# Completions of affine spaces
into Mori fiber spaces with non-rational fibers

Adrien Dubouloz  IMB UMR5584, CNRS, Univ. Bourgogne Franche-Comté, EIPHI
Graduate School ANR-17-EURE-0002, F-21000 Dijon, France.
<EMAIL_ADDRESS>,  Takashi Kishimoto  Department of
Mathematics, Faculty of Science, Saitama University, Saitama 338-8570, Japan
<EMAIL_ADDRESS>and  Karol Palka  Institute of Mathematics,
Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland
<EMAIL_ADDRESS>
###### Abstract.

We describe a method to construct completions of affine spaces into total
spaces of $\mathbb{Q}$-factorial terminal Mori fiber spaces over the
projective line. As an application we provide families of examples with non-
rational, birationally rigid and non-stably rational general fibers.

The authors were supported by the National Science Centre, Poland, grant
number 2015/18/E/ST1/00. The first author was partially supported by the
French ANR project ”FIBALGA” ANR-18-CE40-0003. The second author was partially
supported by JSPS KAKENHI, grant number JP19K03395.

##  1\. Introduction

We work with complex algebraic varieties. A _completion_ of a given variety
$U$ is any complete variety containing a Zariski open subset isomorphic to
$U$. In this article we consider the problem of describing minimal completions
of affine spaces $\mathbb{A}^{n}$. Since the Kodaira dimension of
$\mathbb{A}^{n}$ is negative, a natural way to define minimality in this
context is to require the completion to be the total space of a Mori fiber
space. Indeed, by [BCHM10, Corollary 1.3.3] such varieties come as outputs of
the Minimal Model Program applied to smooth projective varieties of negative
Kodaira dimension. We call them _Mori fiber completions_. From the viewpoint
of the Minimal Model Program it is also natural to consider not only smooth
but also mildly singular varieties and their completions, namely those which
are $\mathbb{Q}$-factorial and have terminal singularities.

If $\pi\colon V\rightarrow B$ is a Mori fiber completion of $\mathbb{A}^{n}$
then $V$ is rational and the base variety $B$ is unirational. The case when
$B$ is a point is especially important, because then $V$ is a Fano variety of
Picard rank one. Fano varieties and their rationality are objects of intensive
studies, see [IP99], [KSC04]. Classifying smooth Fano completions of
$\mathbb{A}^{n}$ of Picard rank one is the projective version of the
celebrated problem of finding minimal analytic completions of complex affine
spaces raised by Hirzebruch [Hir54, Problem 27] and studied by many authors.
In case $n=1,2$ there is only $\mathbb{P}^{n}$. The first difficult case,
$n=3$, was completed in a series of papers [PS88, Fur90, Pro91, Fur93]. For
$n=4$ there are some partial results [Pro94, PZ18].

In this article we focus on the situation where the base $B$ is a curve, hence
is isomorphic to the projective line $\mathbb{P}^{1}$. Simple examples of Mori
fiber completions of $\mathbb{A}^{n}$ of this type are given by locally
trivial $\mathbb{P}^{n-1}$-bundles over $\mathbb{P}^{1}$. Another series of
examples can be constructed by taking the product of $\mathbb{P}^{1}$ with any
$\mathbb{Q}$-factorial terminal Fano variety of Picard rank one which is a
completion of $\mathbb{A}^{n-1}$, see Example 6.1. These examples are special
in the sense that general fibers are completions of $\mathbb{A}^{n-1}$. In
general, the following interesting problem arises:

######  Problem.

Let $\pi\colon V\rightarrow\mathbb{P}^{1}$ be a Mori fiber completion of
$\mathbb{A}^{n}$. What can be said about the geometry of general fibers of
$\pi$?

A basic observation is that a general fiber of $\pi$ is a Fano variety of
dimension $n-1$ with terminal singularities. The property of being a Mori
fiber space implies in particular that the generic fiber of $\pi$ has Picard
rank one over the function field of $\mathbb{P}^{1}$. On the contrary, general
fibers do not necessarily have Picard rank one, as can be seen for instance
for del Pezzo fibrations of some threefolds completing $\mathbb{A}^{3}$, see
Example 6.2. Still, it is a restrictive condition for a Fano variety to be a
general fiber of a Mori fiber space, see [CFST16, CFST18]. For $n=2,3$ a
general fiber of $\pi$, being terminal and Fano, is either $\mathbb{P}^{1}$ or
a smooth del Pezzo surface, hence is a completion of $\mathbb{A}^{2}$. In
contrast, our first result implies that in higher dimensions general fibers of
Mori fiber completions of $\mathbb{A}^{n}$ can be very far from being
rational.

######  Theorem 1.1.

Let $H$ be a hyperplane in $\mathbb{P}^{n}$, $n\geq 2$. For every integral
hypersurface $F\subseteq\mathbb{P}^{n}$ of degree $d\leq n$ such that $F\cap
H$ is irreducible and contained in the smooth locus of $F$ there exists a Mori
fiber completion $\pi\colon V\rightarrow\mathbb{P}^{1}$ of the affine
$n$-space $\mathbb{A}^{n}\cong\mathbb{P}^{n}\setminus H$ such that all
hypersurfaces other than $dH$ in the pencil of divisors $\langle F,dH\rangle$
generated by $F$ and $dH$ appear as fibers of $\pi$.

By Bertini’s theorem a general member of a pencil as in Theorem 1.1 is smooth.
For $(n,d)=(4,3)$ it is a smooth cubic threefold, hence is unirational but not
rational [CG72]. For $d=n\geq 4$ a general member is birationally super-rigid
(see Definition 2.9) by [dF13, IM71, Puk98]. This gives the following
corollary.

######  Corollary 1.2.

For every $n\geq 4$ there exists a Mori fiber completion of $\mathbb{A}^{n}$
over $\mathbb{P}^{1}$ whose general fibers are smooth birationally super-rigid
Fano varieties of Picard rank one.

Another corollary concerns completions of polynomial morphisms
$f\colon\mathbb{A}^{n}\rightarrow\mathbb{A}^{1}$ of low degree into Mori fiber
spaces over $\mathbb{P}^{1}$.

######  Corollary 1.3.

Assume that $f\colon\mathbb{A}^{n}\rightarrow\mathbb{A}^{1}$ is a morphism
given by a polynomial of total degree at most $n$ and that in the natural open
embedding $\mathbb{A}^{n}\subseteq\mathbb{P}^{n}$ the intersection of the
closure of the zero locus of $f$ with $\mathbb{P}^{n}\setminus\mathbb{A}^{n}$
is smooth. Then $f\colon\mathbb{A}^{n}\rightarrow\mathbb{A}^{1}$ can be
completed into a Mori fiber space over $\mathbb{P}^{1}$.

