}F\times\\{y_{i}\\}=\tilde{h}^{*}\vartheta^{*}x=\tilde{h}^{*}\sum
e_{i}y_{i}=\sum e_{i}F\times\\{y_{i}\\}$. It follows, perhaps after
renumbering the $y_{i}$, that $a_{i}=e_{i}$ for all $i$. Moreover, $d=\sum
a_{i}\deg(F\times\\{p_{i}\\}\to G)=\sum e_{i}\deg(F\times\\{p_{i}\\}\to
G)\geq\sum e_{i}=d$. Thus $\deg(F\times\\{p_{i}\\}\to G)=1$ for all $i$. ∎

######  Lemma 2.20.

Let $g\colon(Z,\Delta_{Z})\rightarrow T$ be a klt-trivial fibration, where
$\Delta_{Z}\geq 0$ and the discriminant $B_{g}$ is a reduced divisor. Assume
we have a base change diagram

$\textstyle{(Z,\Delta_{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{(\overline{Z},\Delta_{\overline{Z}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{g}}$$\scriptstyle{\overline{\alpha}}$$\textstyle{T}$$\textstyle{\overline{T},\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$

where $\alpha$ and $\overline{\alpha}$ are finite morphisms and $\overline{g}$
is weakly semistable in codimension $1$. Let $B_{\overline{g}}$ be the
discriminant of $\overline{g}$ and assume that
$\Delta_{Z,v}=(g^{*}B_{g})_{\mathrm{red}}$. Then there exists an open subset
$U\subseteq\overline{T}$ with complement of codimension at leat 2 in
$\overline{T}$ such that:

  1. (i)

$(\overline{g}^{*}B_{\overline{g}})|_{\overline{g}^{-1}(U)}=\big{(}(\overline{\alpha}^{*}\Delta_{Z,v})_{\mathrm{red}}-\overline{g}^{*}R^{\prime}_{T}\big{)}|_{\overline{g}^{-1}(U)}$,
where $R^{\prime}_{T}$ is an effective divisor supported on the ramification
divisor of $\alpha$ and having no common components with $\alpha^{*}B_{g}$;

  2. (ii)

$\Delta_{\overline{Z}}|_{\overline{g}^{-1}(U)}=\big{(}\overline{\alpha}^{*}\Delta_{Z,h}+(\overline{\alpha}^{*}\Delta_{Z,v})_{\mathrm{red}}-\overline{g}^{*}R^{\prime}_{T}\big{)}|_{\overline{g}^{-1}(U)}$.

In particular,
$(\Delta_{\overline{Z}}-\overline{g}^{*}B_{\overline{T}})|_{\overline{g}^{-1}U}\geq
0$ and if $T$ is a curve, then
$\Delta_{\overline{Z}}-\overline{g}^{*}B_{\overline{T}}\geq 0$.

###### Proof.

_Step 1._ Note that $B_{g}\geq 0$ since $\Delta_{Z}\geq 0$. Let
$R_{T}\subseteq\overline{T}$ and $R_{Z}\subseteq\overline{Z}$ be the
ramification divisors of the finite maps $\alpha$ and $\overline{\alpha}$,
respectively. We have

(3)
$\operatorname{Supp}R_{Z}\subseteq\overline{g}^{-1}(\operatorname{Supp}R_{T})$

since the base change by an étale map is étale. We can write

(4) $R_{T}=R^{\prime}_{T}+\alpha^{*}B_{T}-(\alpha^{*}B_{T})_{\mathrm{red}},$

where $R^{\prime}_{T}\geq 0$, and $R_{T}^{\prime}$ and $\alpha^{*}B_{T}$ have
no common components. By [Amb04, Lemma 5.1] we have

(5)
$K_{\overline{T}}+B_{\overline{g}}=\alpha^{*}(K_{T}+B_{g})\quad\text{and}\quad
M_{\overline{g}}=\alpha^{*}M_{g},$

where $M_{\overline{g}}$ is the moduli part of $\overline{g}$. Then (4) gives

(6)
$B_{\overline{g}}=\alpha^{*}B_{g}-R_{T}=(\alpha^{*}B_{g})_{\mathrm{red}}-R^{\prime}_{T}.$

Similarly, we can write

(7)
$R_{Z}=R_{Z}^{\prime}+\overline{\alpha}^{*}\Delta_{Z,v}-(\overline{\alpha}^{*}\Delta_{Z,v})_{\mathrm{red}},$

where $R^{\prime}_{Z}\geq 0$, and $R_{Z}^{\prime}$ and
$\overline{\alpha}^{*}\Delta_{Z,v}$ have no common components. Then (7)
implies

(8) $\displaystyle\Delta_{\overline{Z}}$
$\displaystyle=\overline{\alpha}^{*}\Delta_{Z}-R_{Z}=\overline{\alpha}^{*}\Delta_{Z,h}+\overline{\alpha}^{*}\Delta_{Z,v}-R_{Z}$
$\displaystyle=\overline{\alpha}^{*}\Delta_{Z,h}+(\overline{\alpha}^{*}\Delta_{Z,v})_{\mathrm{red}}-R_{Z}^{\prime}.$

We claim that for a prime divisor
$P\subseteq\alpha(\operatorname{Supp}R^{\prime}_{T})$,

(9) $g^{*}P$ is reduced over the generic point of $P$.

Indeed, otherwise we would have $P\subseteq\operatorname{Supp}B_{g}$ by the
definition of the discriminant. However, this would contradict the fact that
$R_{T}^{\prime}$ and $\alpha^{*}B_{g}$ have no common components.

Let $U\subseteq\overline{T}$ be a big open subset with the following property:
$\overline{g}$ is weakly semistable over $U$, and if a prime divisor
$D\subseteq\overline{g}^{-1}(\operatorname{Supp}R_{T}^{\prime})$ is
$\overline{g}$-exceptional, then $\overline{g}(D)\cap U=\emptyset$. We show in
Steps 2 and 3 that $U$ is satisfies (i) and (ii).

_Step 2._ To show (i), by (6) it is enough to prove

(10)
$\overline{g}^{*}\big{(}(\alpha^{*}B_{T})_{\mathrm{red}}\big{)}|_{\overline{g}^{-1}(U)}=(\overline{\alpha}^{*}\Delta_{Z,v})_{\mathrm{red}}|_{\overline{g}^{-1}(U)}.$

For (10), we have

$(\overline{\alpha}^{*}\Delta_{Z,v})_{\mathrm{red}}=(\overline{\alpha}^{*}g^{*}B_{g})_{\mathrm{red}}=(\overline{g}^{*}\alpha^{*}B_{g})_{\mathrm{red}},$

where the first equality follows by pulling back the relation
$\Delta_{Z,v}=(g^{*}B_{g})_{\mathrm{red}}$ by $\overline{\alpha}$ and taking
the reduced part, and the second equality by the base change diagram. Since
$\overline{g}^{*}\big{(}(\alpha^{*}B_{g})_{\mathrm{red}}\big{)}|_{\overline{g}^{-1}(U)}$
is reduced, we have

$(\overline{g}^{*}\alpha^{*}B_{g})_{\mathrm{red}}|_{\overline{g}^{-1}(U)}=\overline{g}^{*}\big{(}(\alpha^{*}B_{g})_{\mathrm{red}}\big{)}|_{\overline{g}^{-1}(U)},$

which proves (i).

_Step 3._ Finally, we show (ii). By (8), it suffices to show

(11)
$R_{Z}^{\prime}|_{\overline{g}^{-1}(U)}=\overline{g}^{*}R^{\prime}_{T}|_{\overline{g}^{-1}(U)}.$

By (3), (4), (7) and (10) we have

$\displaystyle(\operatorname{Supp}R^{\prime}_{Z})|_{\overline{g}^{-1}(U)}$
$\displaystyle\cup(\operatorname{Supp}\overline{\alpha}^{*}\Delta_{Z,v})|_{\overline{g}^{-1}(U)}$
$\displaystyle=(\operatorname{Supp}R_{Z})|_{\overline{g}^{-1}(U)}\subseteq\overline{g}^{-1}(\operatorname{Supp}R_{T}|_{U})$
$\displaystyle\subseteq\overline{g}^{-1}(\operatorname{Supp}R_{T}^{\prime}|_{U})\cup(\operatorname{Supp}\overline{g}^{*}\alpha^{*}B_{T})_{\overline{g}^{-1}(U)}$
$\displaystyle=\overline{g}^{-1}(\operatorname{Supp}R_{T}^{\prime}|_{U})\cup(\operatorname{Supp}\overline{\alpha}^{*}\Delta_{Z,v})|_{\overline{g}^{-1}(U)}.$

Since $R_{Z}^{\prime}$ and $\overline{\alpha}^{*}(\Delta_{Z,v})$ have no
common components, this implies

$(\operatorname{Supp}R_{Z}^{\prime})|_{\overline{g}^{-1}(U)}\subseteq\overline{g}^{-1}(\operatorname{Supp}R_{T}^{\prime}|_{U}).$

Therefore, for (11) it is enough to show – by the definition of $U$ – that for
each prime divisor
$D\subseteq\overline{g}^{-1}(\operatorname{Supp}R_{T}^{\prime})$ such that
$\overline{g}(D)$ is a divisor in $\overline{T}$ we have

(12)
$\operatorname{mult}_{D}R_{Z}^{\prime}=\operatorname{mult}_{D}\overline{g}^{*}R^{\prime}_{T}.$

Fix such a prime divisor $D$. Denote $Q:=\overline{g}(D)$ and $P:=\alpha(Q)$,
and let $e_{Q}=\operatorname{mult}_{Q}\alpha^{*}P$. Then

(13) $\displaystyle\operatorname{mult}_{D}\overline{g}^{*}R^{\prime}_{T}$
$\displaystyle=\operatorname{mult}_{D}\overline{g}^{*}\big{(}\alpha^{*}P-(\alpha^{*}P)_{\mathrm{red}}\big{)}$
$\displaystyle=(e_{Q}-1)\operatorname{mult}_{D}\overline{g}^{*}Q=e_{Q}-1,$

where the last equality follows since $\overline{g}^{*}Q$ is reduced over the
generic point of $Q$ by the assumption on weak semistability. Furthermore, by
the commutativity of the base change diagram, we also have

$\operatorname{mult}_{D}\overline{\alpha}^{*}g^{*}P=\operatorname{mult}_{D}\overline{g}^{*}\alpha^{*}P=e_{Q}\operatorname{mult}_{D}\overline{g}^{*}Q=e_{Q}.$

Since $g^{*}P$ is reduced over the generic point of $P$ by (9), this shows
that the ramification index of $\overline{\alpha}$ along $D$ is $e_{Q}$, which
together with (13) gives

$\operatorname{mult}_{D}R_{Z}=\operatorname{mult}_{D}\overline{g}^{*}R^{\prime}_{T}.$

To finish the proof of (12) and of (ii), by (7) we only need to show that
$D\not\subseteq\operatorname{Supp}\overline{\alpha}^{*}\Delta_{Z,v}$. Assume
otherwise: then $Q\subseteq\operatorname{Supp}\alpha^{*}B_{T}$ by (10), hence
$Q$ would not be a component of $R_{T}^{\prime}$ by the construction of
$R_{T}^{\prime}$ in Step 1, a contradiction. ∎

######  Proposition 2.21.

Let $f\colon(X,\Delta)\rightarrow Y$ be a klt-trivial fibration of normal
projective varieties with $X$ $\mathbb{Q}$-factorial. Assume $\Delta$
effective over the generic point of $Y$ and $\Delta-f^{*}B_{f}\geq 0$. Assume
that $Y$ is an Ambro model and $M_{f}$ is semiample, let $\phi\colon Y\to V$
be the fibration induced by $M_{f}$. Let $Y_{r}$ be the set of points $x\in Y$
such that $f^{-1}x$ is reduced. Then there are a non empty open set
$V_{0}\subseteq V$, an open subset $Y_{0}\subseteq Y$ with complement of
codimension at least 2 and a set $I(Y)\supseteq\phi^{-1}V_{0}\cap Y_{0}\cap
Y_{r}$ with the following property: for every $x_{1},x_{2}\in I(Y)$ such that
$\phi(x_{1})=\phi(x_{2})$, if $(F_{i},\Delta_{i})$ is the fibre over $x_{i}$
with $\Delta_{i}=\Delta^{h}|_{F_{i}}$, then
$(F_{1},\Delta_{1})\cong(F_{2},\Delta_{2})$.

###### Proof.

We apply Theorem 2.16 and find $\vartheta$ and $\chi$ and a diagram such that
$\vartheta^{*}M_{f}=\chi^{*}M_{f^{+}}$. In particular both
$\vartheta^{*}M_{f}$ and $M_{f^{+}}$ are semiample. After passing to the Stein
factorisation we can assume that $\chi$ has connected fibres. Let
$\widetilde{X}$ be the main component of the normalisation of
$X\times_{Y}\widetilde{Y}$ with the natural morphism
$\tau\colon\widetilde{X}\to X$ and let $\widetilde{\Delta}$ be defined by
$K_{\widetilde{X}}+\widetilde{\Delta}=\tau^{*}(K_{X}+\Delta)$. By Theorem 2.16
there is an open set $\widetilde{U}\subseteq\widetilde{Y}$ and an isomorphism

$(\widetilde{X},\widetilde{\Delta})\times_{\widetilde{Y}}\widetilde{U}\cong(X^{+},\Delta^{+})\times_{Y^{+}}\widetilde{U}.$

By Proposition 2.18, the isomorphism extends to

$(\widetilde{X},\widetilde{\Delta}-\tilde{f}^{*}B_{\tilde{f}})\times_{\widetilde{Y}}\widetilde{Y}_{0}\to(X^{+},\Delta^{+})\times_{Y^{+}}\widetilde{Y}_{0}$

with
$\widetilde{Y}_{0}=\widetilde{Y}\setminus\tilde{f}(\operatorname{Supp}(\widetilde{\Delta}_{v}-\tilde{f}^{*}B_{\tilde{f}})^{<0})$.

There is a diagram

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\widetilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\phi}}$$\scriptstyle{\vartheta}$$\scriptstyle{\chi}$$\textstyle{Y^{+}}$$\textstyle{V}$$\textstyle{\widetilde{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$

where $\sigma\circ\tilde{\phi}$ is the Stein factorisation of
$\phi\circ\theta$.

By Lemma 2.17 there is a generically finite map $\lambda\colon
Y^{+}\to\widetilde{V}$, and it is birational because $\tilde{\phi}$ is a
fibration.

After passing to an open set $U^{+}$ of $Y^{+}$ we can assume that $\lambda$
is an isomorphism and let $\widetilde{V}_{0}=\lambda(U^{+})$.

By Lemma 2.17 and Lemma 2.20 after possibly shrinking $\widetilde{V}_{0}$
further, we can assume that the complement of
$\tilde{f}(\operatorname{Supp}(\widetilde{\Delta}_{v}-\tilde{f}^{*}B_{\tilde{f}})^{<0})\cap\tilde{\phi}^{-1}\widetilde{V}_{0}$
has codimension at least 2 in $\tilde{\phi}^{-1}\widetilde{V}_{0}$.

Then, for $\tilde{x}_{1},\tilde{x}_{2}\in\tilde{\phi}^{-1}\widetilde{V}_{0}$
in the same fibre of $\tilde{\phi}$, the two corresponding fibres are
isomorphic, together with the boundaries.

Let $x_{1},x_{2}\in
Y_{r}\cap\vartheta^{-1}\tilde{\phi}^{-1}\widetilde{V}_{0}=Y_{r}\cap\phi^{-1}\sigma^{-1}\widetilde{V}_{0}$.
If $\phi(x_{1})=\phi(x_{2})$, then there are
$\tilde{x}_{1},\tilde{x}_{2}\in\phi^{-1}\widetilde{V}_{0}$ such that
$\theta(\tilde{x}_{i})=x_{i}$ and
$\tilde{\phi}(\tilde{x}_{1})=\tilde{\phi}(\tilde{x}_{2})$. By Lemma 2.19, the
restriction of $\tau$ to $\tilde{f}^{-1}(\tilde{x}_{i})$ is an isomorphism,
concluding the proof. ∎

We also need the following [Amb05, Theorem 3.5]; see also [Flo14, Theorem 1.2]
for a sharper version.

######  Theorem 2.22.

Let $f\colon(X,\Delta)\to Y$ be a klt-trivial fibration, and assume that the
moduli part $M_{Y}$ descends to $Y$. If $M_{Y}\equiv 0$, then
$M_{Y}\sim_{\mathbb{Q}}0$.

###  2.4. Variation of a klt-trivial fibration

In this section we give the definition and some properties of the variation of
a fibration. For the original definition with $\Delta=0$ and some further
discussion of the properties see [Vie83, Kol87, Fuj03].