General fibers of polynomial morphisms
$f\colon\mathbb{A}^{n}\rightarrow\mathbb{A}^{1}$ as above of degree $\deg
f\leq n-1$ are smooth affine Fano varieties in the sense of [CDP18] (see
Definition 2.10). We note that for $n\geq 6$ and $\deg f=n-1$ general fibers
of the Mori fiber space $\pi\colon V\rightarrow\mathbb{P}^{1}$ are then
completions of so-called _super-rigid affine Fano varieties_ (general fibers
of $f$), see Definition 2.11 and Example 6.9.

We obtain even more families of possible general fibers by considering
singular ambient spaces instead of $\mathbb{P}^{n}$. For a definition of
quasi-smooth hypersurfaces in weighted projective spaces, see Section 6C.

######  Theorem 1.4.

Let $n\geq 4$ and let $\mathbb{P}=\mathbb{P}(1,a_{1},\ldots,a_{n})$ be a
weighted projective space for some positive integers $a_{1},\ldots,a_{n}$,
such that the description of the hyperplane $H=\mathbb{P}(a_{1},\ldots,a_{n})$
is well-formed. Then for every quasi-smooth terminal hypersurface $F\neq H$ of
$\mathbb{P}$ of degree $d\leq a_{1}+\ldots+a_{n}$ there exists a Mori fiber
completion $\pi\colon V\rightarrow\mathbb{P}^{1}$ of
$\mathbb{A}^{n}\cong\mathbb{P}\setminus H$ such that all hypersurfaces other
than $dH$ in the pencil $\langle F,dH\rangle$ generated by $F$ and $dH$ appear
as fibers of $\pi$.

For $n=4$ we get a class of Mori fiber completions of $\mathbb{A}^{4}$ over
$\mathbb{P}^{1}$ whose general fibers are quasi-smooth terminal weighted Fano
threefold hypersurfaces in the 95 families of Fletcher and Reid [IF00, Rei80],
see Corollary 6.11. By [CP17, Main Theorem], see also [CPR00], all such
threefolds are birationally rigid and some of them are even known to be
birationally super-rigid [CP17, Theorem 1.1.10]. In a similar vein, we deduce
from [Oka19] the following result, see Example 6.12.

######  Corollary 1.5.

There exists a Mori fiber completion of $\mathbb{A}^{4}$ over $\mathbb{P}^{1}$
whose very general fibers are Fano varieties of Picard rank one which are not
stably rational.

We now briefly describe our approach. A natural way to obtain completions of a
given quasi-projective variety $U$ with $\mathbb{Q}$-factorial terminal
singularities ($\mathbb{A}^{n}$ in particular) is to find some normal
projective completion whose singularities are not worse than those of $U$, and
then to run a Minimal Model Program on it. If $U$ is smooth then we may take a
smooth completion using resolution of singularities. But in general, finding
appropriate completions form which to run the program is already a nontrivial
task. Moreover, each step, whether it is a divisorial contraction or a flip,
may change the isomorphism type of the image of $U$. Preventing this to happen
is one of the key problems. To gain more control over the successive steps of
the program we study completions which are resolutions of specific pencils of
divisors on terminal Fano varieties, namely of those pencils whose general
members are Fano varieties of Picard rank one with terminal singularities. We
call them _terminal rank one Fano pencils_. This assumption on one hand allows
to find a completion with mild singularities and on the other it gives a
chance to analyze the MMP runs in a more detail. We introduce the notion of a
“compatible thrifty resolution” of such pencils, characterized essentially by
the property that it keeps the isomorphism type of general members unchanged.
We give sufficient criteria for the existence of compatible thrifty
resolutions and we show that terminal rank one Fano pencils which admit such
resolutions yield interesting Mori fiber spaces over $\mathbb{P}^{1}$.

The article is organized as follows: In Section 2 we recall basic notions
concerning varieties and their singularities in the framework of the Minimal
Model Program. Section 3 reviews properties of pencils of Weil divisors on
normal varieties. Section 4 is devoted to the study of terminal rank one Fano
pencils, their resolutions and the outputs of relative MMP’s ran from these.
In Section 5 we consider a class of pencils on Fano varieties with class
groups $\mathbb{Z}$. It provides a big supply of terminal rank one Fano
pencils admitting compatible thrifty resolutions. Applications to the
construction of Mori fiber completions of $\mathbb{A}^{n}$ over
$\mathbb{P}^{1}$ are given in Section 6. This section contains proofs of
Theorems 1.1 and 1.4 and a series of examples.

Acknowledgements. We thank the Institute of Mathematics of Burgundy, the
Saitama University and the Institute of Mathematics of the Polish Academy of
Sciences for excellent working conditions.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2A Singularities in the context of MMP
    2. 2B Inversion of adjunction and a $\mathbb{Q}$-factorial terminalization
    3. 2C Fano varieties and Mori fiber spaces
  3. 3 Pencils and their resolutions
    1. 3A Divisorial sheaves
    2. 3B Pencils of Weil divisors
    3. 3C The graph of a pencil
  4. 4 Terminal rank one Fano pencils and associated Mori fiber spaces
    1. 4A $\mathbb{Q}$-factorial terminal resolutions
    2. 4B Terminal rank one Fano pencils and relative MMPs
  5. 5 Obtaining Mori fiber completions from special pencils
    1. 5A Integrity of members
    2. 5B $H$-special pencils of Cartier divisors
  6. 6 Mori fiber completions of affine spaces over $\mathbb{P}^{1}$
    1. 6A Some known examples
    2. 6B Pencils on smooth Fano varieties
    3. 6C Pencils on weighted projective spaces

##  2\. Preliminaries

We summarize basic notions concerning varieties and their singularities in the
framework of the Minimal Model Program which are used in the article.

We use the following standard terminology: The _domain of the definition_ of a
dominant rational map $f\colon X\dashrightarrow Y$ between algebraic varieties
is the largest open subset $\operatorname{dom}(f)$ of $X$ on which $f$ is
represented by a morphism. Its complement is called the _indeterminacy locus_
of $f$. The _exceptional locus_ $\operatorname{Exc}(f)$ of a proper birational
morphism $f\colon X\rightarrow Y$ is the indeterminacy locus of the birational
map $f^{-1}\colon Y\dashrightarrow X$. A _resolution_ (of indeterminacy) of a
rational map $f\colon X\dashrightarrow Y$ is a proper birational morphism
$\tau\colon X^{\prime}\rightarrow X$ such that $f\circ\tau\colon
X^{\prime}\dashrightarrow Y$ is a morphism.

###  2A. Singularities in the context of MMP

Let $X$ be a normal variety and let $j\colon X_{\mathrm{reg}}\hookrightarrow
X$ be the embedding of the smooth locus. The induced restriction on Picard
groups
$j^{*}\colon\operatorname{Pic}(X)\rightarrow\operatorname{Pic}(X_{\mathrm{reg}})$
is injective and the restriction on class groups
$\operatorname{Cl}(X)\rightarrow\operatorname{Cl}(X_{\mathrm{reg}})\cong\operatorname{Pic}(X_{\mathrm{reg}})$
is an isomorphism. This gives a natural injection
$\operatorname{Pic}(X)\rightarrow\operatorname{Cl}(X)$; see [Gro67, Corollaire
21.6.10]. A canonical divisor of $X$ is a Weil divisor $K_{X}$ on $X$ whose
class in $\operatorname{Pic}(X_{\mathrm{reg}})$ is the class of the canonical
invertible sheaf $\det(\Omega^{1}_{X_{\mathrm{reg}}})$ of $X_{\mathrm{reg}}$.
A Weil divisor on $X$ is called _$\mathbb{Q}$ -Cartier_ if it has a positive
multiple which is Cartier. We say that $X$ is _$\mathbb{Q}$ -factorial_ if
every Weil divisor on $X$ is $\mathbb{Q}$-Cartier.

We now recall some basic facts about singularities of pairs. We refer the
reader to [KM98, Chapter 2] for details. A _log pair_ $(X,D)$ consists of a
normal variety $X$ and a Weil $\mathbb{Q}$-divisor $D=\sum d_{i}D_{i}$ on it,
whose coefficients $d_{i}$ belong to $[0,1]\cap\mathbb{Q}$, and such that the
divisor $K_{X}+D$ is $\mathbb{Q}$-Cartier. Given a proper birational morphism
$f\colon Y\rightarrow X$ from a normal variety $Y$, we denote by
$\mathcal{E}(f)$ the set of prime divisors $E$ on $Y$ contained in the
exceptional locus $\operatorname{Exc}(f)$ of $f$. We call the image of $E$ the
_center of $E$ on $X$_.

Given a log pair $(X,D)$ and a birational proper morphism from a normal
variety $f\colon Y\rightarrow X$ we have a linear equivalence of
$\mathbb{Q}$-divisors

$K_{Y}+f_{*}^{-1}D\sim f^{*}(K_{X}+D)+\sum_{E\in\mathcal{E}(f)}a_{X,D}(E)E,$

where $a_{X,D}(E)\in\mathbb{Q}$. The number $a_{X,D}(E)$ is called the
_discrepancy_ of $E$ with respect to $(X,D)$. It does not depend on $Y$ in the
sense that if $Y^{\prime}\rightarrow Y$ is a proper birational morphism and
$E^{\prime}$ is the proper transform of $E$ on $Y^{\prime}$ then
$a_{X,D}(E)=a_{X,D}(E^{\prime})$. The _discrepancy_ of the log pair $(X,D)$ is
defined as

(2.1)
$\operatorname{Discrep}(X,D)=\underset{f,E\in\mathcal{E}(f)}{\operatorname{inf}}a_{X,D}(E),$

where the infimum is taken over all $f\colon Y\rightarrow X$ as above and all
$E\in\mathcal{E}(f)$.

A log pair $(X,D)$ is _terminal_ if $\operatorname{Discrep}(X,D)>0$, _purely
log terminal (plt)_ if $\operatorname{Discrep}(X,D)>-1$ and _Kawamata log
terminal (klt)_ if it is plt and $\lfloor D\rfloor=0$. A normal variety $X$ is
_terminal_ (respectively klt) if the log pair $(X,0)$ is terminal
(respectively klt). The property of being terminal for a variety $X$ and being
klt for any log pair $(X,D)$ can be verified by computing the infimum (2.1) on
a single proper birational morphism $f\colon Y\rightarrow X$ which is a log
resolution of the log pair $(X,D)$, that is, for which $Y$ is smooth and
$\operatorname{Exc}(f)\cup f^{-1}_{*}(D)$ is a divisor with simple normal
crossings, see [Kol13, Corollaries 2.12, 2.13].

Given a normal variety $X$ and a closed subset $Z$ of it, we say that $X$ is
_terminal (respectively klt) in a neighborhood of $Z$_ if there exists an open
neighborhood $U$ of $Z$ which is terminal (respectively klt). Finally, we say
that a log pair $(X,S)$, where $S$ is a prime Weil divisor on $X$, is _plt in
a neighborhood of $S$_ if there exists an open neighborhood $U$ of $S$ such
that the log pair $(U,S)$ is plt. Equivalently, if for every log resolution
$f\colon Y\rightarrow X$ of $(X,S)$ every exceptional divisor $E$ of $f$ whose
center on $X$ intersects $S$ has discrepancy $a_{X,S}(E)>-1$.

We will need the following known result.

######  Lemma 2.1.

Let $(X,S)$ be a log-pair such that $S$ is a normal prime Cartier divisor.
Assume that $(X,S)$ is plt in a neighborhood of $S$. Then $X$ is terminal in a
neighborhood of $S$ if and only if for every log-resolution $f\colon
Y\rightarrow X$ of $(X,S)$ every exceptional divisor $E$ of $f$ whose center
on $X$ meets $S$ but is not contained in $S$ has positive discrepancy.

###### Proof.

Let $f\colon Y\rightarrow X$ be a log resolution of $(X,S)$. Replacing $X$ by
an open neighborhood of $S$, if necessary, we can assume that the center on
$X$ of every exceptional divisor $E_{i}$ of $f$ meets $S$. Since $S$ is a
prime Cartier divisor, we have $f^{*}S=f_{*}^{-1}(S)+\sum c_{i}E_{i}$, where
$c_{i}$ is a positive integer if $f(E_{i})\subseteq S$ and $c_{i}=0$
otherwise. Writing $K_{Y}\sim f^{*}(K_{X})+\sum b_{i}E_{i}$ we get

$K_{Y}+f_{*}^{-1}(S)\sim f^{*}(K_{X}+S)+\sum_{E_{i},f(E_{i})\not\subseteq
S}b_{i}E_{i}+\sum_{E_{i},f(E_{i})\subseteq S}(b_{i}-c_{i})E_{i}.$

Since $(X,S)$ is plt in a neighborhood of $S$, $a_{X,S}(E_{i})>-1$ for every
$E_{i}$. It follows that for every $E_{i}$ such that $f(E_{i})\subseteq S$ we
have $b_{i}=a_{X,S}(E_{i})+c_{i}>0$. Thus $X$ is terminal in a neighborhood of
$S$ if and only if for every $E_{i}$ such that $f(E_{i})\not\subseteq S$ the
discrepancy $b_{i}=a_{X,S}(E_{i})$ is positive. ∎

Let $(R,\mathfrak{m})$ be a Noetherian local ring. Recall that a sequence
$(x_{1},\ldots,x_{r})$ of elements of $\mathfrak{m}$ is _regular_ if for every
$i=1,\ldots,r$ the element $x_{i}$ is not a zero divisor in
$R/(x_{1},\ldots,x_{i-1})$. The _depth_ and the _(Krull) dimension_ of $R$ are
defined respectively as the maximal length of a regular sequence and as the
maximal number of strict inclusions of prime ideals. The ring is called
_Cohen-Macaulay_ if $\mathfrak{m}$ contains a regular sequence of length $\dim
R$. Equivalently, $R$ satisfies Serre’s conditions $S_{i}$:
$\operatorname{depth}R\geq\min(\dim R,i)$, for all $i\geq 1$. The ring is
_regular_ if $\mathfrak{m}$ contains a regular sequence of length $\dim R$
generating $\mathfrak{m}$. We say that a prime ideal $I\subseteq\mathfrak{m}$
of $R$ is a _complete intersection_ if it is generated by a regular sequence
of length equal to its height.