######  Definition 2.23.

Let $(X,\Delta)$ be a pair and let $f\colon(X,\Delta)\to Y$ be a fibration. We
define the variation of $f$, denoted by $Var(f)$ as

$\min\left\\{\dim
Y^{+}\left|\begin{array}[]{l}\exists\vartheta\colon\widetilde{Y}\to Y\text{
generically finite}\\\ \exists\chi\colon\widetilde{Y}\to
Y^{+},f^{+}\colon(X^{+},\Delta^{+})\to Y^{+}\text{ fibrations}\\\ \text{such
that the fibration induced by}\,f,\vartheta\text{ by fibre}\\\ \text{product
and the fibration induced by}\,f,\chi\text{ by fibre}\\\ \text{product are
birational over }\widetilde{Y}.\end{array}\right.\right\\}$

The following is a generalisation of [Fuj03, Theorem 3.8] to the case
$\Delta\neq 0$, the proof is essentially the same.

######  Proposition 2.24.

Let $f\colon(X,\Delta)\to Y$ be an lc-trivial fibration, assume $Y$ is an
Ambro model. Then $\kappa(M_{f})\leq Var(f)$.

###### Proof.

Thet $\vartheta,\chi$ be such that $Var(f)=transdeg_{k}k(Y^{+})$. Then, after
perhaps passing to higher models of $\widetilde{Y}$ and $Y^{+}$, we have
$\theta^{*}M_{f}=\chi^{*}M_{f^{+}}$. Therefore
$\kappa(M_{f})=\kappa(M_{f^{+}})\leq\dim Y^{+}=Var(f)$.

∎

######  Proposition 2.25.

Let $(X,\Delta)$ be a pair with $X$ $\mathbb{Q}$-factorial and $\Delta\geq 0$
and with coefficients in $\mathbb{Q}$. Let $f\colon(X,\Delta)\to Y$ be a
fibration such that $Var(f)=\dim Y$. Then there is a countable union $E$ of
closed subsets of $Y$ and an open set $U\subseteq Y$ such that for every $y\in
Y\setminus E$ the set

$\\{z\in
U|\;(f^{-1}y,\Delta^{h}|_{f^{-1}y})\cong(f^{-1}z,\Delta^{h}|_{f^{-1}z})\\}$

is a finite set.

###### Proof.

Set $F=f^{-1}y$. We fix a polarisation $A=p_{F}^{*}A_{F}+p_{X}^{*}A_{X}$ on
$F\times X$, where $p_{F}$ and $p_{X}$ are the two projections. Then there is
a quasi projective scheme
$\operatorname{Emb}(F,X)\subseteq\operatorname{Hilb}(F\times X)$ representing
the functor $\mathcal{H}ilb_{P}(F\times X)$ where $P$ is the Hilbert
polynomial of the graph of $F\to f^{-1}y\subseteq X$. There is also a
universal family $u\colon\operatorname{Univ}(F,X)\to\operatorname{Emb}(F,X)$
and a diagram

$\textstyle{\operatorname{Univ}(F,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{F\times
X\times\operatorname{Emb}(F,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{X}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\operatorname{Emb}(F,X)}$$\textstyle{Y.}$

After perhaps replacing the polarisation $A$ with $A+p_{X}^{*}f^{*}A_{Y}$ for
a sufficiently ample divisor $A_{Y}$ on $Y$, all the fibres of $u$ are
contracted by $f\circ p_{X}$. By the rigidity lemma there is
$\phi\colon\operatorname{Emb}(F,X)\to Y$. Its image is $\\{z\in
Y|\;f^{-1}z\cong F\\}$.

Let $k\in\mathbb{N}$ be such that $k\Delta^{h}$ is a Cartier divisor. Set
$D=k\Delta^{h}|_{F}$. Then there is a locally closed subscheme
$\operatorname{Emb}\left((F,D),(X,\mathcal{D})\right)$ of
$\operatorname{Emb}(F,X)$ representing the functor
$\mathcal{E}mb\left((F,D),(X,\mathcal{D})\right)$ with

$\mathcal{E}mb\left((F,D),(X,\mathcal{D})\right)(Z)=\left\\{\begin{array}[]{l}Z-\text{morphisms
}\varphi\colon F\to X,\psi\colon D\to\mathcal{D}\\\ \text{flat over }Z\\\
\text{such that }\varphi\circ i=j\circ\psi,\varphi\text{
embedding}\end{array}\right\\}$

together with a universal family

$u\colon\operatorname{Univ}\left((F,D),(X,\mathcal{D})\right)\to\operatorname{Emb}\left((F,D),(X,\mathcal{D})\right).$

Therefore

$\\{z\in
U|\;(f^{-1}y,\Delta^{h}|_{f^{-1}y})\cong(f^{-1}z,\Delta^{h}|_{f^{-1}z})\\}=\phi(\operatorname{Emb}\left((F,D),(X,\mathcal{D})\right))$

is the image of an algebraic set. By [Kol87, Theorem 2.6] for $y$ in the
complement of a countable union of closed sets in $Y$ the left hand side is at
most countable. Therefore it is a finite set. ∎

##  3\. Semiample line bundles on simple normal crossings surfaces

In this section we establish a criterion of semiampleness of certain line
bundles on simple normal crossings surfaces. For later use and for this
section we introduce different notions of simple normal crossings varieties

Let $\mathcal{Z}$ be a variety with irreducible components $\\{Z_{i}:i\in
I\\}$. Assume that $\dim Z_{i}=k$ for every $i\in I$. We say that
$\mathcal{Z}$ is a simple normal crossing variety [Kol14, Definition 6] if the
$Z_{i}$ are smooth and every point $p\in\mathcal{Z}$ has an open (Euclidean)
neighborhood $p\in U_{p}\subseteq\mathcal{Z}$ and an embedding
$U_{p}\to\mathbb{C}^{k+1}$ such that the image of $U_{p}$ is an open subset of
the union of coordinate hyperplanes $(z_{1}\cdot\ldots\cdot z_{n}=0)$ with
$n\leq k+1$. A stratum of $\mathcal{Z}$ is any irreducible component of an
intersection $\cap_{i\in J}Z_{i}$ for some $J\subseteq I$.

Assume now $\mathcal{Z}=\cup_{k}\mathcal{Z}^{(k)}$ where $\mathcal{Z}^{(k)}$
is the union of irreducible components of dimension $k$. We say that
$\mathcal{Z}$ is a simple normal crossing variety if $\mathcal{Z}^{(k)}$ is
simple normal crossings in the above sense for every $k$ and for every $k$,
for every stratum $Z$ of $\mathcal{Z}_{k-1}=\cup_{j<k}\mathcal{Z}^{(j)}$ we
have that $\mathcal{Z}^{(k)}$ does not contain $Z$ and every point
$p\in\mathcal{Z}^{(k)}\cap Z$ has an open (Euclidean) neighborhood $p\in
U_{p}\subseteq\mathcal{Z}^{(k)}\cup Z$ and an embedding
$U_{p}\to\mathbb{C}^{k+1}$ such that he image of $U_{p}$ is an open subset of
the union of coordinate hyperplanes $(z_{1}\cdot\ldots\cdot z_{n}=0)$ and the
image of $Z$ is an open subset of $(z_{n+1}=\ldots=z_{m}=0)$ with $n<m\leq
k+1$ [Kol07b, Definition 3.24].

Finally, a curve is said to be seminormal if every point has a neighbourhood
if and only if it is analytically isomorphic to the union of the n coordinate
axes in $\mathbb{A}^{n}$ [Kol13, Example 10.12].

######  Remark 3.1.

If $\mathcal{Z}\subseteq Y$ has pure codimension 1 and is a simple normal
crossings variety, then it is a simple normal crossings divisor.

If $\mathcal{Z}$ is a connected simple normal crossings variety of pure
dimension 1, then it is a semistable curve.

A divisor on a simple normal crossings variety can be recovered from its
restrictions to its irreducible components plus a gluing condition on the
intersections:

Let $\mathcal{Z}$ be a simple normal crossings variety of pure dimension $k$.
A divisor on $\mathcal{Z}$ is the data of a divisor $D_{Z}$ on every
irreducible component $Z$ of $\mathcal{Z}$ with the property that, if $Z_{1}$
and $Z_{2}$ are two irreducible components of $\mathcal{Z}$, then
$D_{Z_{1}}|_{Z_{1}\cap Z_{2}}=D_{Z_{2}}|_{Z_{1}\cap Z_{2}}$.

From now on, we assume that $\mathcal{S}$ is a simple normal crossings variety
of pure dimension 2. We refer to $\mathcal{S}$ as a simple normal crossings
surface.

######  Lemma 3.2.

Let $\mathcal{S}$ be a connected simple normal crossings surface. Assume that
there is an integral curve $Q$ and a surjective morphism with connected fibres
$\varphi\colon\mathcal{S}\to Q$, that for every $S\subseteq\mathcal{S}$
irreducible component $\varphi(S)$ is an irreducible curve. For an irreducible
component $S$ of $\mathcal{S}$, we denote by $\varphi|_{S}\colon
S\overset{f_{S}}{\rightarrow}C(S)\to Q$ the Stein factorisation. Let $D$ be an
effective divisor on $\mathcal{S}$ such that
$\varphi(\operatorname{Supp}D)\subseteq Q^{\text{smooth}}$. Then there is a
positive integer $m$ such that $mD$ is the pullback of a Cartier divisor in
$Q$ if and only if for every irreducible component $S$ of $\mathcal{S}$ there
is a positive integer $d$ such that the restriction of $dD$ to $S$ is the
pullback of a divisor in $C(S)$.

###### Proof.

If there is a positive integer $m$ such that $mD$ is the pullback of a divisor
in $Q$, then the statement on the restrictions of $D$ to the irreducible
components of $\mathcal{S}$ is obvious.

We assume now that for every irreducible component $S$ of $\mathcal{S}$ there
is a positive integer $d$ such that the restriction of $dD$ to $S$ is the
pullback of a divisor in $C$.

By hypothesis, there are $p_{1},\ldots,p_{k}$ in the smooth locus of $Q$ such
that the support of $D$ is contained in
$\varphi^{-1}\\{p_{1},\ldots,p_{k}\\}$. We prove the statement by induction on
$k$. If $k=0$, there is nothing to prove. Assume now that the statement holds
for $k-1$. Let $Q_{1}$ be the irreducible component of $Q$ such that $p_{1}\in
Q_{1}$. Let $S$ be an irreducible component of $\mathcal{S}$ such that
$\varphi(S)=Q_{1}$. We set $D|_{S}=\sum_{\ell}\sum_{j}a_{\ell,j}F_{\ell,j}$
where for every $\ell$ the union $\cup_{j}F_{\ell,j}$ is a connected component
of $\operatorname{Supp}D|_{S}$. Without loss of generality, we can assume that
$\cup_{j}F_{1,j}$ is contained in $\varphi^{-1}p_{1}$. Let $\alpha$ be such
that $\varphi^{*}(\alpha
p_{1})|_{S}=\sum_{j}a_{1,j}F_{1,j}+\sum_{h}\sum_{j}b_{h,j}F_{h,j}$. We want to
prove that $\operatorname{Supp}D-\varphi^{*}(\alpha
p_{1})\subseteq\varphi^{-1}\\{p_{2},\ldots,p_{k}\\}$.

Assume that this is not the case, that is, assume that there is $S^{\prime}$
and an irreducible component $F$ of $\varphi^{-1}p_{1}\cap S^{\prime}$ such
that $\text{coeff}_{F}(D-\varphi^{*}(\alpha p_{1}))$ is not zero. The fibre
$\varphi^{-1}p_{1}$ is connected, thus there are
$S=S_{0},S_{1},\ldots,S_{N}=S^{\prime}$ and for every $i$ a subvariety
$\cup_{j}F^{i}_{j}$ of $S_{i}$ and a point $q_{i}\in S_{i}$ with the following
properties:

  * •

$\cup_{j}F^{i}_{j}$ is the support of a fibre of $f_{S_{i}}$;

  * •

$\cup_{j}F^{0}_{j}=\cup_{j}F_{1,j}$;

  * •

$F\subseteq\cup_{j}F^{N}_{j}$;

  * •

$q_{i}\in(\cup_{j}F^{i}_{j})\cap(\cup_{j}F^{i+1}_{j})$.

We have that for every $j$

$\text{coeff}_{F^{0}_{j}}(D-\varphi^{*}(\alpha
p_{1}))=0\;\;\;\;\text{and}\;\;\;\;\text{coeff}_{F^{N}_{j}}(D-\varphi^{*}(\alpha
p_{1}))\neq 0.$

Then there is $i$ such that for every $j$

$\text{coeff}_{F^{i}_{j}}(D-\varphi^{*}(\alpha
p_{1}))=0\;\;\;\;\text{and}\;\;\;\;\text{coeff}_{F^{i+1}_{j}}(D-\varphi^{*}(\alpha
p_{1}))\neq 0.$

This is a contradiction as $D-\varphi^{*}(\alpha p_{1})$ is a divisor on
$\mathcal{S}$ and $D-\varphi^{*}(\alpha p_{1})|_{S_{i}}$ but
$D-\varphi^{*}(\alpha p_{1})|_{S_{i+1}}$ do not coincide on the intersection
$S_{i}\cap S_{i+1}$. ∎

######  Lemma 3.3.

Let $\mathcal{S}$ be a connected simple normal crossings surface. Assume that
there is a seminormal curve $Q$ and a surjective morphism with connected
fibres $\varphi\colon\mathcal{S}\to Q$ and that for every
$S\subseteq\mathcal{S}$ irreducible component $\varphi(S)$ is an irreducible
curve. Let $\mathcal{L}$ be a line bundle on $\mathcal{S}$ such that for every
fibre $F$ of $\varphi$ the restriction $\mathcal{L}|_{(F)_{\text{red}}}$ has a
nowhere vanishing section. Then there is a positive integer $m$ and a line
bundle $\mathcal{M}$ on $Q$ such that $\mathcal{L}^{\otimes
m}\sim\varphi^{*}\mathcal{M}$.

###### Proof.

Let $Q=\cup\overline{Q}_{i}$ be the decomposition of $Q$ into irreducible
components. Let $\mathcal{S}_{i}$ be the union of the irreducible components
$S$ of $\mathcal{S}$ such that $\varphi(S)=\overline{Q}_{i}$ and let
$\varphi\colon\mathcal{S}_{i}\overset{f_{i}}{\longrightarrow}Q_{i}\overset{\nu_{i}}{\longrightarrow}\overline{Q}_{i}$
be the Stein factorisation. The morphism $\nu_{i}$ is birational and finite.
We prove that $Q_{i}$ is normal. Indeed for every irreducible component
$S_{i,j}$ of $\mathcal{S}_{i}$ the restriction of $f_{i}$ to $S_{i,j}$ factors
through the normalisation $Q_{i}^{\nu}$ of $Q_{i}$ and there is $f_{i,j}\colon
S_{i,j}\to Q_{i}^{\nu}$. As $\mathcal{S}_{i}$ has simple normal crossings, the
restriction of $f_{i}$ to $S_{i,h}\cap S_{i,k}$ factors through $Q_{i}^{\nu}$.
Thus, if and if $x\in S_{i,h}\cap S_{i,k}$, we have $f_{i,h}(x)=f_{i,k}(x)$
and there is a morphism $f^{\prime}_{i}\colon\mathcal{S}_{i}\to Q_{i}^{\nu}$.
By the uniqueness of the Stein factorisation $Q_{i}=Q_{i}^{\nu}$.

Since it is a curve, it is a smooth projective curve and the morphism $f_{i}$
is flat.

The sheaf $f_{i*}(\mathcal{L}^{\vee}|_{\mathcal{S}_{i}})$ has generically rank
1 on $Q_{i}$ and, by semicontinuity, all its stalks are non zero. Let $A_{i}$
be an ample line bundle on $Q_{i}$. After possibly replacing $A_{i}$ with a
multiple, we can assume that $f_{i*}(\mathcal{L}^{\vee})\otimes A_{i}$ is
globally generated and has therefore a non-zero global section. Moreover, by
the projection formula, as $A_{i}$ is locally free, we have

$H^{0}(\mathcal{S}_{i},\mathcal{L}^{\vee}\otimes
f_{i}^{*}A_{i})=H^{0}(Q_{i},f_{i*}(\mathcal{L}^{\vee}\otimes
f_{i}^{*}A_{i}))=H^{0}(Q_{i},f_{i*}(\mathcal{L}^{\vee})\otimes A_{i}).$

Then there is a non-zero global section $s\in
H^{0}(\mathcal{S}_{i},\mathcal{L}^{\vee}\otimes f_{i}^{*}A_{i})$ inducing an
isomorphism of line bundles $\mathcal{L}^{\vee}\otimes
f_{i}^{*}A_{i}\sim\mathcal{O}(D_{i})$ with $D_{i}$ an effective Cartier
divisor on $\mathcal{S}_{i}$. On the general fibre the morphism
$\mathcal{L}\to f_{i}^{*}A_{i}$ is an isomorphism, therefore $D_{i}$ is
supported on fibres of $f_{i}$.