A scheme $X$ is called Cohen-Macaulay if all its local rings are Cohen-
Macaulay. An irreducible closed subscheme $Y$ of a scheme $X$ is called a
_local complete intersection in $X$_ if the sheaf of ideals of $Y$ is a
complete intersection in all local rings of $X$. Finally, we say that an
irreducible scheme $X$ is a _local complete intersection_ if it is locally
isomorphic to a local complete intersection in a smooth scheme.

Let us also recall that a normal variety $X$ has _rational singularities_ if
for every resolution $f\colon\widetilde{X}\rightarrow X$ of the singularities
of $X$ we have $R^{i}f_{*}\mathcal{O}_{\widetilde{X}}=0$ for $i\geq 1$. We
will use the following known facts about rational singularities. Note that
Kawamata log terminal singularities are rational, see e.g. [Ish18, Theorem
6.2.12].

######  Lemma 2.2 (Rational singularities).

Let $X$ be a normal variety with rational singularities. Then the following
hold:

  1. (a)

$X$ is Cohen-Macaulay and hence every local complete intersection $Y$ in $X$
is Cohen-Macaluay. In particular, $Y$ is normal if and only if its singular
locus has codimension at least $2$.

  2. (b)

The group $\operatorname{Cl}(X)/\operatorname{Pic}(X)$ is finitely generated.

  3. (c)

If $X$ is projective and $\mathcal{N}$ is a big and nef invertible sheaf on
$X$ then $H^{i}(X,\mathcal{N}^{\vee})=0$ for $0\leq i\leq\dim X-1$.

###### Proof.

(a) By [Ish18, Theorem 6.2.14], $X$ is Cohen-Macaulay. Then by [Ish18,
Proposition 5.3.12], $Y$ is Cohen-Macaulay, too. In particular, by Serre’s
criterion [Mat89, Theorem 23.8], $Y$ is normal if and only if it is regular in
codimension $1$.

To prove (b) and (c) we first make a cohomological observation. Let $f\colon
Y\rightarrow X$ be a morphism of schemes, $\mathcal{E}$ a locally free sheaf
of finite rank on $X$ and $\mathcal{F}$ a quasi-coherent sheaf on $Y$. By the
projection formula [Har77, Exercise III.8.3],
$R^{j}f_{*}(\mathcal{F}\otimes_{\mathcal{O}_{Y}}f^{*}\mathcal{E})\cong
R^{j}f_{*}(\mathcal{F})\otimes_{\mathcal{O}_{X}}\mathcal{E}$, so the Leray
spectral sequence gives
$H^{i}(X,R^{j}f_{*}(\mathcal{F})\otimes_{\mathcal{O}_{X}}\mathcal{E})\Rightarrow
H^{i+j}(Y,\mathcal{F}\otimes_{\mathcal{O}_{Y}}f^{*}\mathcal{E}).$ If
$R^{j}f_{*}(\mathcal{F})=0$ for $j\geq 1$ then the sequence degenerates, which
gives natural isomorphisms for all $i\geq 0$ (cf. [Har77, Exercise III.8.1]):

(2.2)
$H^{i}(X,f_{*}(\mathcal{F})\otimes_{\mathcal{O}_{X}}\mathcal{E})\xrightarrow{\cong}H^{i}(Y,\mathcal{F}\otimes_{\mathcal{O}_{Y}}f^{*}\mathcal{E}).$

(b) This is proved in [Kaw88, Lemma 1.1], see also [KM92, Propositions 12.1.4,
12.1.6]. Let us recall the argument. Denote the resolution of singularities of
$X$ by $f\colon\widetilde{X}\rightarrow X$. By assumption
$R^{j}f_{*}(\mathcal{O}_{\widetilde{X}})=0$ for $j\geq 1$, so since
$f_{*}\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{X}$, for every locally free
sheaf of finite rank $\mathcal{E}$ on $X$, (2.2) gives natural isomorphisms

(2.3)
$H^{i}(X,\mathcal{E})\xrightarrow{\cong}H^{i}(\widetilde{X},f^{*}\mathcal{E}),\quad{i\geq
0}.$

Let $f_{\mathrm{an}}\colon\widetilde{V}\rightarrow V$ be the associated
holomorphic map between the respective analytifications of $\widetilde{X}$ and
$X$. (We note that if $X$ is projective then one may argue with $f$, without
passing to $f_{\mathrm{an}}$). By [Kaw88, Lemma 1.10],
$\operatorname{Cl}(X)/\operatorname{Pic}(X)$ embeds into
$\operatorname{Cl}(V)/\operatorname{Pic}(V)$. The groups of principal divisors
on $\widetilde{V}$ and $V$ are naturally isomorphic and we have a surjection
$\operatorname{Cl}(\widetilde{V})\rightarrow\operatorname{Cl}(V)$, hence

$\operatorname{Cl}(V)/\operatorname{Pic}(V)\cong\operatorname{Cl}(\widetilde{V})/f_{\mathrm{an}}^{*}\operatorname{Pic}(V)\cong\operatorname{Pic}(\widetilde{V})/f_{\mathrm{an}}^{*}\operatorname{Pic}(V).$

The exponential sequence induces an exact sequence which reads as

$H^{1}(V,\mathcal{O}_{V})\rightarrow
H^{1}(V,\mathcal{O}_{V}^{*})\cong\operatorname{Pic}(V)\rightarrow
H^{2}(V,\mathbb{Z})\rightarrow H^{2}(V,\mathcal{O}_{V}),$

and similarly for $\widetilde{V}$. By the analytic analogue of (2.3),
$f_{\mathrm{an}}$ induces isomorphisms $H^{i}(V,\mathcal{O}_{V})\cong
H^{i}(\widetilde{V},\mathcal{O}_{\widetilde{V}})$, so comparing the exact
sequences for $V$ and $\widetilde{V}$ we obtain an embedding

$\operatorname{Pic}(\widetilde{V})/f_{\mathrm{an}}^{*}\operatorname{Pic}(V)\hookrightarrow
H^{2}(\widetilde{V},\mathbb{Z})/f_{\mathrm{an}}^{*}H^{2}(V,\mathbb{Z}).$

Since $\widetilde{V}$ is the analytification of a smooth algebraic variety,
the group $H^{2}(\widetilde{V},\mathbb{Z})$ is finitely generated (see e.g.
[Dim92, 6.10 Corollary]) and so,
$\operatorname{Pic}(\widetilde{V})/f_{\mathrm{an}}^{*}\operatorname{Pic}(V)$
is finitely generated.