For an irreducible component $S$ of $\mathcal{S}_{i}$, we denote by
$f_{i}|_{S}\colon S\overset{f_{S}}{\longrightarrow}C\to Q_{i}$ the Stein
factorisation. By Zariski’s lemma [BPVdV84, Lemma 8.2], for every irreducible
component $S$ of $\mathcal{S}_{i}$, the restriction $D_{i}|_{S}$ is
proportional to fibres of $f_{S}$. By Lemma 3.2, the divisor $D_{i}$ is
proportional to fibres of $f_{i}$ and
$\mathcal{O}(D_{i})\sim_{\mathbb{Q}}f_{i}^{*}\delta_{i}$ with $\delta_{i}\geq
0$. After tensoring $A_{i}$ with a higher multiple, we can assume that
$\operatorname{Supp}\delta_{i}\subseteq\nu_{i}^{-1}Q^{\text{reg}}$.

For $p\not\in Q^{\text{reg}}$ let $F_{p}=\varphi^{-1}p$. We notice that
$F_{p}$ is a semistable curve. As $Q$ is seminormal, there are $A$ and
$\delta$ on $Q$ such that $A|_{Q_{i}}=A_{i}$ and $\delta|_{Q_{i}}=\delta_{i}$
for every $i$ such that
$f_{i}^{*}A_{i}|_{F_{p}}=\mathcal{L}|_{F_{p}}=f_{i}^{*}\delta_{i}|_{F_{p}}$
for every $p\not\in Q^{\text{reg}}$.

It follows that $\mathcal{L}\sim\varphi^{*}A(-\delta)$. ∎

######  Remark 3.4.

To prove that $Q_{i}$ is normal we could also have argued in the following
way. Let $\sqcup S_{i,j}$ be the normalisation of $\mathcal{S}_{i}$ and let
for every $i,j$ be $f_{i}\colon S_{i,j}\to
V_{i,j}\overset{\sigma_{i,j}}{\longrightarrow}Q_{i}$ be the Stein
factorisation of $\varphi_{S_{i,j}}$. Then $Q_{i}$ is the quotient of $\sqcup
V_{i,j}$ by the relation, for $x\in V_{ih}$ and $y\in V_{ik}$, $x\sim y$ if
and only if $\sigma_{ih}(x)=\sigma_{ik}(y)$. This equivalence relation is
finite, equidimensional and $\sqcup V_{i,j}$ is normal. By [Kol13, Proposition
9.14] the curve $Q_{i}$ is normal as well.

######  Theorem 3.5.

Let $\mathcal{S}$ be a connected simple normal crossings surface. Assume that
there is an integral seminormal curve $Q$ and a surjective morphism with
connected fibres $\varphi\colon\mathcal{S}\to Q$. Let $\mathcal{L}$ be a line
bundle on $\mathcal{S}$ such that for every $S\subseteq\mathcal{S}$ the
restriction $\mathcal{L}|_{S}$ is semiample and the Stein factorisation of
$\varphi|_{S}$ is the morphism induced by $\mathcal{L}|_{S}$. Assume that for
every fibre $F$ of $\varphi$ the restricted line bundle
$\mathcal{L}|_{(F)_{\text{red}}}$ has a non-zero section. Then $\mathcal{L}$
is semiample.

###### Proof.

Let $x\in\mathcal{S}$ be a point. We want to prove that there is a global
section of $\mathcal{L}$ non zero along $x$.

We write $\mathcal{S}=\mathcal{S}_{0}\cup\mathcal{S}_{1}$ where

$\mathcal{S}_{0}=\\{S|\;\mathcal{L}|_{S}\text{ has Kodaira dimension 0}\\}$
$\mathcal{S}_{1}=\\{S|\;\mathcal{L}|_{S}\text{ has Kodaira dimension 1}\\}$

Let
$\varphi\colon\mathcal{S}_{1}\overset{f}{\longrightarrow}Q^{\prime}\overset{\nu}{\longrightarrow}\overline{Q}_{i}$
be the Stein factorisation. The morphism $\nu$ is birational and finite. By
Lemma 3.3 there is a positive integer $m$ and a line bundle $\mathcal{M}$ on
$Q^{\prime}$ such that $\mathcal{L}^{\otimes m}\sim f^{*}\mathcal{M}$. The
line bundle $\mathcal{M}$ is ample on $Q^{\prime}$. After maybe taking a
multiple of $m$, there is a global section $s$ of $\mathcal{M}$ which is non
zero on every irreducible component of $Q^{\prime}$, such that if
$\nu(p_{1})=\nu(p_{2})$ then $s(p_{1})=s(p_{2})$ and such that
$s(\varphi(x))\neq 0$.

Set $\varphi(\mathcal{S}_{0})=\\{q_{1},\ldots,q_{k}\\}$ and
$F_{i}=\varphi^{-1}q_{i}$ taken with the reduced structure. Thus for every $i$
we chose a global section $s_{i}$ of $\mathcal{L}|_{F_{i}}$ agreeing with
$\varphi^{*}s$ on $F_{i}\cap\mathcal{S}_{1}$. Thus the data
$s_{1},\ldots,s_{k},f^{*}s$ define a global section of $\mathcal{L}$ which
does not vanish on $x$.

∎

##  4\. Profinite equivalence relations

Let $X$ be a scheme. A relation on $X$ is the data of a scheme $\mathcal{S}$
and an embedding $\sigma\colon\mathcal{S}\to X\times X$ [Kol13, Definition
9.1]. It is finite if the projections $\sigma_{i}\colon\mathcal{S}\to X$ are
finite for $i=1,2$. A set theoretic equivalence relation, or equivalence
relation for short, is a relation such that $\sigma$ is geometrically
injective, $\mathcal{S}$ contains the diagonal (reflexive), is invariant by
the involution of $X\times X$ exchanging the two factors (symmetric) and is
transitive, that is, if we consider

$\textstyle{\mathcal{S}\times_{X}\mathcal{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{2}}$$\textstyle{\mathcal{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{1}}$$\textstyle{X}$

then there is a natural morphism
$\sigma\colon\mathcal{S}\times_{X}\mathcal{S}\to X\times X\times X$ and
$(\pi_{1},\pi_{2})\circ\sigma\left(\mathrm{red}(\mathcal{S}\times_{X}\mathcal{S})\right)\to
X\times X$ factors through $\mathcal{S}$ [Kol13, Definition 9.2].

######  Remark 4.1.

If $\mathcal{R}$ is a finite equivalence relation on an algebraic variety (not
necessarily irreducible) and $Z\subset X$ a subvariety then
$\mathcal{S}Z=\\{z\in X|\;\text{ there is}\;z^{\prime}\in Z\text{
with}\;(z,z^{\prime})\in\mathcal{R}\\}$ is a finite union of subvarieties of
$X$. Indeed, we have $\mathcal{R}Z=\sigma_{2}\sigma_{1}^{-1}Z$.

######  Definition 4.2.

Let $\mathcal{R}$ be an equivalence relation on $X$. A subset $Z\subseteq X$
is invariant by $\mathcal{R}$ if one of the following equivalent condition is
verified:

  * •

for every $x\in X$, if there is $z\in Z$ which is equivalent to $x$, then
$x\in Z$;

  * •

for every $x\in X$, if there is $z\in Z$ such that $(x,z)\in\mathcal{R}$ then
$x\in Z$;

  * •

$\sigma_{2}\sigma_{1}^{-1}Z\subseteq Z$;

  * •

$\sigma_{1}\sigma_{2}^{-1}Z\subseteq Z$.

######  Construction 4.3.

(Equivalence closure) The equivalence closure $\langle\mathcal{S}\rangle$ of a
relation $\mathcal{S}$ is the smallest equivalence relation containing it. We
refer to [Kol13, 9.3 ] for the complete construction, which consists in making
$\mathcal{S}$ reflexive, symmetric and transitive. We recall just that if
$S_{1},S_{2}\subseteq\mathcal{S}$ are irreducible components, then in order to
make $\mathcal{S}$ transitive, we “add” to $\mathcal{S}$ the variety
$S_{3}=(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})(S_{1}\times_{X}S_{2})$,
where $\pi_{i}$ are the natural projections

$\textstyle{S_{1}\times_{X}S_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$$\textstyle{S_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{1}}$$\scriptstyle{\sigma_{2}}$$\textstyle{S_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{1}}$$\scriptstyle{\sigma_{2}}$$\textstyle{X}$$\textstyle{X}$$\textstyle{X}$

An equivalence relation is called profinite if it is the closure of a finite
relation.

######  Notation 4.4.

We denote by $\mathcal{S}_{K}$ (resp. $\mathcal{S}_{\leq K}$) the union of all
the irreducible components of $\mathcal{S}$ of dimension $K$ (resp. $\leq K$).

######  Lemma 4.5.

Let $X$ be a normal variety of maximal dimension $D$. Notation as in
Construction 4.3. Then

  1. (1)

$\dim S_{3}\leq\min\\{\dim S_{1},\dim S_{2}\\}$;

  2. (2)

if $S_{1},S_{2}\subseteq\mathcal{S}_{D}$ and
$\sigma_{2}(S_{1})=\sigma_{1}(S_{2})$, then every component of $S_{3}$ has
dimension $\dim S_{1}=\dim S_{2}$;

  3. (3)

$\sigma_{1}(S_{3})\subseteq\sigma_{1}(S_{1})$,
$\sigma_{2}(S_{3})\subseteq\sigma_{2}(S_{2})$.

###### Proof.

Both $\pi_{1}$ and $\pi_{2}$ are finite morphisms as they are the base change
of $\sigma_{1}$ and $\sigma_{2}$ respectively, which are finite. Thus

$\dim S_{1}\times_{X}S_{2}\leq\min\\{\dim S_{1},\dim S_{2}\\}.$

If $\sigma_{2}(S_{1})=\sigma_{1}(S_{2})$, then $\dim S_{1}=\dim S_{2}$ because
$\sigma_{i}$ is finite for $i=1,2$. Since $\pi_{i}$ is surjective, $\dim
S_{1}\times_{X}S_{2}=\min\\{\dim S_{1},\dim S_{2}\\}$.
Assume that $S_{1},S_{2}$ have dimension $D$. Then their image $X_{1}$ in $X$
is an irreducible component, and therefore normal. By Chevalley’s criterion
[Gro67, 14.4.4] and [Kol13, Definition 1.44] the morphism $\sigma_{2}\colon
S_{1}\to X_{1}$ is universally open. Then $\sigma^{\prime}_{2}\colon
S_{1}\times_{X_{1}}S_{2}\to S_{1}$ is open and finite.
Let $S_{1}\times_{X_{1}}S_{2}=W_{1}\cup\ldots\cup W_{\ell}$ be the
decomposition into irreducible components. Then
$U_{1}=W_{1}\cap(W_{2}\cup\ldots\cup W_{\ell})^{c}$ is open in
$S_{1}\times_{X_{1}}S_{2}$ and its image $\sigma^{\prime}_{2}(U_{1})$ is open
in $S_{2}$. Thus $\dim W_{1}=\dim U_{1}=\dim\sigma^{\prime}_{2}(U_{1})=\dim
S_{2}$.

Since $\pi_{i}$ and $\sigma_{i}$ are finite for $i=1,2$,
$(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})$ is finite as well, proving
(1) and (2).

As for (3), we have
$\sigma_{i}(S_{3})=\sigma_{i}((\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})S_{1}\times_{X}S_{2})=(\sigma_{i}\circ\pi_{i})(S_{1}\times_{X}S_{2})\subseteq\sigma_{i}(S_{i})$.

∎

######  Remark 4.6.

We denote by $\Delta_{X}$ the diagonal of $X\times X$ By Lemma 4.5, if
$D=\dim(\mathcal{S}\setminus\Delta_{X})$, then
$\langle\mathcal{S}_{D}\rangle_{D}=\langle\mathcal{S}\rangle_{D}$.

The following lemma is a slight generalisation of [ByB04, 2.7]. It is a
consequence of Lemma 4.5(2) for which we followed closely the proof of [Kol13,
Lemma 9.14].

######  Lemma 4.7.

Let $X$ be a normal variety of dimension $D$. Let $\mathcal{S}$ be a finite
relation on $X$. Then
$\langle\mathcal{S}_{D}\rangle=\langle\mathcal{S}\rangle_{D}$. In particular
$\langle\mathcal{S}\rangle_{D}$ is an equivalence relation.

###### Proof.

By Remark 4.6 it is enough to prove that
$\langle\mathcal{S}_{D}\rangle_{D}=\langle\mathcal{S}_{D}\rangle$. One
inclusion is obvious. For the other one, if $S_{1},S_{2}$ are two components
of $\mathcal{S}_{D}$, since $X$ is normal and $D=\dim X$, then either
$\sigma_{1}(S_{1})=\sigma_{2}(S_{2})$ or
$\sigma_{1}(S_{1})\cap\sigma_{2}(S_{2})=\emptyset$. Therefore by Lemma 4.5(2),
every irreducible component of $S_{1}\times_{X}S_{2}$ and of its projection in
$X\times X$ has dimension $D$.

∎

######  Lemma 4.8.

Let $X$ be a normal variety of dimension $D$. Let $\mathcal{S}$ be a finite
reflexive and symmetric relation on $X$ and $\mathcal{R}$ the equivalence
closure of $\mathcal{S}$. Assume that $\mathcal{R}_{D}$ is finite. Then
$X_{1}=\mathcal{R}_{D}(\sigma_{1}(\mathcal{S}_{\leq
D-1})\cup\sigma_{2}(\mathcal{S}_{\leq D-1}))$ is $\mathcal{R}$-invariant.

###### Proof.

The set $X_{1}$ is $\mathcal{R}_{D}$-invariant. It is enough to prove that
$\sigma_{j}(\mathcal{R}_{\leq D-1})\subseteq X_{1}$ for $j=1,2$. Set
$\mathcal{S}^{i}=(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})\mathcal{S}^{i-1}\times_{X}\mathcal{S}^{i-1}$.
By [Kol13, 9.3], the equivalence closure of $\mathcal{S}$ is
$\mathcal{R}=\cup\mathcal{S}^{i}$. We will prove by induction on $i$ that
$\sigma_{j}((\mathcal{S}^{i})_{\leq D-1})\subseteq X_{1}$ for $j=1,2$.

We have

$\begin{split}\mathcal{S}^{i}&=(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})\mathcal{S}^{i-1}\times_{X}\mathcal{S}^{i-1}\\\
&=(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})(\mathcal{S}^{i-1}_{D}\times_{X}\mathcal{S}^{i-1}_{D})\cup\\\
&(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})(\mathcal{S}^{i-1}_{D}\times_{X}\mathcal{S}^{i-1}_{\leq
D-1}\cup\mathcal{S}^{i-1}_{\leq
D-1}\times_{X}\mathcal{S}^{i-1}_{D}\cup\mathcal{S}^{i-1}_{\leq
D-1}\times_{X}\mathcal{S}^{i-1}_{\leq D-1})\end{split}$

By Lemma 4.5(2) we have
$(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})(\mathcal{S}^{i-1}_{D}\times_{X}\mathcal{S}^{i-1}_{D})\subseteq\mathcal{R}_{D}$
and by Lemma 4.5(1)

$(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})(\mathcal{S}^{i-1}_{D}\times_{X}\mathcal{S}^{i-1}_{\leq
D-1}\cup\mathcal{S}^{i-1}_{\leq
D-1}\times_{X}\mathcal{S}^{i-1}_{D}\cup\mathcal{S}^{i-1}_{\leq
D-1}\times_{X}\mathcal{S}^{i-1}_{\leq D-1})\subseteq\mathcal{R}_{\leq D-1}.$

Therefore

$\mathcal{S}^{i}_{\leq
D-1}=(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})(\mathcal{S}^{i-1}_{D}\times_{X}\mathcal{S}^{i-1}_{\leq
D-1}\cup\mathcal{S}^{i-1}_{\leq
D-1}\times_{X}\mathcal{S}^{i-1}_{D}\cup\mathcal{S}^{i-1}_{\leq
D-1}\times_{X}\mathcal{S}^{i-1}_{\leq D-1})$

By Lemma 4.5(3) we have
$\sigma_{2}(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})(\mathcal{S}^{i-1}_{D}\times_{X}\mathcal{S}^{i-1}_{\leq
D-1})\subseteq\sigma_{2}S^{i-1}_{\leq D-1}$ and by induction
$\sigma_{2}S^{i-1}_{\leq D-1}\subseteq X_{1}$, proving

(14)
$\sigma_{2}(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})(\mathcal{S}^{i-1}_{D}\times_{X}\mathcal{S}^{i-1}_{\leq
D-1})\subseteq X_{1}.$

As for
$\sigma_{1}(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})(\mathcal{S}^{i-1}_{D}\times_{X}\mathcal{S}^{i-1}_{\leq
D-1})$, we have

$\begin{split}\sigma_{1}(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})(\mathcal{S}^{i-1}_{D}\times_{X}\mathcal{S}^{i-1}_{\leq
D-1})\subseteq\sigma_{1}\left(\sigma_{2}|_{\mathcal{R}_{D}}\right)^{-1}\sigma_{1}\mathcal{S}^{i-1}_{\leq
D-1}\\\
\subseteq\sigma_{1}\left(\sigma_{2}|_{\mathcal{R}_{D}}\right)^{-1}X_{1}\subseteq
X_{1}\end{split}$

where the first inclusion is because

$\begin{split}(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})(\mathcal{S}^{i-1}_{D}\times_{X}\mathcal{S}^{i-1}_{\leq
D-1})\\\ =\\{(x,y)|\;\exists z\in
X,\;(x,z)\in\mathcal{S}^{i-1}_{D}\;(z,y)\in\mathcal{S}^{i-1}_{\leq D-1}\\}\\\
\subseteq\\{(x,y)|\;\exists z\in\sigma_{1}(S^{i-1}_{\leq
D-1}),\;(x,z)\in\mathcal{R}_{D}\\}\end{split}$

and the image via $\sigma_{1}$ of the last set coincides with
$\sigma_{1}\left(\sigma_{2}|_{\mathcal{R}_{D}}\right)^{-1}\sigma_{1}\mathcal{S}^{i-1}_{\leq
D-1}$. The second inclusion follows by induction and the third because $X_{1}$
is $\mathcal{R}_{D}$-invariant.