(c) Put $\widetilde{\mathcal{N}}=f^{*}\mathcal{N}$. By (2.3),
$H^{i}(\widetilde{X},\widetilde{\mathcal{N}}^{\vee})\cong
H^{i}(X,\mathcal{N}^{\vee})$. Since $\widetilde{\mathcal{N}}$ is big, nef and
invertible, $H^{i}(\widetilde{X},\widetilde{\mathcal{N}}^{\vee})=H^{\dim
X-i}(\widetilde{X},\mathcal{O}(K_{\widetilde{X}})\otimes\widetilde{\mathcal{N}})=0$
for $i<\dim X$ by Serre’s duality and the Kawamata-Viehweg vanishing theorem
for smooth varieties. ∎

###  2B. Inversion of adjunction and a $\mathbb{Q}$-factorial terminalization

We recall the following version of adjunction and inversion of adjunction, see
[KM98, Remark 5.47 and Theorem 5.50], cf. [Kol13, Chapter 4] and [FA92,
Chapters 16 and 17].

######  Lemma 2.3 (Inversion of adjunction).

Let $(X,S)$ be a log pair such that $S$ is a normal prime Weil divisor which
is Cartier in codimension 2. Then the adjunction formula
$K_{S}=(K_{X}+S)|_{S}$ holds. Furthermore, $(X,S)$ is plt in a neighborhood of
$S$ if and only if $S$ is klt.

A _$\mathbb{Q}$ -factorial terminalization_ of a normal quasi-projective
variety $X$ is a proper birational morphism $f\colon X^{\prime}\rightarrow X$
such that $X^{\prime}$ is a quasi-projective $\mathbb{Q}$-factorial terminal
variety and $K_{X^{\prime}}$ is $f$-nef.

######  Lemma 2.4 ($\mathbb{Q}$-factorial terminalization).

Every normal quasi-projective variety $X$ has a $\mathbb{Q}$-factorial
terminalization $f\colon X^{\prime}\rightarrow X$. Furthermore, the
restriction of $f$ over every $\mathbb{Q}$-factorial terminal open subset of
$X$ is an isomorphism.

###### Proof.

By [Kol13, Theorem 1.33] there exists a proper birational morphism $g\colon
Y\rightarrow X$ such that $Y$ is quasi-projective and terminal and $K_{Y}$ is
$g$-nef. By [Kol13, Corollary 1.37] there exists a proper birational morphism
$h\colon X^{\prime}\rightarrow Y$ such that $X^{\prime}$ is quasi-projective
terminal and $\mathbb{Q}$-factorial and $h$ is small, i.e. does not contract
any divisor. Then $h\circ g$ is a $\mathbb{Q}$-factorial terminalization.

Given an open subset $U\subseteq X$, the restriction $f|_{f^{-1}(U)}$ is a
$\mathbb{Q}$-factorial terminalization of $U$, so without loss of generality
we may assume that $U=X$. By assumption $K_{X^{\prime}}$ is $f$-nef, so the
divisor $K_{X^{\prime}}-f^{*}K_{X}=\sum_{E\in\mathcal{E}(f)}a_{X}(E)E$ is
$f$-nef. Since $X^{\prime}$ is $\mathbb{Q}$-factorial and the latter divisor
is contracted by $f$, the Negativity Lemma [KM98, 3.39(1)] gives $a_{X}(E)\leq
0$ for each $E\in\mathcal{E}(f)$. Since $X$ is terminal, we infer that
$\mathcal{E}(f)=\emptyset$, that is, $\operatorname{Exc}(f)$ has codimension
at least $2$. But $X$ is also $\mathbb{Q}$-factorial, so [Kol96, VI.1, Theorem
1.5] implies that $\operatorname{Exc}(f)=\emptyset$. Thus $f$ is an
isomorphism. ∎

###  2C. Fano varieties and Mori fiber spaces

######  Definition 2.5 (Fano variety and its index).

A _Fano variety_ is a normal projective variety whose anti-canonical divisor
is ample (in particular, $\mathbb{Q}$-Cartier). For a Fano variety $X$ with
class group $\operatorname{Cl}(X)\cong\mathbb{Z}$ the _Fano index_ of $X$ is
the positive integer $i_{X}$ such that $-K_{X}\sim i_{X}H$ for some ample
generator $H$ of $\operatorname{Cl}(X)$.

Recall [Ful98, §2.1 and Example 19.1.4] that for a (possibly non-normal)
complete algebraic variety $X$ the quotient $\operatorname{NS}(X)$ of the
Picard group of $X$ by the relation of numerical equivalence is a finitely
generated free abelian group, whose rank $\rho(X)$ is called the _Picard rank_
of $X$. For a surjective morphism of complete varieties $f\colon X\rightarrow
B$ we put $\rho(X/B)=\rho(X)-\rho(B)$.

A morphism $f\colon X\rightarrow B$ between quasi-projective varieties is
called a _contraction_ if it is proper, surjective and
$f_{*}\mathcal{O}_{X}=\mathcal{O}_{B}$. The latter condition implies
connectedness of fibers of $f$ and in case $B$ is normal it is equivalent to
it.

######  Remark 2.6 (Contractions from varieties with $\rho=1$).

A projective variety of Picard rank one has only trivial contractions. Indeed,
let $f\colon X\rightarrow B$ be a contraction from such a variety onto a
positive dimensional variety $B$. Note that since $f$ is proper and $B$ is
quasi-projective, $B$ is projective. Let $H$ be an effective ample Cartier
divisor on $B$. By Kleiman’s criterion a divisor numerically equivalent to an
ample divisor is ample, so since $\rho(X)=1$, $f^{*}H$ is an effective ample
Cartier divisor on $X$. For every irreducible curve $C$ on $X$, it follows
from the projection formula [Ful98, Proposition 2.3(c)] that $H\cdot
f_{*}(C)=(f^{*}H)\cdot C>0$. Thus, $f\colon X\rightarrow B$ is a proper
morphism which does not contract any curve, hence is a finite morphism. Since
$f_{*}\mathcal{O}_{X}=\mathcal{O}_{B}$ by assumption, $f$ is an isomorphism.

We note the following information on the geometry and topology of klt Fano
varieties. Recall that a variety is _rationally connected_ if every two points
on it are connected by a rational curve contained in the variety.

######  Remark 2.7 (Basic geometry of Fano varieties).

Every klt Fano $X$ variety is simply connected, rationally connected and
linear equivalence on it coincides with numerical and homological equivalence.
In particular, $\operatorname{Pic}X\cong
H^{2}(X,\mathbb{Z})\cong\operatorname{NS}(X)$.

###### Proof.

The Kawamata-Viehweg vanishing on klt proper varieties, see [KM98, Theorem
2.70] or [Mat02, Theorem 5.2.7], implies the vanishing of
$H^{i}(X,\mathcal{O}_{X})$ for all $i>0$. Now the natural isomorphism
$\operatorname{Pic}(X)\cong H^{2}(X,\mathbb{Z})$ follows from the exact
sequence associated to the exponential sequence, and the isomorphism
$H^{2}(X,\mathbb{Z})\cong\operatorname{NS}(X)$ from Riemann-Roch for a
numerically trivial divisor, see [IP99, Proposition 2.1.2]. Simple
connectedness of $X$ was proved in [Tak00]. Rational connectedness is proved
in [Zha06, Theorem 1]. Let us recall how to infer simple connectedness from
rational connectedness. By [Kol95, Theorem 4.13] the fundamental group of
every rationally (chain) connected variety is finite, so the universal cover
$f\colon\widetilde{X}\rightarrow X$ is algebraic. Since $f$ is finite,
$-f^{*}K_{X}$ is ample, and since $f$ is étale, we have
$K_{\widetilde{X}}=f^{*}K_{X}$. We infer that $\widetilde{X}$ is a Fano
variety. It is also klt (see [Kol13, Proposition 2.15]), so
$\chi(X,\mathcal{O}_{X})=\chi(\widetilde{X},\mathcal{O}_{\tilde{X}})=1$. Hence
by the multiplicativity of the Euler characteristic [Ful98, Example 18.3.9],
$|\pi_{1}(X)|=\deg
f=\chi(\widetilde{X},\mathcal{O}_{\widetilde{X}})/\chi(X,\mathcal{O}_{X})=1$.
∎

######  Definition 2.8 (Mori fiber spaces and completions).

  1. (a)

A _Mori fiber space_ is a $\mathbb{Q}$-factorial terminal projective variety
$X$ endowed with a contraction $f\colon X\rightarrow B$ onto a lower-
dimensional normal variety $B$ such that $\rho(X/B)=1$ and $-K_{X}$ is
$f$-ample.

  2. (b)

Two Mori fiber spaces $f_{i}\colon X_{i}\rightarrow B_{i}$, $i=1,2$, are
called _weakly square birational equivalent_ if there exist birational maps
$\varphi\colon X_{1}\dashrightarrow X_{2}$ and $\varphi^{\prime}\colon
B_{1}\dashrightarrow B_{2}$ such that $f_{2}\circ\varphi=\varphi^{\prime}\circ
f_{1}$.

  3. (c)

Given a quasi-projective variety $U$, a _Mori fiber completion of $U$_ is a
Mori fiber space whose total space is a completion of $U$.

It follows from the definition that general fibers of a Mori fiber space are
Fano varieties. Since the total space is assumed to be terminal, by [Kol97,
Proposition 7.7] general fibers are terminal too. We note that weakly square
birational equivalent Mori fiber spaces are _square birational equivalent_ in
the sense of [Cor00, Definition 1.2] if the induced morphism on generic fibers
is an isomorphism.

We have the following notion of rigidity of varieties. See [Che05] and [Puk13]
for related results.

######  Definition 2.9 (Birationally rigid varieties).

A Fano variety is called _birationally rigid_ if it has no birational maps to
Mori fiber spaces other than its own birational automorphisms. It is called
_super-rigid_ if additionally all its birational automorphisms are regular.

In particular, positive-dimensional birationally rigid varieties are non-
rational. We have the following analogous affine notions, see [CDP18].

######  Definition 2.10 (Affine Fano varieties).

An _affine Fano variety_ is an affine variety which admits a completion by a
purely log terminal log pair $(X,S)$ such that $X$ is a (normal projective)
$\mathbb{Q}$-factorial variety of Picard rank one, $S$ is prime and
$-(K_{X}+S)$ is ample.

######  Definition 2.11 (Affine super-rigid Fano varieties).

An affine Fano variety $U$ is _super-rigid_ if it satisfies the following
conditions:

  1. (a)

$U$ does not contain Zariski open subsets which are relative affine Fano
varieties over varieties of positive dimension.

  2. (b)

For every completion $(X,S)$ of $U$ and every log pair
$(X^{\prime},S^{\prime})$ as in Definition 2.10, if there exists an
isomorphism $U\cong X^{\prime}\setminus S^{\prime}$ then it extends to an
isomorphism $X\cong X^{\prime}$ mapping $S$ onto $S^{\prime}$.

Note that $\mathbb{A}^{1}$ is the only affine Fano curve and it is super-
rigid. It follows from the definition that a super-rigid affine Fano variety
of dimension $\geq 2$ does not contain open $\mathbb{A}^{1}$-cylinders, that
is, open subsets isomorphic to the product of $\mathbb{A}^{1}$ with a variety
of smaller dimension.

##  3\. Pencils and their resolutions

We recall the correspondence between dominant rational maps $\psi\colon
X\dashrightarrow\mathbb{P}^{1}$ on a normal variety $X$ and linear systems of
Weil divisors of projective dimension $1$ on $X$. In the smooth case it
restricts to the well-known correspondence between such maps and one-
dimensional linear systems of Cartier divisors, see e.g. [Dol12]. For this
purpose we use the correspondence between Weil divisors and coherent reflexive
sheaves of rank one, see [Ish18, Section 5.2], [Rei80] or [BS13, Appendix].
For a sheaf $\mathcal{F}$ on $X$ we denote the sheaf
$\operatorname{\mathcal{H}om}_{\mathcal{O}_{X}}(\mathcal{F},\mathcal{O}_{X})$,
dual to $\mathcal{F}$, by $\mathcal{F}^{\vee}$.

###  3A. Divisorial sheaves

Let $X$ be a normal variety and let $\mathcal{K}_{X}$ be its sheaf of rational
functions. For a Weil divisor $D$ on $X$ the _divisorial sheaf associated to_
$D$ is the unique subsheaf of $\mathcal{O}_{X}$-modules of $\mathcal{K}_{X}$
whose sections over every open subset $U$ of $X$ are

$\mathcal{O}_{X}(D)(U)=\\{f\in\mathcal{K}_{X}^{*},\
\operatorname{div}(f)|_{U}+D|_{U}\geq 0\\}\cup\\{0\\}.$

The sheaf $\mathcal{O}_{X}(D)$ is a coherent reflexive sheaf of rank one. It
is invertible if and only if $D$ is Cartier. If $D$ is effective then
$\mathcal{O}_{X}(-D)=\mathcal{O}_{X}(D)^{\vee}$ is the ideal sheaf
$\mathcal{I}_{D}$ of $D$, which is a coherent reflexive subsheaf of
$\mathcal{O}_{X}$ or rank one. Conversely, every coherent reflexive sheaf of
rank one $\mathcal{F}$ on $X$ embeds into $\mathcal{K}_{X}$ and for each
embedding $i\colon\mathcal{F}\hookrightarrow\mathcal{K}_{X}$ there is a unique
Weil divisor $D$ on $X$ such that $\operatorname{Im}(i)=\mathcal{O}_{X}(D)$.
We henceforth use the term _divisorial sheaf_ to refer to any coherent
reflexive sheaf or rank one on $X$. We note that
$\mathcal{O}_{X}(0)=\mathcal{O}_{X}$, because on a normal variety a rational
function with no poles is regular, [Mat89, Theorem 11.5]. More generally,
given a divisorial sheaf $\mathcal{F}$ and an open subset $j\colon
U\hookrightarrow X$ such that $\operatorname{codim}(X\setminus U)\geq 2$, the
natural homomorphism $\mathcal{F}\rightarrow j_{*}(\mathcal{F}|_{U})$ is an
isomorphism.