A very similar proof implies that
$\sigma_{j}(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})\mathcal{S}^{i-1}_{\leq
D-1}\times_{X}\mathcal{S}^{i-1}_{D}\subseteq X_{1}$ for $j=1,2$.

Again by Lemma 4.5(3) we have
$\sigma_{j}(\sigma_{1}\circ\pi_{1},\sigma_{2}\circ\pi_{2})(\mathcal{S}^{i-1}_{\leq
D-1}\times_{X}\mathcal{S}^{i-1}_{\leq D-1})\subseteq\sigma_{2}S^{i-1}_{\leq
D-1}$ for $j=1,2$ and $\sigma_{2}S^{i-1}_{\leq D-1}\subseteq X_{1}$ by
induction. ∎

######  Definition 4.9.

Let $\mathcal{S}\to X\times X$ be a finite relation and $g\colon\tilde{X}\to
X$ a finite morphism. The pullback of $\mathcal{S}$ by $g$ is
$g^{*}\mathcal{S}=\mathcal{S}\times_{X\times X}\tilde{X}\times\tilde{X}$.

######  Lemma 4.10.

Let $\mathcal{S}$ be a finite relation on a variety $X$ and let
$D=\dim\mathcal{S}\setminus\Delta_{X}$. Let $g\colon\tilde{X}\to X$ be a
finite surjective morphism. If $\langle\mathcal{S}_{D}\rangle_{D}$ is
infinite, then $\langle g^{*}\mathcal{S}_{D}\rangle_{D}$ is infinite.

###### Proof.

For every $D$-dimensional component $S$ of $\langle\mathcal{S}_{D}\rangle$ the
pull back $\widetilde{S}$ in $\widetilde{X}\times\widetilde{X}$ has dimension
$D$.

∎

######  Definition 4.11.

A profinite equivalence relation $\mathcal{R}$ on an equidimensional variety
$X$ is equidimensional if every irreducible component of $\mathcal{R}$
projects onto a connected component of $X$.

The definition coincides with what is called wide in [ByB04, Definition 2.1].

######  Proposition 4.12.

Let $\mathcal{S}$ be a finite relation on a normal variety $X$, let
$\mathcal{R}$ be the equivalence closure of $\mathcal{S}$. If $\mathcal{R}$ is
not finite then there are

  1. (1)

a subrelation $\mathcal{R}^{\prime}\subseteq\mathcal{R}$

  2. (2)

$Z_{1},\ldots,Z_{k}$ subvarieties of $X$

such that $\cup Z_{i}$ is $\mathcal{R}^{\prime}$-invariant,
$\mathcal{R}^{\prime}|_{\cup Z_{i}}$ is an infinite equidimensional relation
and the set of infinite equivalence classes is dense in $\cup Z_{i}$.

###### Proof.

We prove the statement by induction on $D=\dim X$. If $\mathcal{R}_{D}$ is not
finite, we let $Z_{i}$ be the irreducible components of $X$ of dimension $D$
which are dominated by infinitely many components of $\mathcal{R}_{D}$ and we
set $\mathcal{R}^{\prime}=\mathcal{R}_{D}$.
We assume now that $\mathcal{R}_{D}$ is finite. We set
$X_{1}=\mathcal{R}_{D}(\sigma_{1}(\mathcal{S}_{\leq
D-1})\cup\sigma_{2}(\mathcal{S}_{\leq D-1}))$. By Lemma 4.8 the subvariety
$X_{1}$ is $\mathcal{R}$-invariant.

By Lemma 4.10 the pullback of the restriction of $\mathcal{R}$ to $X_{1}$ via
the normalisation of $X_{1}$ is not finite. We conclude by induction as the
dimension of the normalisation of $X_{1}$ is at most $D-1$. ∎

##  5\. Gluing bases of fibrations

Throughout this section, $\mathcal{L}$ will be a line bundle with the property
that $\mathcal{L}|_{T}$ is semiample for every irreducible component
$T\subseteq\mathcal{T}$. For every $T$ we denote by $\phi_{T}\colon T\to V$
the fibration induced by a multiple of $\mathcal{L}$.

######  Definition 5.1.

The equivalence relation $R_{\mathcal{L}}$ on the set
$\bigsqcup_{T\in\mathcal{T}}V$ is the closure of the relation

$x_{1}\sim x_{2}\Leftrightarrow\exists
T,T^{\prime}\subseteq\mathcal{T},\;\exists y\in T\cap
T^{\prime}\;\;\phi_{T}(y)=x_{1},\phi_{T^{\prime}}(y)=x_{2}.$

######  Remark 5.2.

Assume that $\mathcal{T}$ is a simple normal crossing divisor. Let
$\nu\colon\sqcup T\to\mathcal{T}$ be the normalisation. Let $\Xi^{n}$ be the
normalisation of the non-normal locus of $\mathcal{T}$. Then there is an
involution $\zeta\colon\Xi^{n}\to\Xi^{n}$ and we have
$(\zeta_{1},\zeta_{2})\colon\Xi^{n}\to\sqcup T\times\sqcup T$. Let
$\upsilon\colon\Xi^{n}\to\sqcup W$ be the fibration induced by
$\mathcal{L}|_{\Xi}$. The morphism $(\zeta_{1},\zeta_{2})$ induces a morphism
$(\xi_{1},\xi_{2})\colon\sqcup W\to\sqcup V$. Then the equivalence relation
$(\xi_{1},\xi_{2})\colon\sqcup W\to\sqcup V$ coincides with $R_{\mathcal{L}}$.

######  Notation 5.3.

Let $\nu\colon\sqcup T\to\mathcal{T}$ be the normalisation. For a subset
$S\subseteq\sqcup V$ we will denote by $\phi^{-1}S$ the set
$\nu\left(\sqcup\phi_{T}^{-1}(S\cap V)\right)$.

######  Remark 5.4.

If the line bundle $\mathcal{L}$ restricted to $\mathcal{T}$ is base point
free, then the relation $\sim$ is finite and $\mathcal{L}$ induces a morphism
$\phi\colon\mathcal{T}\to(\bigsqcup_{T\in\mathcal{T}}V)/R_{\mathcal{L}}$

######  Definition 5.5.

Let $\mathcal{T}\subseteq Y$ be a divisor and let $\mathcal{L}$ be a line
bundle such that $\mathcal{L}|_{T}$ is base point free for every
$T\subseteq\mathcal{T}$ for every irreducible component. Let $\phi_{T}\colon
T\to V$ be the morphism induced by $\mathcal{L}$. For an equivalence class
$[x]$ of $R_{\mathcal{L}}$ we set the pseudo-fibre as

$\mathcal{T}_{[x]}=\cup_{x^{\prime}\in[x]}\phi_{T}^{-1}(x^{\prime})=\phi^{-1}[x].$

######  Remark 5.6.

The relation $R_{\mathcal{L}}$ is finite if and only if $\mathcal{T}_{[x]}$ is
an algebraic variety for every $[x]$. Indeed $R_{\mathcal{L}}$ is finite if
and only if $[x]$ is a finite set for every $x$.

######  Proposition 5.7.

Let $Y$ be a normal variety and let $\mathcal{T}\subseteq Y$ be a divisor. Let
$\mathcal{L}$ be a line bundle on $Y$ which is semiample on the irreducible
components of $\mathcal{T}$. Let $\tau\colon\overline{Y}\to Y$ be a finite map
and $\overline{\mathcal{T}}=\tau^{-1}\mathcal{T}$. Then there is a commutative
diagram

$\textstyle{\bigsqcup_{\overline{T}\subseteq\overline{\mathcal{T}}}\overline{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\left(\phi_{\overline{T}}\right)}$$\textstyle{\bigsqcup_{T\subseteq\mathcal{T}}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left(\phi_{T}\right)}$$\textstyle{\bigsqcup\overline{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{\bigsqcup
V.}$

with $\sigma$ a finite map. Moreover for every $x\in\sqcup V$ we have
$\sigma^{-1}[x]=\sqcup_{\sigma(\bar{x})=x}[\bar{x}]$.

In particular
$\mathcal{R}_{\tau^{*}\mathcal{L}}=\sigma^{*}\mathcal{R}_{\mathcal{L}}$

###### Proof.

Assume that $\tau(\overline{T})=T$. There is a commutative diagram

$\textstyle{\overline{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\phi_{\overline{T}}}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{T}}$$\textstyle{\overline{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{V}}$$\textstyle{V}$

where $\sigma_{V}\circ\phi_{\overline{T}}=\phi_{T}\circ\tau$. And the
$\sigma_{V}$ define a finite map
$\sigma\colon\bigsqcup\overline{V}\to\bigsqcup V$.

Let $\bar{x},\bar{x}^{\prime}\in\bigsqcup\overline{V}$ such that
$\bar{x}\sim\bar{x}^{\prime}$. Then there is
$\bar{y}\in\overline{T}\cap\overline{T}^{\prime}$ such that
$\phi_{\overline{T}}(\bar{y})=\bar{x}$ and
$\phi_{\overline{T}^{\prime}}(\bar{y})=\bar{x}^{\prime}$. By the commutativity
of the diagram $\phi_{T}(\tau\bar{y})=\sigma(\bar{x})$ and
$\phi_{T^{\prime}}(\tau\bar{y})=\sigma(\bar{x}^{\prime})$. Therefore
$\sigma[\bar{x}]\subseteq[\sigma(\bar{x})]$.

On the other hand let $\bar{x}\in\sigma^{-1}[x]$. We want to prove that
$\bar{x}$ is equivalent to a point in $\sigma^{-1}x$. The point
$\sigma\bar{x}$ is equivalent to $x$. Therefore there are $\sigma\bar{x}\sim
x_{1}\sim\ldots\sim x_{k}=x$. We prove our statement by induction on $k$. If
$k=1$, the statement is obvious. We assume from now on that $k>1$. Then
$\sigma\bar{x}\sim x_{1}$ if and only if
$\phi_{T}^{-1}(\sigma\bar{x})\cap\phi_{T^{\prime}}^{-1}(x_{1})\neq\emptyset$.
Therefore
$\phi_{\overline{T}}^{-1}(\bar{x})\cap\tau^{-1}\phi_{T^{\prime}}^{-1}(x_{1})\neq\emptyset$.
Let
$\bar{y}_{1}\in\phi_{\overline{T}}^{-1}(\bar{x})\cap\tau^{-1}\phi_{T^{\prime}}^{-1}(x_{1})$
and $\bar{x}_{1}=\phi_{\overline{T}^{\prime}}\bar{y}_{1}$. Then
$\sigma(\bar{x}_{1})=x_{1}$ and we can conclude by the inductive hypothesis.

∎

######  Proposition 5.8.

Let $Y$ be a normal variety and let $\mathcal{T}\subseteq Y$ be a divisor. Let
$\mathcal{L}$ be a line bundle on $Y$ which is semiample on the irreducible
components of $\mathcal{T}$. Let $\varepsilon\colon\overline{Y}\to Y$ be a
birational map which is an isomorphism on the generic points of $T\cap
T^{\prime}$ for every $T,T^{\prime}$ irreducible component of $\mathcal{T}$.
Let $\overline{\mathcal{T}}$ be the strict transform of $\mathcal{T}$. Then
there is a commutative diagram

$\textstyle{\bigsqcup_{\overline{T}\subseteq\overline{\mathcal{T}}}\overline{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\scriptstyle{\left(\phi_{\overline{T}}\right)}$$\textstyle{\bigsqcup_{T\subseteq\mathcal{T}}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left(\phi_{T}\right)}$$\textstyle{\bigsqcup
V.}$

with $\phi_{\overline{T}}=\phi_{T}\circ\varepsilon$. Then
$\mathcal{R}_{\varepsilon^{*}\mathcal{L}}=\mathcal{R}_{\mathcal{L}}$.

###### Proof.

It is enuogh to prove that the equivalence classes coincide. Let
$x_{1},x_{2}\in\bigsqcup V$ be such that there is
$\bar{y}\in\overline{T}_{1}\cap\overline{T}_{2}$ with
$\phi_{\overline{T}_{i}}(\bar{y})=x_{i}$. The divisor $\overline{T}_{i}$ is
the strict transform of $T_{i}\subseteq Y$. Then $y=\varepsilon(\bar{y})$ is
such that $\phi_{\overline{T}_{i}}(\bar{y})=x_{i}$. This proves that
$\mathcal{R}_{\varepsilon^{*}\mathcal{L}}\subseteq\mathcal{R}_{\mathcal{L}}$.

Let $x_{1},x_{2}\in\bigsqcup V$ be such that there is $y\in T_{1}\cap T_{2}$
with $\phi_{T_{i}}(y)=x_{i}$. Let $\overline{T}_{i}$ be the strict transform
of $T_{i}\subseteq Y$. As $\varepsilon$ is an isomorphism on the generic point
of $T_{1}\cap T_{2}$, the intersection $T_{1}\cap T_{2}\cap\varepsilon^{-1}y$
is non empty. If $\bar{y}$ is in the intersection, then
$\phi_{\overline{T}_{i}}(\bar{y})=\phi_{T_{i}}(\varepsilon y)=x_{i}$. This
proves that
$\mathcal{R}_{\varepsilon^{*}\mathcal{L}}\supseteq\mathcal{R}_{\mathcal{L}}$,
concluding the proof. ∎

######  Corollary 5.9.

Let $Y$ be a normal variety and let $\mathcal{T}\subseteq Y$ be a divisor. Let
$\mathcal{L}$ be a line bundle on $Y$ which is semiample on the irreducible
components of $\mathcal{T}$. Let $\theta\colon\overline{Y}\to Y$ be a
generically finite map such that $\theta Exc(\theta)$ does not contain the
generic points of $T\cap T^{\prime}$ for every $T,T^{\prime}$ irreducible
component of $\mathcal{T}$. Let $\overline{\mathcal{T}}$ be the strict
transform of $\mathcal{T}$.

Then there is a commutative diagram

$\textstyle{\bigsqcup_{\overline{T}\subseteq\overline{\mathcal{T}}}\overline{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\scriptstyle{\left(\phi_{\overline{T}}\right)}$$\textstyle{\bigsqcup_{T\subseteq\mathcal{T}}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left(\phi_{T}\right)}$$\textstyle{\bigsqcup\overline{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{\bigsqcup
V.}$

with $\sigma$ a finite map such that
$\mathcal{R}_{\theta^{*}\mathcal{L}}=\sigma^{*}\mathcal{R}_{\mathcal{L}}$.

###### Proof.

Let $\theta=\varepsilon\circ\tau$ be the Stein factorisation. Let
$\widetilde{\mathcal{T}}=\tau^{-1}\mathcal{T}$. By Proposition 5.7 there is a
diagram

$\textstyle{\bigsqcup_{\widetilde{T}\subseteq\widetilde{\mathcal{T}}}\widetilde{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\left(\phi_{\widetilde{T}}\right)}$$\textstyle{\bigsqcup_{T\subseteq\mathcal{T}}T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left(\phi_{T}\right)}$$\textstyle{\bigsqcup\widetilde{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{\bigsqcup
V.}$

with $\sigma$ a finite map such that
$\mathcal{R}_{\tau^{*}\mathcal{L}}=\sigma^{*}\mathcal{R}_{\mathcal{L}}$. By
Proposition 5.8 there is a diagram

$\textstyle{\bigsqcup_{\overline{T}\subseteq\overline{\mathcal{T}}}\overline{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\scriptstyle{\left(\phi_{\overline{T}}\right)}$$\textstyle{\bigsqcup_{\widetilde{T}\subseteq\widetilde{\mathcal{T}}}\widetilde{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left(\phi_{\widetilde{T}}\right)}$$\textstyle{\bigsqcup\widetilde{V}.}$

∎

######  Lemma 5.10.

Let $\mathcal{T}\subseteq Y$ be a reduced and connected divisor and let
$\mathcal{L}$ be a line bundle such that $\mathcal{L}|_{T}$ is base point free
for every $T\subseteq\mathcal{T}$ for every irreducible component. Let
$\phi_{T}\colon T\to V$ be the morphism induced by $\mathcal{L}$. Then
$\mathcal{T}_{[x]}$ is connected.

###### Proof.

Let $y_{1},y_{2}\in\mathcal{T}_{[x]}$. Then there are $x_{2},\ldots,x_{r}$
such that $\phi_{1}(y_{1})=x_{1}\sim x_{2}\ldots x_{r}\sim
x_{r+1}=\phi_{2}(y_{2})$. Let $V_{i}$ be such that $x_{i}\in V_{i}$ and
$T_{i}$ with $\phi_{i}\colon T_{i}\to V_{i}$. Then there are $y_{i,i+1}\in
T_{i,i+1}$ such that $\phi_{i}(y_{i,i+1})=x_{i}$ and
$\phi_{i+1}(y_{i,i+1})=x_{i+1}$. Thus

$y_{1},y_{2}\in\bigcup_{i=1}^{r+1}\phi_{i}^{-1}(x_{i})\subseteq\mathcal{T}_{[x]}$

and $\cup_{i=1}^{r+1}\phi_{i}^{-1}(x_{i})$ is connected as for every $i$ there
is $y_{i,i+1}\in\phi_{i}^{-1}(x_{i})\cap\phi_{i+1}^{-1}(x_{i+1})$.

∎

##  6\. Graph theory

We recall here a few basic notions of graph theory. We follow the presentation
of [Sta83].

A graph $\Gamma$ consists of two sets $E$ and $V$ (edges and vertices), and
two functions $E\to E$, $e\mapsto\bar{e}$ and $E\to V$, $e\mapsto i(e)$: for
each $e\in E$, there is an element $\bar{e}\in E$, and an element $i(e)\in V$.
The function $\bar{\cdot}$ is such that $\bar{\bar{e}}=e$ and $\bar{e}\neq e$.
The vertex $i(e)$ is called the initial vertex of $e$, the vertex
$t(e)=i(\bar{e})$ is called the terminal vertex of $e$.

We call a graph finite if both $V$ and $E$ are finite sets.

A map of graphs $f\colon\Gamma_{1}\to\Gamma_{2}$ consists of a pair of
functions, edges to edges, vertices to vertices, preserving the structure. A
map of graphs is surjective if it is surjective on vertices and on edges.

We recall that pull-backs exist in the category of graphs: given
$f_{1}\colon\Gamma_{1}\to\Delta$ and $f_{2}\colon\Gamma_{2}\to\Delta$ two maps
of graphs, there is a graph $\Gamma_{1}\times_{\Delta}\Gamma_{2}$ together
with surjective maps
$g_{i}\colon\Gamma_{1}\times_{\Delta}\Gamma_{2}\to\Gamma_{i}$ such that
$f_{1}\circ g_{1}=f_{2}\circ g_{2}$.

A path in a graph $\Gamma$ is an $n$-tuple of edges $(e_{1},\ldots,e_{n})\in
E^{n}$ such that $t(e_{i})=i(e_{i+1})$. The vertices $i(e_{1})$ and $t(e_{n})$
are the initial vertex and terminal vertex of the path.

A circuit is a path whose initial and terminal vertex coincide. Equivalently,
we define $C_{n}$ the standard circuit of length $n$ as the regular polygone
with $n$ edges and a circuit in $\Gamma$ is a map of graphs $C_{n}\to\Gamma$.
A circuit is proper if the map $C_{n}\to\Gamma$ is injective on the vertices.
The standard arc of length $n$ $A_{n}$ can be described as the interval
$[0,n]$ subdivided at the integral points. The vertices are
$V=\\{0,\ldots,n\\}$, the edges are the oriented segments $[i,i+1]$ and
$[i+1,i]$ between $i$ and $i+1$. The involution $\bar{\cdot}$ exchanges
$[i,i+1]$ and $[i+1,i]$.

The homotopy equivalence on paths is the relation generated by

$(e_{1},\ldots,e_{n})\sim(e_{1},\ldots,e_{i},e,\bar{e},e_{i+1},\ldots,e_{n})$

and the set of paths starting and ending at a same vertex $v$ modulo homotopy
is denoted by $\pi_{1}(\Gamma,v)$ and called the fundamental group of
$\Gamma$. It has a natural group structure with respect to the concatenation
of paths.

A path is reduced if it contains no sub-paths of the form $e\bar{e}$ and one
can prove that every path is homotopic to a reduced one.

Let $v$ be a vertex of the graph $\Gamma$. The star of $v$ in $\Gamma$ is the
set

$St(v,\Gamma)=\\{e\in E|\;i(e)=v\\}.$

A map of graphs $f\colon\Gamma_{1}\to\Gamma_{2}$ is a covering if for each
vertex $v$ of $\Gamma_{1}$ the natural function

$f_{v}\colon St(v,\Gamma_{1})\to St(f(v),\Gamma_{2})$

is bijective. By [Sta83, 4.1(d)] if $f\colon\Gamma_{1}\to\Gamma_{2}$ is a
covering, then $f\colon\pi_{1}(\Gamma_{1},v)\to\pi_{1}(\Gamma_{2},f(v))$ is an
injective homomorphism and if the graphs are finite then
$f\pi_{1}(\Gamma_{1},v)\subseteq\pi_{1}(\Gamma_{2},f(v))$ has finite index
equal to the cardinality of $f^{-1}f(v)$.

This last remark combined with [Sta83, 3.3] and [Sta83, 4.4], gives the
following proposition

######  Proposition 6.1.

If $f\colon\Gamma_{1}\to\Gamma_{2}$ is a surjective maps of finite graphs,
then $f\pi_{1}(\Gamma_{1},v)\subseteq\pi_{1}(\Gamma_{2},f(v))$ has finite
index $i\leq|f^{-1}f(v)|$.

We conclude this section with an easy but useful lemma.

######  Lemma 6.2.

Let $\Gamma$ be a finite graph. Then there is a standard circuit $C_{N}$ and a
surjective morphism $C_{N}\to\Gamma$.

###### Proof.

We construct recursively a morphism $f\colon A_{N}\to\Gamma$. We notice that
if $f$ is surjective on the edges then it is surjective on the vertices and
that it is enough to show that for every $e\in E$ either there is $i$ such
that $e=f[i,i+1]$ or such that $\bar{e}=f[i,i+1]$. We set $\pi\colon
E\to\hat{E}$ the quotient by the action of $\mathbb{Z}/2\mathbb{Z}$ sending
$e$ to $\bar{e}$. Let $e\in E$. We set $f[0,1]=e$. Assume we have $f\colon
A_{k}\to\Gamma$. If $\hat{E}\setminus\pi\left(\\{f[i,i+1]\\}_{i}\right)$ is
not empty, then we pick $e\in
E\setminus\\{f[i,i+1],\overline{f[i,i+1]}\\}_{i}$, we pick a path
$(e_{1},\ldots,e_{n})$ from $t(f[k-1,k])$ to $i(e)$ and we set

$\left\\{\begin{array}[]{lcl}f[k+i-1,k+i]&=&e_{i}\;for\;i\leq n\\\
f[k+n+1,k+n+2]&=&e.\end{array}\right.$

If $\hat{E}\setminus\pi\left(\\{f[i,i+1]\\}_{i}\right)$ is empty, then we pick
a path $(e_{1},\ldots,e_{n})$ from $t(f[k-1,k])$ to $i(f[0,1])$ and we set
$f[k+i-1,k+i]=e_{i}$ for $i\leq n$.

∎

##  7\. Trivial line bundles on simple normal crossings varieties

In this section we discuss a triviality condition for line bundles on
reducible varieties and develop the tools for the proof of Theorem 10.1. We
are mostly concerned with the case of simple normal crossings varieties in the
sense of Definition 3. Lemmas 7.5, 7.8 and 7.10 can be seen as a refinement of
[BLR90, Example 9.2.8].

######  Definition 7.1.

Let $\mathcal{Z}=\cup Z$ be a reducible variety. We define the incidence graph
$\Gamma^{i}(\mathcal{Z})$ of $\mathcal{Z}$ by
$V^{i}=\\{Z|\;Z\;irreducible\;component\;of\;\mathcal{Z}\\}$ with an edge
between $Z$ and $Z^{\prime}$ for every connected component of $Z\cap
Z^{\prime}$.

######  Notation 7.2.

A circuit $\mathcal{C}$ in $\Gamma^{i}(\mathcal{Z})$ will be denoted by

$(\\{Z_{1},\ldots,Z_{k}\\},Z_{1,2}\ldots Z_{k,1})\text{ or
}(\\{Z_{i}\\},Z_{i,i+1})\text{ for short}$

where the $Z_{i}$ are irreducible components of $\mathcal{Z}$ and for every
the varietiy $Z_{i,i+1}$ is a connected component of $Z_{i}\cap Z_{i+1}$, and
$Z_{k,1}$ is a connected component of $Z_{1}\cap Z_{k}$.

We will refer to $\cup Z_{i}$ as the support of the circuit $\mathcal{C}$.

######  Remark 7.3.

If $\mathcal{Z}$ is a divisor with simple normal crossing support, then
$\Gamma^{i}(\mathcal{Z})$ coincides with the 1-skeleton of the dual complex of
$\mathcal{Z}$ (see [dFKX17, Section 2]).

Throughout this subsection $Y$ will be a normal connected variety and
$\mathcal{Z}\subseteq Y$ a reducible reduced and connected subvariety of $Y$.
We will consider $\mathcal{L}$ a line bundle on $Y$ such that
$\mathcal{L}|_{Z}\sim\mathcal{O}_{Z}$ for every irreducible component $Z$ of
$\mathcal{Z}$.

######  Definition 7.4.

Let $\mathcal{C}=(\\{Z_{i}\\},Z_{i,i+1})$ be a circuit in
$\Gamma^{i}(\mathcal{Z})$. A section of the restriction of $\mathcal{L}$ to
$\mathcal{C}$ (or of $\mathcal{L}|_{\mathcal{C}}$) is the data of $s_{i}\in
H^{0}(Z_{i},\mathcal{L})$ such that

$s_{i}|_{Z_{i,i+1}}=s_{i+1}|_{Z_{i,i+1}}.$

######  Lemma 7.5.

Let $\mathcal{Z}$ be a connected reduced simple normal crossings variety of
pure dimension $k$. Let $\mathcal{L}$ be a line bundle on $\mathcal{Z}$ such
that $\mathcal{L}|_{Z}\sim\mathcal{O}_{Z}$ for every irreducible component $Z$
of $\mathcal{Z}$. Then
$\mathcal{L}|_{\mathcal{Z}}\sim\mathcal{O}_{\mathcal{Z}}$ if and only if for
every circuit $\mathcal{C}$ in $\Gamma^{i}(\mathcal{Z})$, the restriction of
$\mathcal{L}$ to $\mathcal{C}$ has a nowhere vanishing global section.

###### Proof.

If $\mathcal{L}$ is trivial, then it has a nowhere vanishing global section
$s\in H^{0}(\mathcal{Z},\mathcal{L})$. Then for any circuit
$\mathcal{C}=(\\{Z_{i}\\},Z_{1,2},\ldots,Z_{k,1})$ in
$\Gamma^{i}(\mathcal{Z})$ it is enough to set $s_{i}=s|_{Z_{i}}$.

Conversely, let $\mathcal{Z}=\bigcup Z_{i}$ be the decomposition of
$\mathcal{Z}$ into its irreducible components. By possibly relabeling, we may
assume that for any $i>1$ the subvariety $Z_{i}$ meets
$\mathcal{Z}_{i-1}:=\bigcup_{j<i}Z_{j}$. Let
$\mathcal{Z}_{i}=\mathcal{Z}_{i-1}\cup Z_{i}$. For each $i\neq j$ with
$Z_{i}\cap Z_{j}\neq\emptyset$, we fix $p_{i,j}\in Z_{i}\cap Z_{j}$. Fix
$s_{1}\in H^{0}(Z_{1},\mathcal{L})\setminus\\{0\\}$. We construct inductively
a nowhere-vanishing section $\sigma_{i}\in H^{0}(\mathcal{Z}_{i},\mathcal{L})$
such that $\sigma_{i}|_{Z_{1}}=s_{1}$.

For $i>1$ we assume there is a section $\sigma_{i-1}\in
H^{0}(\mathcal{Z}_{i-1},\mathcal{L})$. Choose the largest $r<i$ such that
$Z_{i}\cap Z_{r}\neq\emptyset$, and let $s_{i}\in
H^{0}(Z_{i},\mathcal{L})\setminus\\{0\\}$ be the unique section such that

(15) $s_{i}(p_{i,r})=\sigma_{i-1}|_{Z_{r}}(p_{i,r}).$

If $Z_{j}\cap Z_{r}=\emptyset$ for all $j<i$ with $j\neq r$ and $Z_{i}\cap
Z_{r}$ is connected, then (15) defines a nowhere-vanishing section
$\sigma_{i}\in H^{0}(\mathcal{Z}_{i},\mathcal{L})$.

Otherwise, there exists $Z_{s}$ with $s<i$ and a point $p_{i,s}\in Z_{i}\cap
Z_{s}$. Then there exists a circuit
$\mathcal{C}=(Z_{i_{1}},\ldots,Z_{i_{k}},Z_{1,2},\ldots,Z_{k,1})$ such that
$Z_{i_{1}}=Z_{s}$, $Z_{i_{k-1}}=Z_{r}$, $Z_{i_{k}}=Z_{i}$, $p_{i,r}\in
Z_{k-1,k}$ and $p_{i,s}\in Z_{k,1}$. By assumption there exists a non-trivial
global section of $\mathcal{L}|_{\mathcal{C}}$, which is the data of
$\theta_{i}\in H^{0}(Z_{i_{j}},\mathcal{L})$ for $j=1,\ldots,k$. By rescaling,
we may assume that $\theta_{1}=\sigma_{i-1}|_{Z_{i_{1}}}$. Then, by the
construction above, for every $1\leq j\leq k-1$ we have
$\theta_{j}=\sigma_{i-1}|_{Z_{j}}$ and $\theta_{k}=s_{i}$, and in particular

(16) $s_{i}(p_{i,s})=\sigma_{i-1}|_{Z_{s}}(p_{i,s}).$

Since this holds for any choice of $p_{i,s}\in Z_{i}\cap Z_{s}$, (15) and (16)
define a nowhere-vanishing section $\sigma_{i}\in
H^{0}(\mathcal{Z}_{i},\mathcal{L})$. ∎

######  Definition 7.6.

Let $\mathcal{Z}$ be a connected reduced simple normal crossings variety of
pure dimension $k$. Let $\mathcal{L}$ be a line bundle on $\mathcal{Z}$ such
that $\mathcal{L}|_{Z}\sim\mathcal{O}_{Z}$ for every irreducible component $Z$
of $\mathcal{Z}$. Let $\mathcal{C}=(\\{Z_{1},\ldots,Z_{k}\\},Z_{i,i+1})$ be a
circuit in $\Gamma^{i}(\mathcal{Z})$. We chose $s_{1}\in
H^{0}(Z_{1},\mathcal{L})\setminus\\{0\\}$ and for every $i>1$ we set $s_{i}\in
H^{0}(Z_{i},\mathcal{L})\setminus\\{0\\}$ as the unique section such that

$s_{i}|_{Z_{i-1,i}}=s_{i-1}|_{Z_{i-1,i}}.$

We define then

$\begin{array}[]{rcl}\Phi_{\mathcal{L},\mathcal{C}}\colon
H^{0}(Z_{1},\mathcal{L})&\to&H^{0}(Z_{1},\mathcal{L})\\\ s&\mapsto&s\cdot
s_{k+1}/s_{1}\end{array}$

######  Remark 7.7.

The map $\Phi_{\mathcal{L},\mathcal{C}}$ is the identity if and only if the
restriction of $\mathcal{L}$ to $\mathcal{C}$ admits a nowhere vanishing
global section.

It is easy to see that this does not depend on the choice of $s_{1}$.
Moreover, if $\mathcal{C},\mathcal{C}^{\prime}$ are circuits based in $Z_{1}$
and they are homotopically equivalent, then
$\Phi_{\mathcal{L},\mathcal{C}}=\Phi_{\mathcal{L},\mathcal{C}^{\prime}}$. If
$\mathcal{C}_{1},\mathcal{C}_{2}$ are circuits based in $Z_{1}$ and
$\mathcal{C}=\mathcal{C}_{1}\star\mathcal{C}_{2}$ is their concatenation, then
$\Phi_{\mathcal{L},\mathcal{C}}=\Phi_{\mathcal{L},\mathcal{C}_{2}}\circ\Phi_{\mathcal{L},\mathcal{C}_{1}}$.
All these remarks prove the following lemma.