The correspondence $D\mapsto\mathcal{O}_{X}(D)$ induces an isomorphism between
the class group $\operatorname{Cl}(X)$ of $X$ and the set of isomorphism
classes of divisorial sheaves endowed with the group law defined by
$\mathcal{F}\hat{\otimes}\mathcal{F}^{\prime}=(\mathcal{F}\otimes\mathcal{F}^{\prime})^{\vee\vee}$.
In case $\mathcal{F}$ or $\mathcal{F}^{\prime}$ is invertible, there is a
canonical isomorphism
$\mathcal{F}\otimes\mathcal{F}^{\prime}\cong(\mathcal{F}\otimes\mathcal{F}^{\prime})^{\vee\vee}$.
The inclusion of the smooth locus $j\colon X_{\mathrm{reg}}\hookrightarrow X$
induces and isomorphism
$j^{*}\colon\operatorname{Cl}(X)\rightarrow\operatorname{Cl}(X_{\mathrm{reg}})\cong\operatorname{Pic}(X_{\mathrm{reg}})$,
whose inverse is given by associating to an invertible sheaf $\mathcal{N}$ on
$X_{\mathrm{reg}}$ the divisorial sheaf $(j_{*}\mathcal{N})^{\vee\vee}$ on
$X$. The canonical sheaf of $X$ is
$\omega_{X}=(j_{*}\det(\Omega^{1}_{X_{\mathrm{reg}}}))^{\vee\vee}\cong\mathcal{O}_{X}(K_{X})$.

######  Example 3.1 (The affine quadric cone in $\mathbb{A}^{3}$).

On the affine quadric cone $X=\\{x^{2}-yz=0\\}\subseteq\mathbb{A}^{3}$
consider the prime Weil divisor $D=\\{x=y=0\\}$ and the associated divisorial
sheaf $\mathcal{O}_{X}(D)$. Put $R=\mathbb{C}[X]$. We have

$H^{0}(X,\mathcal{O}_{X}(-D))=R\langle x,y\rangle\quad\text{and}\quad
H^{0}(X,\mathcal{O}_{X}(D))=R\langle 1,\frac{z}{x}\rangle.$

Since the ideal $(x,y)$ is non-principal, $D$ is not Cartier. It is
$\mathbb{Q}$-Cartier, because $\operatorname{div}(y)=2D$. Put
$D^{\prime}=\\{z=x=0\\}$ and $s=\tfrac{z}{x}$. We have
$\operatorname{div}(s)=\operatorname{div}(z)-\operatorname{div}(x)=2D^{\prime}-(D^{\prime}+D)=D^{\prime}-D$.
In particular, $s\in H^{0}(X,\mathcal{O}_{X}(D))$ and $D^{\prime}\sim D$. Note
that the natural $\mathcal{O}_{X}$-linear homomorphism
$\mathcal{O}_{X}(-D)\otimes\mathcal{O}_{X}(-D)\rightarrow\mathcal{O}_{X}(-2D)$
is not an isomorphism, as on global sections it gives the inclusion of ideals
$(xy,y^{2},x^{2})=(y)\cdot(x,y,z)\hookrightarrow(y)$.

Let now $\mathbb{P}$ be the weighted projective plane $\mathbb{P}(1,1,2)$ with
weighted homogeneous coordinates $x_{0}$, $x_{1}$, $x_{2}$ and let
$H=\\{x_{0}=0\\}$; see Section 6C. Since $H\cong\mathbb{P}^{1}$ is irreducible
and $\mathbb{P}\setminus H\cong\mathbb{A}^{2}$ has a trivial class group, the
class group of $\mathbb{P}$ is isomorphic to $\mathbb{Z}$ and is generated by
$H$. We see that $2H$ is Cartier. But $H$ is not. Indeed, after restricting to
the open subset $U=\mathbb{P}\setminus\\{x_{2}=0\\}$ we have
$(\mathbb{P}|_{U},H|_{U})\cong(X,D)$. The Picard group of $\mathbb{P}$ is
generated by the class of $2H$.

###  3B. Pencils of Weil divisors

Let $X$ be a normal variety and let $\psi\colon
X\dashrightarrow\mathbb{P}^{1}$ be a dominant rational map. Let $j\colon
U=\operatorname{dom}(\psi)\hookrightarrow X$ be the inclusion of the domain of
definition of $\psi$ and let $\psi_{U}:=\psi|_{U}\colon
U\rightarrow\mathbb{P}^{1}$. Since $X$ is normal, $X\setminus U$ is a closed
subset of codimension $\geq 2$ of $X$. Put
$V=H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(1))$. The invertible
sheaf $\mathcal{N}=\psi_{U}^{*}\mathcal{O}_{\mathbb{P}^{1}}(1)$ on $U$ extends
to a unique divisorial sheaf $\mathcal{F}=(j_{*}\mathcal{N})^{\vee\vee}$ on
$X$ and the sections $\psi_{U}^{*}(s)$, $s\in V$, extend uniquely to global
sections of $\mathcal{F}$. The obtained homomorphism $\psi^{*}\colon
V\rightarrow H^{0}(X,\mathcal{F})$ is injective and its image is a
$2$-dimensional linear subspace $\mathcal{L}\subseteq H^{0}(X,\mathcal{F})$.

The scheme-theoretic fibers $U_{p}=\psi_{U}^{*}(p)$ of $\psi_{U}$ over the
closed points $p$ of
$\mathbb{P}^{1}=\operatorname{Proj}(\operatorname{Sym}^{\bm{\cdot}}(V))$ are
linearly equivalent Cartier divisors on $U$ for which
$\mathcal{O}_{U}(U_{p})\cong\psi_{U}^{*}\mathcal{O}_{\mathbb{P}^{1}}(1)$. For
every $p\in\mathbb{P}^{1}$ we denote by $\mathcal{L}_{p}$ the scheme-theoretic
closure of $\psi_{U}^{*}(p)$ in $X$. Since $X$ is normal and
$\operatorname{codim}_{X}(X\setminus U)\geq 2$, these are linearly equivalent
Weil divisors on $X$ for which
$\mathcal{O}_{X}(\mathcal{L}_{p})\cong\mathcal{F}$. Denote by
$\mathcal{I}_{U_{p}}\subseteq\mathcal{O}_{U}$ the ideal sheaf of $U_{p}$. Then
the ideal sheaf $\mathcal{I}_{\mathcal{L}_{p}}\subseteq\mathcal{O}_{X}$ of
$\mathcal{L}_{p}$ is equal to
$(j_{*}\mathcal{I}_{U_{p}})^{\vee\vee}\cong\mathcal{O}_{X}(-\mathcal{L}_{p})\cong\mathcal{F}^{\vee}$.