######  Lemma 7.8.

Let $\mathcal{Z}$ be a connected reduced simple normal crossings variety of
pure dimension $k$, let $Z_{1}\subseteq\mathcal{Z}$ be an irreducible
component. Let $\mathcal{L}$ be a line bundle on $\mathcal{Z}$ such that
$\mathcal{L}|_{Z}\sim\mathcal{O}_{Z}$ for every irreducible component $Z$ of
$\mathcal{Z}$. There is a group homomorphism

$\begin{array}[]{rcl}\Phi_{\mathcal{L}}\colon\pi_{1}(\Gamma^{i}(\mathcal{Z}),Z_{1})&\to&GL(H^{0}(Z_{1},\mathcal{L}))\cong\mathbb{C}^{*}\\\
\mathcal{C}&\mapsto&\Phi_{\mathcal{L},\mathcal{C}}\end{array}$

which is trivial if and only if $\mathcal{L}\sim\mathcal{O}_{\mathcal{Z}}$.

######  Remark 7.9.

In the context of the previous definition, for all $m$ we have

$\Phi_{\mathcal{L}^{\otimes
m},\mathcal{C}}=\underbrace{\Phi_{\mathcal{L},\mathcal{C}}\circ\dots\circ\Phi_{\mathcal{L},\mathcal{C}}}_{m\text{
times}}.$

######  Lemma 7.10.

Let $\mathcal{Z}$ be a connected reduced simple normal crossings variety of
pure dimension $k$, let $Z_{1}\subseteq\mathcal{Z}$ be an irreducible
component. Let $\mathcal{L}$ be a line bundle on $\mathcal{Z}$ such that
$\mathcal{L}|_{Z}\sim\mathcal{O}_{Z}$ for every irreducible component $Z$ of
$\mathcal{Z}$. Then $\mathcal{L}$ is a torsion line bundle if and only if the
image of $\Phi_{\mathcal{L}}$ is a finite subgroup of $\mathbb{C}^{*}$.

###### Proof.

If $\mathcal{L}$ is torsion, then there is a positive integer $m$ such that
$\mathcal{L}^{m}\sim\mathcal{O}_{\mathcal{Z}}$. Therefore for every circuit
$\mathcal{C}$ based in $Z_{1}$ the map $\Phi_{\mathcal{L}^{\otimes
m},\mathcal{C}}$ is the identity. The conclusion follows from Remark 7.9.

Conversely, let $m$ be a positive integer such that the image of
$\Phi_{\mathcal{L}}$ is contained in the $m$-th roots of 1. Then for every
circuit $\mathcal{C}$ in $\Gamma^{i}(\mathcal{Z})$ based in $Z_{1}$ the map
$\Phi_{\mathcal{L},\mathcal{C}}\circ\dots\circ\Phi_{\mathcal{L},\mathcal{C}}$
(composition $m$ times) is the identity. By Remark 7.9 this map is
$\Phi_{\mathcal{L}^{\otimes m},\mathcal{C}}$, and then the restriction of
$\mathcal{L}^{\otimes m}$ to every circuit $\mathcal{C}$ in
$\Gamma^{i}(\mathcal{Z})$ based in $Z_{1}$ admits a global section. The
statement follows from Lemma 7.5 and from the fact that every circuit is
homotopic to a circuit based in $Z_{1}$. ∎

###  7.1. Trivial line bundles on divisors

######  Definition 7.11.

Let $\tau\colon\overline{Y}\to Y$ be a finite map of normal projective
varieties and let $\mathcal{Z}$ be a connected subvariety of $Y$. Let
$\overline{\mathcal{Z}}$ be the preimage of $\mathcal{Z}$ under $\tau$. Set
$\overline{\mathcal{Z}}=\tau^{-1}\mathcal{Z}$. We define a graph
$\Gamma^{i}(\overline{\mathcal{Z}},\tau)\subseteq\Gamma^{i}(\overline{\mathcal{Z}})$
having as vertices the vertices of $\Gamma^{i}(\overline{\mathcal{Z}})$ and
having an edge between $\overline{Z}$ and $\overline{Z}^{\prime}$ if and only
if there is an edge between $\overline{Z}$ and $\overline{Z}^{\prime}$ in
$\Gamma^{i}(\overline{\mathcal{Z}})$ and
$\tau(\overline{Z})\neq\tau(\overline{Z}^{\prime})$.

######  Construction 7.12.

Let $\tau\colon\overline{Y}\to Y$ be a finite map of normal projective
varieties and let $\mathcal{Z}$ be a connected subvariety of $Y$. Let
$\overline{\mathcal{Z}}$ be the preimage of $\mathcal{Z}$ under $\tau$. Then
there is a natural map of graphs

$\tau\colon\Gamma^{i}(\overline{\mathcal{Z}},\tau)\to\Gamma^{i}(\mathcal{Z})$

defined on vertices by $\tau(v_{\overline{Z}})=v_{\tau\overline{Z}}$. To an
edge $e$ of $\Gamma^{i}(\overline{\mathcal{Z}})$ corresponding to a connected
component $\overline{Z}_{0}$ of $\overline{Z}\cap\overline{Z}^{\prime}$ the
map $\tau$ associates the unique connected component of
$\tau\overline{Z}\cap\tau\overline{Z}^{\prime}$ containing
$\tau\overline{Z}_{0}$.

######  Lemma 7.13.

Let $\tau\colon\overline{Y}\to Y$ be a finite map of normal projective
varieties and let $\mathcal{Z}$ be a connected subvariety of $Y$. Let
$\overline{\mathcal{Z}}$ be the preimage of $\mathcal{Z}$ under $\tau$. The
map in Construction 7.12 is surjective and for every $v$ vertex of
$\Gamma^{i}(\mathcal{Z})$ we have $|\tau^{-1}v|\leq\deg\tau$.

###### Proof.

The map is clearly surjective on vertices. Let $Z_{0}$ be a connected
component of $Z\cap Z^{\prime}$. Let $\overline{Z}$ be an irreducible
component of $\overline{\mathcal{Z}}$ such that $\tau\overline{Z}=Z$. The set
$\tau|_{\overline{Z}}^{-1}Z_{0}$ is not empty and it is contained in
$\overline{Z}\cap\tau^{-1}Z^{\prime}$. Then there is an irreducible component
$\overline{Z}^{\prime}$ of $\tau^{-1}Z^{\prime}$ meeting $\overline{Z}$. Let
$\overline{Z}_{0}$ be a connected component of
$\overline{Z}\cap\overline{Z}^{\prime}$. Then $\tau$ sends the edge
corresponding to $\overline{Z}_{0}$ to the edge corresponding to $Z_{0}$. ∎

Combining Lemma 7.13 and Proposition 6.1 we get

######  Corollary 7.14.

Let $\tau\colon\overline{Y}\to Y$ be a finite map of normal projective
varieties and let $\mathcal{Z}$ be a connected subvariety of $Y$. Let
$\overline{\mathcal{Z}}$ be the preimage of $\mathcal{Z}$ under $\tau$ and fix
an irreducible component $\overline{Z}_{1}$ of $\overline{\mathcal{Z}}$. Then
$\tau\pi_{1}(\Gamma^{i}(\overline{\mathcal{Z}},\tau),\overline{Z}_{1})$ has
finite index $k$ in $\pi_{1}(\Gamma^{i}(\mathcal{Z}),\tau\overline{Z}_{1})$.
Moreover $k\leq\deg\tau$.

######  Lemma 7.15.

Let $\tau\colon\overline{Y}\to Y$ be a finite map of degree $d$ of normal
projective varieties and let $\mathcal{Z}\subseteq Y$ be a simple normal
crossings divisor. Let $\overline{\mathcal{Z}}$ be the preimage of
$\mathcal{Z}$ under $\tau$. Let $\mathcal{L}$ be a line bundle on $Y$ such
that $\mathcal{L}|_{Z}\sim\mathcal{O}_{Z}$ for every component $Z$ of
$\mathcal{Z}$. If $\mathcal{L}|_{\mathcal{Z}}\sim\mathcal{O}_{\mathcal{Z}}$
then
$\tau^{*}\mathcal{L}|_{\overline{\mathcal{Z}}}\sim\mathcal{O}_{\overline{\mathcal{Z}}}$.
If
$\tau^{*}\mathcal{L}|_{\overline{\mathcal{Z}}}\sim\mathcal{O}_{\overline{\mathcal{Z}}}$,
then $\mathcal{L}^{d!}|_{\mathcal{Z}}\sim\mathcal{O}_{\mathcal{Z}}$

###### Proof.

If $\mathcal{L}|_{\mathcal{Z}}\sim\mathcal{O}_{\mathcal{Z}}$, then the
pullback of the nowhere vanishing global section of
$\mathcal{L}|_{\mathcal{Z}}$ by $\tau$ gives a nowhere vanishing global
section of $\tau^{*}\mathcal{L}|_{\overline{\mathcal{Z}}}$, settling the first
part of the statement.

Conversely, we assume that
$\tau^{*}\mathcal{L}|_{\overline{\mathcal{Z}}}\sim\mathcal{O}_{\overline{\mathcal{Z}}}$.
Fix an irreducible component $\overline{Z}_{1}$ of $\overline{\mathcal{Z}}$
and set $Z_{1}=\tau\overline{Z}_{1}$. We want to prove that for every circuit
$\mathcal{C}$ in $\Gamma^{i}(\mathcal{Z})$ based on $Z_{1}$, the morphism
$\Phi_{\mathcal{L},\mathcal{C}}^{d!}$ is the identity. As
$\pi_{1}(\Gamma^{i}(\mathcal{Z}),Z_{1})$ is finitely generated, the result
will follow from Lemma 7.10.

By Corollary 7.14, the group
$\tau\pi_{1}(\Gamma^{i}(\overline{\mathcal{Z}},\tau),\overline{Z}_{1})$ is a
subgroup of $\pi_{1}(\Gamma^{i}(\mathcal{Z}),Z_{1})$ of index $k\leq d$.
Therefore there exist a circuit
$\overline{\mathcal{C}}=(\\{\overline{Z}_{i}\\},\overline{Z}_{i,i+1})$ in
$\Gamma^{i}(\overline{\mathcal{Z}})$ such that, if we denote by
$\mathcal{C}^{k}$ the concatenation of $\mathcal{C}$ with itself $k$ times,
the circuits $\mathcal{C}^{k}$ and $\tau\overline{\mathcal{C}}$ are
homotopically equivalent. By Remark 7.9 it is enough to prove that
$\Phi_{\mathcal{L},\tau\overline{\mathcal{C}}}=\Phi_{\mathcal{L},\mathcal{C}}^{k}$
is the identity. We notice that if $Z_{i}=\tau(\overline{Z}_{i})$ for $i=1,2$
and $Z_{1,2}=\tau(\overline{Z}_{1,2})$, we have a commutative diagram

$\textstyle{H^{0}(Z_{1},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{H^{0}(Z_{1,2},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{H^{0}(Z_{2},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{H^{0}(\overline{Z}_{1},\tau^{*}\mathcal{L})}$$\textstyle{H^{0}(\overline{Z}_{1,2},\tau^{*}\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\overline{Z}_{2},\tau^{*}\mathcal{L})}$

where the horizontal arrows are the restriction isomorphisms and the vertical
arrows are isomorphisms induced by the pullback by $\tau$.

Les $s$ be a global section for
$\tau^{*}\mathcal{L}|_{\overline{\mathcal{Z}}}$. Then, with the
identifications of the previous diagram, the linear map
$\Phi_{\mathcal{L},\tau\overline{\mathcal{C}}}$ is the multiplication by
$s/s=1$, therefore it is the identity.

Then $\Phi_{\mathcal{L},\mathcal{C}}^{k}$ is the identity, and so is
$\Phi_{\mathcal{L},\mathcal{C}}^{d!}$, proving the statement. ∎

###  7.2. Trivial line bundles on semistable curves

In this subsection we present an analog of Construction 7.12 and Lemma 7.15
for curves.

######  Definition 7.16.

Let $\tau\colon\overline{Y}\to Y$ be a generically finite map of normal
projective varieties and let $\mathcal{Z}$ be a connected curve in $Y$. Let
$\overline{Y}$ be the a curve in $\mathcal{Z}$ such that
$\tau\overline{\mathcal{Z}}=\mathcal{Z}$. We define a graph
$\Gamma^{i}(\overline{\mathcal{Z}},\tau)\subseteq\Gamma^{i}(\overline{\mathcal{Z}})$
having as vertices the vertices of $\Gamma^{i}(\overline{\mathcal{Z}})$ and
having an edge between $\overline{Z}$ and $\overline{Z}^{\prime}$ if and only
if there is an edge between $\overline{Z}$ and $\overline{Z}^{\prime}$ in
$\Gamma^{i}(\overline{\mathcal{Z}})$ and either

  * •

$\tau{\overline{Z}}$ and $\tau{\overline{Z}}^{\prime}$ are curves in $Y$, or

  * •

$\tau{\overline{Z}}$ is a curve in $Y$, and
$\tau{\overline{Z}}^{\prime}=p\in\tau{\overline{Z}}$, or

  * •

$\tau{\overline{Z}}=\tau{\overline{Z}}^{\prime}=p\in Y$.

######  Construction 7.17.

Let $\tau\colon\overline{Y}\to Y$ be a generically finite map of normal
projective varieties and let $\mathcal{Z}$ be a a simple normal crossings
curve in $Y$. Let $\overline{\mathcal{Z}}$ be a simple normal crossings curve
in $\mathcal{Z}$ such that $\tau\overline{\mathcal{Z}}=\mathcal{Z}$. Let
$\overline{Z}_{1}\subseteq\overline{\mathcal{Z}}$ be such that
$Z_{1}=\tau\overline{Z}_{1}$ is a curve in $Y$.

Then there a homomorphism of groups

$\tau\colon\\{\text{circuits in }\Gamma^{i}(\overline{\mathcal{Z}},\tau)\text{
based at }\overline{Z}_{1}\\}\to\\{\text{circuits in
}\Gamma^{i}(\mathcal{Z})\text{ based at }Z_{1}\\}$

defined in the following way. Let
$\overline{\mathcal{C}}=(\\{\overline{Z}_{1},\ldots,\overline{Z}_{k}\\},\overline{Z}_{i,i+1})$
be a circuit in $\Gamma^{i}(\overline{\mathcal{Z}},\tau)$. If
$\tau(\overline{Z}_{i})$ is a curve for every $i$, we set
$\tau\overline{\mathcal{C}}=(\\{\tau\overline{Z}_{1},\ldots,\tau\overline{Z}_{k}\\},\tau\overline{Z}_{i,i+1})$.
Otherwise let $i_{j}$ and $h_{j}$ be such that $h_{j}>0$ and
$i_{j}+h_{j}+1=i_{j+1}$, and

  * •

for every $s=1,\ldots,h_{j}$ we have
$\tau(\overline{Z}_{i_{j}+1})=\tau(\overline{Z}_{i_{j}+s})$ is a point in $Y$,

  * •

$\tau(\overline{Z}_{i_{j}})$ and $\tau(\overline{Z}_{i_{j}+h_{j}+1})$ are
curves in $Y$.

If $\tau(\overline{Z}_{i_{j}})=\tau(\overline{Z}_{i_{j}+h_{j}+1})$ we set

$(\overline{Z}_{i_{j}},\overline{Z}_{i_{j}+1},\ldots,\overline{Z}_{i_{j}+h_{j}+1})\mapsto(\tau(\overline{Z}_{i_{j}})).$

If $\tau(\overline{Z}_{i_{j}})\neq\tau(\overline{Z}_{i_{j}+h_{j}+1})$ we set

$(\overline{Z}_{i_{j}},\overline{Z}_{i_{j}+1},\ldots,\overline{Z}_{i_{j}+h_{j}+1})\mapsto(\tau(\overline{Z}_{i_{j}}),\tau(\overline{Z}_{i_{j}+h_{j}+1}))$

with the edge $\tau(\overline{Z}_{i_{j}+1})$ between
$\tau(\overline{Z}_{i_{j}})$ and $\tau(\overline{Z}_{i_{j}+h_{j}+1})$.

######  Lemma 7.18.

Notation as in Construction 7.17. The map of Construction 7.17 respects the
homotopy of loops and defines thus a homomorphism of groups
$\tau\colon\pi_{1}(\Gamma^{i}(\overline{\mathcal{Z}},\tau),\overline{Z}_{1})\to\pi_{1}(\Gamma^{i}(Z),Z_{1})$.

###### Proof.

It is enough to prove that the two circuits

$\overline{\mathcal{C}}=(\\{\overline{Z}_{1},\ldots,\overline{Z}_{k}\\},\overline{Z}_{i,i+1})\text{
and }$
$\overline{\mathcal{C}}^{\prime}=(\\{\overline{Z}_{1},\ldots,\overline{Z}_{j},\overline{Z},\overline{Z}_{j}\ldots,\overline{Z}_{k}\\},\overline{Z}_{1,2},\ldots,\overline{Z}_{j-1,j},\overline{Z}^{\prime},\overline{Z}^{\prime},\overline{Z}_{j,j+1},\ldots,\overline{Z}_{k,1})$

have homotopically equivalent images. If $\tau\overline{Z}_{j}$ and
$\tau\overline{Z}$ are curves, then it is clear. If $\tau\overline{Z}$ is a
point, then the path $(\overline{Z}_{j},\overline{Z},\overline{Z}_{j})$ has
the same image as $(\overline{Z}_{j})$. If $\tau\overline{Z}$ is a curve and
$\tau\overline{Z}_{j}$ a point, then let $h<j<k$ be such that
$\tau\overline{Z}_{h}$ and $\tau\overline{Z}_{k}$ are curves and
$\tau\overline{Z}_{i}$ is a point for every $h<i<k$.

If $\tau\overline{Z}_{h}=\tau\overline{Z}_{k}$, then

$(\overline{Z}_{h},\ldots,\overline{Z}_{k})\mapsto(\tau\overline{Z}_{h}).$

If moreover $\tau\overline{Z}_{h}=\tau\overline{Z}$, then

$(\overline{Z}_{h},\ldots,\overline{Z}_{j},\overline{Z},\overline{Z}_{j}\ldots,\overline{Z}_{k})\mapsto(\tau\overline{Z}_{h}).$

If $\tau\overline{Z}_{h}\neq\tau\overline{Z}$, then

$(\overline{Z}_{h},\ldots,\overline{Z}_{j},\overline{Z},\overline{Z}_{j}\ldots,\overline{Z}_{k})\mapsto(\tau\overline{Z}_{h},\tau\overline{Z},\tau\overline{Z}_{h}).$

In both cases we get homothopically equivalent circuits.

If $\tau\overline{Z}_{h}\neq\tau\overline{Z}_{k}$, then

$(\overline{Z}_{h},\ldots,\overline{Z}_{k})\mapsto(\tau\overline{Z}_{h},\tau\overline{Z}_{k}).$

Then either $\tau\overline{Z}_{h}=\tau\overline{Z}$ or
$\tau\overline{Z}_{k}=\tau\overline{Z}$, and in both cases

$(\overline{Z}_{h},\ldots,\overline{Z}_{j},\overline{Z},\overline{Z}_{j}\ldots,\overline{Z}_{k})\mapsto(\tau\overline{Z}_{h},\tau\overline{Z}_{k}).$

∎

###  7.3. Trivial line bundles and pullbacks

We prove in this subsection that, if we have a generically finite morphism
between two immersed simple normal crossings varieties, then a line bundle is
trivial on the first variety if and only if its pullback is trivial on the
second.

######  Lemma 7.19.

Let $\mathcal{Z}$ be a connected simple normal crossings variety of dimension
at least 1 and let $\mathcal{L}$ be a line bundle on $\mathcal{Z}$ which is
trivial for every irreducible component of $\mathcal{Z}$. Then there is a
simple normal crossings curve $\mathcal{K}\subseteq\mathcal{Z}$ such that the
restriction of $\mathcal{L}$ to $\mathcal{K}$ is trivial if and only if
$\mathcal{L}$ is trivial.

###### Proof.

If $\mathcal{L}$ is trivial, then for every curve
$\mathcal{K}\subseteq\mathcal{Z}$, the restriction of $\mathcal{L}$ to
$\mathcal{K}$ is trivial.

For the other implication, we proceed by induction on

$\dim\mathcal{Z}=\max\\{\dim Z|\;Z\text{ irreducible component of
}\mathcal{Z}\\}.$

If $\dim\mathcal{Z}=1$, then we set $\mathcal{K}=\mathcal{Z}$. We assume now
the existence of such a curve for connected simple normal crossings varieties
of dimension $k-1$. Let $\mathcal{Z}$ be a connected simple normal crossings
variety of dimension $k$. Let $\mathcal{Z}^{(k)}$ be the union of all the
irreducible component of dimension $k$ and let $\mathcal{Z}_{k-1}$ be the
union of all the irreducible component of dimension at most $k-1$. Let $A$ be
a section of a very ample divisor on $\mathcal{Z}^{(k)}$ such that
$H^{1}(\mathcal{Z}^{(k)},\mathcal{O}(-A))=\\{0\\}$ and
$H^{1}(\mathcal{Z}^{(k)},\mathcal{L}(-A))=\\{0\\}$. In particular, for every
connected component $Z$ of $\mathcal{Z}^{(k)}$ the intersection $A\cap Z$ is
connected. Assume moreover that
$A\supseteq\mathcal{Z}^{(k)}\cap\mathcal{Z}^{(1)}$ and that
$A\cup\mathcal{Z}_{k-1}$ is a simple normal crossings variety.

We set $\mathcal{W}=A\cup\mathcal{Z}_{k-1}$. For every irreducible component
$Z$ of $\mathcal{Z}_{k-1}$ such that $Z\cap\mathcal{Z}^{(k)}\neq\emptyset$, we
have $Z\cap A\neq\emptyset$. Indeed, if $\dim Z\cap\mathcal{Z}^{(k)}\geq 1$,
it is true because $A$ is ample. If $\dim Z\cap\mathcal{Z}^{(k)}=0$, then it
is true by construction of $A$.

Then $\mathcal{W}$ is a connected simple normal crossings variety of dimension
$k-1$. In order to conclude, it is enough to prove that if the restriction of
$\mathcal{L}$ to $\mathcal{W}$ is trivial, then $\mathcal{L}$ is trivial.

If the restriction of $\mathcal{L}$ to $\mathcal{W}$ is trivial, then there is
a section $\bar{s}\in H^{0}(\mathcal{W},\mathcal{L})\setminus\\{0\\}$. As
$H^{1}(\mathcal{Z}^{(k)},\mathcal{L}(-A))=\\{0\\}$, there is a section
$s^{k}\in H^{0}(\mathcal{Z}^{(k)},\mathcal{L})$ such that
$s^{k}|_{A}=\bar{s}|_{A}$.

We want to show that $s^{k}$ and $\bar{s}|_{\mathcal{Z}_{k-1}}$ glue to a
section of $\mathcal{Z}$. This happens if and only if
$(s^{k},\bar{s}|_{\mathcal{Z}_{k-1}})$ is in the kernel of

$\begin{array}[]{rcl}\alpha\colon H^{0}(\mathcal{Z}^{(k)},\mathcal{L})\oplus
H^{0}(\mathcal{Z}_{k-1},\mathcal{L})&\to&H^{0}(\mathcal{Z}^{(k)}\cap\mathcal{Z}_{k-1},\mathcal{L})\\\
(s_{1},s_{2})&\mapsto&s_{1}-s_{2}.\end{array}$

We have a commutative diagram

$\textstyle{H^{0}(\mathcal{Z}^{(k)},\mathcal{L})\oplus
H^{0}(\mathcal{Z}_{k-1},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\scriptstyle{\alpha}$$\textstyle{H^{0}(\mathcal{Z}^{(k)}\cap\mathcal{Z}_{k-1},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(A,\mathcal{L})\oplus
H^{0}(\mathcal{Z}_{k-1},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{H^{0}(A\cap\mathcal{Z}_{k-1},\mathcal{L})}$

Since $(\bar{s}|_{A},\bar{s}|_{\mathcal{Z}_{k-1}})$ is in the kernel of
$\beta$, it follows that $(s^{k},\bar{s}|_{\mathcal{Z}_{k-1}})$ is in the
kernel of $\alpha$.

∎

######  Lemma 7.20.

Let $\varepsilon\colon\overline{Y}\to Y$ be a generically finite map of normal
projective varieties and let $\mathcal{Z}$ be a connected simple normal
crossings subvariety of $Y$. Assume that the preimage $\overline{\mathcal{Z}}$
of $\mathcal{Z}$ under $\varepsilon$ is a simple normal crossings variety. Let
$\mathcal{L}$ be a line bundle on $Y$ such that
$\mathcal{L}|_{Z}\sim\mathcal{O}_{Z}$ for every irreducible component $Z$ of
$\mathcal{Z}$. Then $\mathcal{L}|_{\mathcal{Z}}$ is torsion if and only if
$\varepsilon^{*}\mathcal{L}|_{\overline{\mathcal{Z}}}$ is torsion.

###### Proof.

If $\mathcal{L}|_{\mathcal{Z}}\sim\mathcal{O}_{\mathcal{Z}}$, then
$\varepsilon^{*}\mathcal{L}|_{\overline{\mathcal{Z}}}\sim\mathcal{O}_{\overline{\mathcal{Z}}}$.

Conversely, we assume that
$\tau^{*}\mathcal{L}|_{\overline{\mathcal{Z}}}\sim\mathcal{O}_{\overline{\mathcal{Z}}}$.
By Lemma 7.19 there is a semistable curve $\mathcal{K}\subseteq\mathcal{Z}$
such that the restriction of $\mathcal{L}$ to $\mathcal{K}$ is trivial if and
only if $\mathcal{L}$ is trivial.

######  Claim 7.21.

There is a semistable curve $\overline{\mathcal{K}}\subseteq\overline{Y}$ such
that $\tau\overline{\mathcal{K}}=\mathcal{K}$ and the image of the
homomorphism
$\tau\colon\pi_{1}(\Gamma^{i}(\overline{\mathcal{K}},\tau),\overline{K}_{1})\to\pi_{1}(\Gamma^{i}(\mathcal{K}),K_{1})$
has finite index in $\pi_{1}(\Gamma^{i}(\mathcal{K}),K_{1})$.

Assuming the claim, we conclude the proof.

Fix an irreducible component $\overline{K}_{1}$ of $\overline{\mathcal{K}}$
such that $K_{1}=\tau\overline{K}_{1}$ is a curve. We want to prove that there
is a positive integer $h$ such that for every circuit $\mathcal{C}$ in
$\Gamma^{i}(\mathcal{K})$ based in $K_{1}$, morphism
$\Phi_{\mathcal{L},\mathcal{C}}^{h}$ is the identity. As
$\pi_{1}(\Gamma^{i}(\mathcal{K}),K_{1})$ is finitely generated, the result
will follow from Lemma 7.10.

By Claim 7.21, the group
$\tau\left(\pi_{1}(\Gamma^{i}(\overline{\mathcal{K}},\tau),\overline{K}_{1})\right)$
is a subgroup of $\pi_{1}(\Gamma^{i}(\mathcal{K}),K_{1})$ of finite index $k$.
Therefore, there exists a circuit $\overline{\mathcal{C}}$ in
$\Gamma^{i}(\overline{\mathcal{K}})$ such that, if we denote by
$\mathcal{C}^{k}$ the concatenation of $\mathcal{C}$ with itself $k$ times,
the circuits $\mathcal{C}^{k}$ and $\tau\overline{\mathcal{C}}$ are
homotopically equivalent. By Remark 7.9, it is enough to prove that
$\Phi_{\mathcal{L},\tau\overline{\mathcal{C}}}=\Phi_{\mathcal{L},\mathcal{C}}^{k}$
is the identity. Let $K_{i},K_{i+1}$ be curves in $\tau\overline{\mathcal{C}}$
with the edge $p_{i}$ between them. Let
$\overline{K}_{i},\overline{K}_{i,j},\overline{K}_{i+1}$ be curves in
$\overline{\mathcal{C}}$ with $K_{h}=\tau\overline{K}_{h}$ for $h=i,i+1$ and
$\tau\overline{K}_{i,j}=p_{i}$ for $j=1,\ldots,\ell_{i}$, where for every
$j=0,\ldots,\ell_{i}$ we denote by $\bar{p}_{i,j}$ the edge between
$\overline{K}_{i,j}$ and $\overline{K}_{i,j+1}$, with
$\overline{K}_{i,0}=\overline{K}_{i}$ and
$\overline{K}_{i,\ell_{i}+1}=\overline{K}_{i+1}$. We have commutative diagrams

$\textstyle{H^{0}(K_{i},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{H^{0}(p_{i},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{H^{0}(\overline{K}_{i},\tau^{*}\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\bar{p}_{i,0},\tau^{*}\mathcal{L})}$$\textstyle{H^{0}(\cup_{j}\overline{K}_{i,j},\tau^{*}\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

and

$\textstyle{H^{0}(p_{i},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{H^{0}(K_{i+1},\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\wr}$$\textstyle{H^{0}(\cup_{j}\overline{K}_{i,j},\tau^{*}\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}(\bar{p}_{i,\ell_{i}},\tau^{*}\mathcal{L})}$$\textstyle{H^{0}(\overline{K}_{i+1},\tau^{*}\mathcal{L})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the horizontal arrows are the restriction isomorphisms and the vertical
arrows are isomorphisms induced by the pullback by $\tau$.

Les $s$ be a global section for
$\tau^{*}\mathcal{L}|_{\overline{\mathcal{Z}}}$. Then, with the
identifications of the previous diagram, the linear map
$\Phi_{\mathcal{L},\tau\overline{\mathcal{C}}}$ is the multiplication by
$s/s=1$, therefore it is the identity.

Then $\Phi_{\mathcal{L},\mathcal{C}}^{k}$ is the identity, proving the
statement.

We are left with the proof of Claim 7.21.

Let

$\begin{array}[]{l}\mathcal{K}_{f}=\\{K\subseteq\mathcal{K}\text{ irreducible
component}|\;K\not\subseteq\tau Exc(\tau)\\}\\\
\mathcal{K}_{e}=\\{K\subseteq\mathcal{K}\text{ irreducible
component}|\;K\subseteq\tau Exc(\tau)\\}.\end{array}$

For $K\in\mathcal{K}_{f}$, let $\widetilde{K}$ be its strict transform.
Let $K\in\mathcal{K}_{e}$. For every irreducible component
$Z\subseteq\tau^{-1}K$ surjecting onto $K$, let $H_{i}$ be hyperplane sections
such that $K_{Z}=Z\cap\bigcap H_{i}$ is a reduced curve. We can moreover find
the $H_{i}$ such that if $\cup Z_{i}$ is connected, then
$\widetilde{K}=\cup_{Z}K_{Z}$ is connected.
If $p\in\mathcal{K}^{sing}\cap\tau Exc(\tau)$, and $\\{p\\}=K_{1}\cap K_{2}$,
for every irreducible component $Z\subseteq\tau^{-1}p$ let $H_{i}$ be
hyperplane sections such that $K_{Z}=Z\cap\bigcap H_{i}$ is a reduced curve
and has the following property: if $\widetilde{K}_{1},\widetilde{K}_{2}$ are
such that $\tau\widetilde{K}_{i}=K_{i}$, then
$\widetilde{K}_{i}\cap\tau^{-1}p\subseteq K_{p}$. We can moreover find the
$H_{i}$ such that if $\cup Z_{i}$ is connected, then the union
$K_{p}=\cup_{Z}K_{Z}$ is connected.
Finally, we set

$\overline{\mathcal{K}}=\cup\\{\widetilde{K}|\;K\in\mathcal{K}_{f}\cup\mathcal{K}_{e}\\}\cup\\{K_{p}|\;p\in\mathcal{K}^{sing}\cap\tau
Exc(\tau)\\}.$

By the generality of $\mathcal{K}$, we can assume that
$\overline{\mathcal{K}}$ is a simple normal crossings curve.

We want to prove now that
$\tau\colon\pi_{1}(\Gamma^{i}(\overline{\mathcal{K}},\tau),\overline{K}_{1})\to\pi_{1}(\Gamma^{i}(\mathcal{K}),K_{1})$
has finite index in $\pi_{1}(\Gamma^{i}(\mathcal{K}),K_{1})$.

Let $\mathcal{C}=(K_{1},\ldots,K_{\ell},p_{i})$ be a circuit in
$\Gamma^{i}(\mathcal{K})$. Let
$\overline{K}_{1}\subseteq\overline{\mathcal{K}}$ be such that
$\tau\overline{K}_{1}=K_{1}$. Let $N$ be the number of curves in
$\overline{\mathcal{K}}$ surjecting onto $K_{1}$. We construct a circuit
$\overline{\mathcal{C}}$ in $\Gamma^{i}(\overline{\mathcal{K}},\tau)$ such
that $\tau\overline{\mathcal{C}}=m\mathcal{C}$ in the group of circuits based
on $K_{1}$ with $m$ dividing $N!$.

We assume now that we have $\overline{K}_{i}$ for $i=1,\ldots,r+q\ell$, and
$\overline{K}_{i,j}$ for $i=1,\ldots,r-1+q\ell$ and $j=1,\ldots,\ell_{i}$ and
edges $\bar{q}_{i,j}\in\overline{K}_{i,j}\cap\overline{K}_{i,j+1}$ for
$j=0,\ldots,\ell_{i}$ such that $\tau\overline{K}_{i}=K_{\bar{i}}$, where
$\bar{i}$ is the remainder of the euclidean division of $i$ by $\ell$, and
$\tau\overline{K}_{i,j}=p_{i}$.

If $q_{r}\not\in\tau Exc(\tau)$, then we let $\overline{K}_{r+1+q\ell}$ be a
curve such that

$\tau\overline{K}_{r+1+q\ell}=K_{r+1}\text{ and
}\tau^{-1}K_{r,r+1}\cap\overline{K}_{r+1+q\ell}\cap\overline{K}_{r+q\ell}\neq\emptyset.$

We set $\bar{q}_{r+q\ell,0}$ as a point in
$\tau^{-1}K_{r,r+1}\cap\overline{K}_{r+1+q\ell}\cap\overline{K}_{r+q\ell}$.

If $q_{r}\in\tau Exc(\tau)$, then let $\overline{K}^{\prime}_{r+1}$ such that
$\tau\overline{K}^{\prime}_{r+1}=K_{r+1}$ and $\overline{K}^{\prime}_{r+1}$
meets a connected component of $\tau^{-1}q_{r}$ meeting
$\overline{K}_{r+q\ell}$.
Let $\overline{K}_{r+q\ell,j}$ be such that

  * •

$\overline{K}_{r+q\ell,j}\cap\overline{K}_{r+q\ell,j+1}\neq\emptyset$,

  * •

$\overline{K}_{r+q\ell}\cap\overline{K}_{r+q\ell,1}\neq\emptyset$, and

  * •

$\overline{K}_{r+q\ell,\ell_{r+q\ell}}\cap\overline{K}^{\prime}_{r+1}\neq\emptyset$.

We set
$\bar{p}_{r+q\ell,0}\in\overline{K}_{r+q\ell}\cap\overline{K}_{r+q\ell,1}$,
$\bar{p}_{r+q\ell,j}\in\overline{K}_{r+q\ell,j}\cap\overline{K}_{r+q\ell,j+1}$
and
$\bar{p}_{r+q\ell,\ell_{i}}\in\overline{K}_{r+q\ell,\ell_{r+q\ell}}\cap\overline{K}^{\prime}_{r+1}$.
Finally, we set $\overline{K}_{r+1+q\ell}=\overline{K}^{\prime}_{r+1}$.

Then there are $q_{1}<q_{2}$ with $q_{2}-q_{1}\leq N$ such that
$\overline{K}_{1+q_{1}\ell}=\overline{K}_{1+q_{2}\ell}$. Then we set
$\gamma=(\overline{K}_{1},\ldots,\overline{K}_{q_{1}\ell})$ and
$\overline{\mathcal{C}}=\gamma*(\overline{K}_{q_{1}\ell},\ldots,\overline{K}_{q_{1}\ell})*\gamma^{-1}$.
We have
$\tau\overline{\mathcal{C}}=(q_{2}-q_{1})\mathcal{C}=\underbrace{\mathcal{C}*\ldots*\mathcal{C}}_{q_{2}-q_{1}\
\text{times}}$. ∎

##  8\. Restriction of the moduli part to log canonical centres

The goal of this section is to describe the restriction of the moduli part to
a log canonical centre of $(Y,\Sigma_{f})$. Part of the results can be seen as
a higher codimensional version of [FL19, Proposition 4.2]. We refer to [Hu20]
for similar results.

######  Definition 8.1.

(Definition 3.12 [FL19]) Let $f\colon(X,\Delta)\to Y$ be an lc-trivial
(respectively klt-trivial) fibration. Then $f$ is _acceptable_ if there exists
another lc-trivial (respectively klt-trivial) fibration
$\bar{f}\colon(\overline{X},\overline{\Delta})\to Y$ such that
$\overline{\Delta}$ is effective on the generic fibre of $\bar{f}$, and a
birational morphism $\mu\colon X\to\overline{X}$ such that $f=\bar{f}\circ\mu$
and
$K_{X}+\Delta\sim_{\mathbb{Q}}\mu^{*}(K_{\overline{X}}+\overline{\Delta})$.
Note that then the horizontal part of $\Delta^{<0}$ with respect to $f$ is
$\mu$-exceptional. Note also that any birational base change of an acceptable
lc-trivial (respectively klt-trivial) fibration is again an acceptable lc-
trivial (respectively klt-trivial) fibration.

$\textstyle{(X,\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\mu}$$\textstyle{(\overline{X},\overline{\Delta})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{f}}$$\textstyle{Y}$

######  Definition 8.2.

(Definition 3.10 [FL19]) Let $f\colon(X,\Delta)\rightarrow Y$ be an lc-trivial
fibration, where $(X,\Delta)$ is log smooth and $Y$ is smooth. Fix a prime
divisor $T$ on $Y$. An _$(f,T)$ -bad divisor_ is any reduced divisor
$\Sigma_{f,T}$ on $Y$ which contains:

  1. (a)

the locus of critical values of $f$,

  2. (b)

the closed set $f(\operatorname{Supp}\Delta_{v})\subseteq Y$, and

  3. (c)

the set $\operatorname{Supp}B_{f}\cup T$.

The next result is a corollary of [FL19, Proposition 4.2].

######  Proposition 8.3.

Let $f\colon(X,\Delta)\rightarrow Y$ be an acceptable klt-trivial fibration,
where $(X,\Delta)$ is a log smooth log canonical pair and $Y$ is a smooth
Ambro model for $f$. Assume that there exists an $(f,0)$-bad divisor
$\Sigma_{f}\subseteq Y$ which has simple normal crossings, and such that the
divisor $\Delta+f^{*}\Sigma_{f}$ has simple normal crossings support. Denote

$\Delta_{X}=\Delta+\sum_{\Gamma\subseteq\Sigma_{f}}\gamma_{\Gamma}f^{*}\Gamma,$

where $\gamma_{\Gamma}$ are the generic log canonical thresholds with respect
to $f$ as in Definition 2.9. Let $Z=T_{1}\cap\ldots\cap T_{k}$ be a log
canonical centre of $(Y,\Sigma_{f})$. Denote

$\Xi_{Z}:=(\Sigma_{f}-\sum T_{i})|_{Z}.$

Let $S$ be a minimal log canonical centre of $(X,\Delta_{X})$ over $Z$, which
exists by [FL19, Lemma 4.1] Let

$f|_{S}\colon
S\overset{h}{\longrightarrow}Z^{\prime}\overset{\tau}{\longrightarrow}Z$

be the Stein factorisation, and let $R$ denote the ramification divisor of
$\tau$ on $Z^{\prime}$. Then:

  1. (i)

if $K_{S}+\Delta_{S}=(K_{X}+\Delta_{X})|_{S}$, then
$h\colon(S,\Delta_{S})\rightarrow Z^{\prime}$ is a klt-trivial fibration with
$B_{h}\geq 0$,

  2. (ii)

$\tau^{*}(M_{f}|_{Z})\sim_{\mathbb{Q}}M_{h}+R^{\prime}+E$, where $M_{f}$ is
chosen so that $Z\not\subseteq M_{f}$ and

$R^{\prime}=\sum\limits_{\Gamma\not\subseteq\tau^{-1}(\Xi_{Z})}(\operatorname{mult}_{\Gamma}R)\cdot\Gamma\quad\text{and}\quad
E=\sum\limits_{\Gamma\not\subseteq\tau^{-1}(\Xi_{Z})}(\operatorname{mult}_{\Gamma}B_{h})\cdot\Gamma.$

###### Proof.

The proof follows the same line as [FL19, Proposition 4.2]. In particular,
Steps 1-5 are the same: we find a birational map
$\rho\colon(X,\Delta_{X})\dasharrow(W,\Delta_{W})$ over $Y$ such that, if
$\psi\colon(W,\Delta_{W})\to Y$ is the induced lc-trivial fibration, then
$(\phi^{*}\Sigma_{f})_{\mathrm{red}}\leq\Delta_{W,v}$. After replacing $T$
with $Z$ in Step 5, the fibration $h\colon(S,\Delta_{S})\rightarrow
Z^{\prime}$ is klt-trivial.

_Step 6._ Let $T_{1},\ldots,T_{h}$ be components of $\Sigma_{f}$ such that
$Z=T_{1}\cap\ldots\cap T_{h}$. By equation [FL19, (13)] every component
$D_{1}$ of $\psi^{*}T_{1}$ which dominates $T_{1}$ has coefficient $1$ in
$\Delta_{W}$. Denote $\Delta_{D_{1}}:=(\Delta_{W}-D_{1})|_{D_{1}}$, so that
the Stein factorisation of $\psi|_{D_{1}}\colon(D_{1},\Delta_{D_{1}})\to
T_{1}$ gives an lc-trivial fibration. Let
$\Xi_{T_{1}}=(\Sigma_{f}-T_{1})|_{T_{1}}$ and let $P$ be a component of
$(\psi|_{D_{1}})^{*}\Xi_{T_{1}}$. Since
$(\psi|_{D_{1}})^{*}\Xi_{T_{1}}=(\psi^{*}\Sigma_{f}-\psi^{*}T_{1})|_{D_{1}}$,
and each component of $\psi^{*}\Sigma_{f}$ is a component of $\Delta_{W}^{=1}$
by [FL19, (12) and (13)], this implies that $P$ is a component of
$(\Delta_{W}^{=1}-D_{1})|_{D_{1}}=\Delta_{D_{1}}^{=1}$. In other words,

$\big{(}(\psi|_{D_{1}})^{*}\Xi_{T_{1}}\big{)}_{\mathrm{red}}\leq\Delta_{D_{1}}^{=1}.$

Assume that for $i>1$ there are components $D_{1},\ldots,D_{i}$ such that
$\phi(D_{j})=T_{j}$ and $\big{(}(\psi|_{D_{1}\cap\ldots\cap
D_{i}})^{*}\Xi_{T_{1}\cap\ldots\cap
T_{i}}\big{)}_{\mathrm{red}}\leq\Delta_{D_{1}\cap\ldots\cap D_{i}}^{=1}$,
where $\Xi_{T_{1}\cap\ldots\cap T_{i}}=(\Sigma_{f}-T_{1}-\ldots-
T_{i})|_{T_{1}\cap\ldots\cap T_{i}}$ and $\Delta_{D_{1}\cap\ldots\cap
D_{i}}^{=1}=(\Delta_{W}^{=1}-D_{1}-\ldots-D_{i})|_{D_{1}\cap\ldots\cap
D_{i}}$.

There is a component $D_{i+1}$ of $\psi^{*}T_{i+1}$ which has coefficient $1$
in $\Delta_{W}$. Denote $\Delta_{D_{1}\cap\ldots\cap
D_{i+1}}=(\Delta_{W}-D_{1}-\ldots-D_{i+1})|_{D_{1}\cap\ldots\cap D_{i+1}}$, so
that the Stein factorisation of $\psi|_{D_{1}\cap\ldots\cap
D_{i+1}}\colon({D_{1}\cap\ldots\cap D_{i+1}},\Delta_{D_{1}\cap\ldots\cap
D_{i+1}})\to T_{1}\cap\ldots\cap T_{i+1}$ gives an lc-trivial fibration. Let
$\Xi_{T_{1}\cap\ldots\cap T_{i+1}}=(\Sigma_{f}-T_{1}-\ldots-
T_{i+1})|_{T_{1}\cap\ldots\cap T_{i+1}}$ and let $P$ be a component of
$(\psi|_{D_{1}\cap\ldots\cap D_{i+1}})^{*}\Xi_{T_{1}\cap\ldots\cap T_{i+1}}$.
As before,

$\big{(}(\psi|_{D_{1}\cap\ldots\cap D_{i+1}})^{*}\Xi_{T_{1}\cap\ldots\cap
T_{i+1}}\big{)}_{\mathrm{red}}\leq\Delta_{D_{1}\cap\ldots\cap D_{i+1}}^{=1}.$

We proved by induction that there are $D_{1},\ldots,D_{h}$ such that

(17) $\big{(}(\psi|_{D_{1}\cap\ldots\cap
D_{h}})^{*}\Xi_{Z}\big{)}_{\mathrm{red}}\leq\Delta_{D_{1}\cap\ldots\cap
D_{h}}^{=1}.$

Now, by [FL19, Proposition 2.6] there are components $D_{1},\ldots,D_{h}$ of
$\Delta_{W}$ and $S_{1},\dots,S_{k}$ of $\Delta_{D}^{=1}$, where
$D=D_{1}\cap\ldots\cap D_{h}$, such that $S_{W}$ is a component of
$S_{1}\cap\dots\cap S_{k}$, and note that the $S_{i}$ dominate $Z$. This and
(17) imply

(18)
$\big{(}(\psi|_{D})^{*}\Xi_{Z}\big{)}_{\mathrm{red}}\leq\Delta_{D}^{=1}-S_{1}-\dots-
S_{k},$

hence

$\big{(}(\psi|_{S_{W}})^{*}\Xi_{Z}\big{)}_{\mathrm{red}}\leq(\Delta_{D}^{=1}-S_{1}-\dots-
S_{k})|_{S_{W}}\leq\Delta_{S_{W}}^{=1}.$

Thus, for every prime divisor $P\subseteq\operatorname{Supp}\tau^{*}\Xi_{Z}$,
the generic log-canonical threshold $\gamma_{P}$ of $(S_{W},\Delta_{S_{W}})$
with respect to $h_{W}^{*}P$ is zero. If we define

$E:=\sum\limits_{\Gamma\not\subseteq\tau^{-1}(\Xi_{Z})}(\operatorname{mult}_{\Gamma}B_{h_{W}})\cdot\Gamma=\sum\limits_{\Gamma\not\subseteq\tau^{-1}(\Xi_{Z})}(\operatorname{mult}_{\Gamma}B_{h})\cdot\Gamma,$

where the second equality follows from [FL19, (17)], then

(19) $B_{h_{W}}=(\tau^{*}\Xi_{Z})_{\mathrm{red}}+E.$

Finally, Steps 7 is the same after replacing $T$ with $Z$. ∎

######  Proposition 8.4.

Let $f\colon(X,\Delta)\rightarrow Y$ be an acceptable klt-trivial fibration.
Assume that $Y$ is an Ambro model for $f$ and that there exists a simple
normal crossings divisor $R$ on $Y$ such that the support of the divisor
$\Delta+f^{-1}\Sigma_{f}$ has simple normal crossings. Assume that $f$ is
semistable. Set

$\Delta_{X}=\Delta+\sum_{\Gamma\subseteq\Sigma_{f}}\gamma_{\Gamma}f^{*}\Gamma$

where $\gamma_{\Gamma}$ are the generic log canonical thresholds with respect
to the klt-fibration $f$ as in Definition 2.9. Then there exists a birational
map $\rho\colon X\dasharrow W$ and a fibration $\psi\colon W\rightarrow Y$
such that:

  1. (a)

the pair $(W,\Delta_{W})$ is $\mathbb{Q}$-factorial dlt, where
$\Delta_{W}:=\rho_{*}\Delta_{X}$, and $\Delta_{W}\geq 0$;

  2. (b)

$\psi\colon(W,\Delta_{W})\rightarrow Y$ is a klt-trivial fibration;

  3. (c)

$\rho\colon(X,\Delta_{X})\dasharrow(W,\Delta_{W})$ is crepant birational;

  4. (d)

the discriminant of $\psi$ is $\Sigma_{f}$ and the moduli part is $M_{f}$;

  5. (e)

$\Delta_{W,v}=\psi^{*}\Sigma_{f}$.

Let $Z$ be a log canonical centre of $(Y,\Sigma_{f})$ and let $S$ be a minimal
log canonical centre of $(W,\Delta_{W})$ over $Z$. Let $\psi|_{S}\colon
S\overset{h}{\longrightarrow}Z^{\prime}\overset{\tau}{\longrightarrow}Z$ be
the Stein factorisation.

  1. (i)

If $K_{S}+\Delta_{S}=(K_{W}+\Delta_{W})|_{S}$, then
$h\colon(S,\Delta_{S})\rightarrow Z^{\prime}$ is a klt-trivial fibration.

  2. (ii)

Assume that $\tau^{*}M_{f}|_{Z}=M_{h}$. Then $\Delta_{S,v}=h^{*}B_{h}$ and
$B_{h}=(\tau^{*}\Xi_{Z})_{\mathrm{red}}$.

  3. (iii)

Let $Z$ be a component of $\Sigma_{f}$ such that $M_{f}|_{Z}\equiv 0$. Then
$\tau^{*}M_{f}|_{Z}=M_{h}\sim_{\mathbb{Q}}0$ and $\Delta_{S,v}=h^{*}B_{h}$.

  4. (iv)

If either $\tau^{*}M_{f}|_{Z}=M_{h}$ or $M_{f}|_{Z}\equiv 0$, then $h$ has
reduced fibres over an open set meeting all the irreducible components of
$B_{h}$.

###### Proof.

_Step 1._ The existence of $\rho$ satisfying (a), (b), (c), (d) follows from