In what follows, we call the subspace $\mathcal{L}\subseteq
H^{0}(X,\mathcal{F})$ the _pencil of (Weil) divisors on $X$_ defining the
rational map $\psi\colon X\dashrightarrow\mathbb{P}^{1}$. The divisors
$\mathcal{L}_{p}$, $p\in\mathbb{P}^{1}$ are called the _members_ of
$\mathcal{L}$. The _base scheme_ of the pencil, denoted by
$\operatorname{Bs}\mathcal{L}$, is the scheme-theoretic intersection in $X$ of
all members of the pencil.

######  Proposition 3.2 (Pencils and their associated rational maps).

Let $X$ be a normal algebraic variety. There exists a natural bijection
between the set of dominant rational maps $\psi\colon
X\dashrightarrow\mathbb{P}^{1}$ and the set of equivalence classes of pairs
$(\mathcal{F},\mathcal{L})$, where $\mathcal{F}$ is a divisorial sheaf on $X$
and $\mathcal{L}\subseteq H^{0}(X,\mathcal{F})$ is a $2$-dimensional space of
global sections generating $\mathcal{F}$ off a closed subset of codimension
$\geq 2$, where two such pairs $(\mathcal{F},\mathcal{L})$ and
$(\mathcal{F}^{\prime},\mathcal{L}^{\prime})$ are equivalent if there exists
an isomorphism $\alpha\colon\mathcal{F}\rightarrow\mathcal{F}^{\prime}$ for
which $H^{0}(\alpha)(\mathcal{L})=\mathcal{L}^{\prime}$.

###### Proof.

We already described above how to associate to a dominant rational map
$\psi\colon X\dashrightarrow\mathbb{P}^{1}$ a pair
$(\mathcal{F},\mathcal{L})=(\mathcal{F}_{\psi},\mathcal{L}_{\psi}):=((j_{*}\mathcal{N})^{\vee\vee},\psi^{*}V)$.
Consider the natural homomorphism

(3.1)
$\mathrm{e}\colon\mathcal{L}\otimes_{\mathbb{C}}\mathcal{O}_{X}\rightarrow\mathcal{F}$

defined by restricting global sections to stalks of $\mathcal{F}$. By
construction, the restriction of $\psi$ to $U=\operatorname{dom}(\psi)$ is
isomorphic to the pull-back by $\psi_{U}\colon U\rightarrow\mathbb{P}^{1}$ of
the canonical surjection
$V\otimes_{\mathbb{C}}\mathcal{O}_{\mathbb{P}^{1}}\rightarrow\mathcal{O}_{\mathbb{P}^{1}}(1)$.
Thus the sections contained in $\mathcal{L}$ generate $\mathcal{F}$ outside
the indeterminacy locus $X\setminus U$ of $\psi$, which is a closed subset of
codimension $\geq 2$ of $X$.

Conversely, let $\mathcal{F}$ be a divisorial sheaf on $X$ and let
$\mathcal{L}\subseteq H^{0}(X,\mathcal{F})$ be a $2$-dimensional space of
global sections such that the support $Z$ of the cokernel of the homomorphism
(3.1) has codimension $\geq 2$ in $X$. Since $\mathcal{F}$ is divisorial and
$X$ is normal, the set $\mathcal{F}_{\mathrm{sing}}$ of points $x$ of $X$ such
that $\mathcal{F}_{x}$ is not a free $\mathcal{O}_{X,x}$-module is a closed
subset of codimension $\geq 2$. Thus,
$W:=W(\mathcal{F},\mathcal{L})=X\setminus(Z\cup\mathcal{F}_{\mathrm{sing}})$
is an open subset of $X$ with a complement of codimension $\geq 2$, on which
$\mathrm{e}$ restricts to a surjection
$\mathrm{e}|_{W}\colon\mathcal{O}_{W}^{\oplus
2}\cong\mathcal{L}\otimes_{\mathbb{C}}\mathcal{O}_{W}\rightarrow\mathcal{F}|_{W}$
onto the invertible sheaf $\mathcal{F}|_{W}$. By [Har77, Proposition 7.12]
there exists a unique dominant morphism $f=f_{\mathcal{F},\mathcal{L}}\colon
W\rightarrow\mathbb{P}^{1}$ such that $\mathrm{e}|_{W}$ is equal to the pull-
back by $f$ of the canonical surjection
$V\otimes_{\mathbb{C}}\mathcal{O}_{\mathbb{P}^{1}}\rightarrow\mathcal{O}_{\mathbb{P}^{1}}(1)$.
This morphism determines in turn a unique rational map $\psi\colon
X\dashrightarrow\mathbb{P}^{1}$ whose domain of definition $U$ contains $W$
and for which $\psi_{U}|_{W}=f$.

Two pairs $(\mathcal{F},\mathcal{L})$ and
$(\mathcal{F}^{\prime},\mathcal{L}^{\prime})$ determine the same dominant
rational map $\psi\colon X\dashrightarrow\mathbb{P}^{1}$ if and only if their
associated morphisms $f_{\mathcal{F},\mathcal{L}}\colon
W(\mathcal{F},\mathcal{L})\rightarrow\mathbb{P}^{1}$ and
$f_{\mathcal{F}^{\prime},\mathcal{L}^{\prime}}\colon
W(\mathcal{F}^{\prime},\mathcal{L}^{\prime})\rightarrow\mathbb{P}^{1}$
coincide on the open subset $\widetilde{W}=W(\mathcal{F},\mathcal{L})\cap
W(\mathcal{F}^{\prime},\mathcal{L}^{\prime})$. This is in turn equivalent to
the existence of an isomorphism
$\alpha_{\widetilde{W}}\colon\mathcal{F}|_{\widetilde{W}}\rightarrow\mathcal{F}^{\prime}|_{\widetilde{W}}$
of sheaves on $\widetilde{W}$ which maps the global sections
$s|_{\widetilde{W}}$ of $\mathcal{F}|_{\widetilde{W}}$, $s\in\mathcal{L}$,
bijectively onto the global sections $s^{\prime}|_{\widetilde{W}}$ of
$\mathcal{F}^{\prime}_{\widetilde{W}}$, $s^{\prime}\in\mathcal{L}^{\prime}$.
Since $X$ is normal, $\operatorname{codim}_{X}(X\setminus\widetilde{W})\geq 2$
and $\mathcal{F}$ and $\mathcal{F}^{\prime}$ are reflexive,
$\alpha_{\widetilde{W}}$ uniquely extends to an isomorphism
$\alpha\colon\mathcal{F}\rightarrow\mathcal{F}^{\prime}$ of sheaves over $X$
such that $H^{0}(\alpha)(\mathcal{L})=\mathcal{L}^{\prime}$. So the
association $(\mathcal{F},\mathcal{L})\mapsto\psi_{\mathcal{F},\mathcal{L}}$
induces a well-defined injective map from the set of equivalence classes of
pairs $(\mathcal{F},\mathcal{L})$ to the set of dominant rational maps
$\psi\colon X\dashrightarrow\mathbb{P}^{1}$. This map is also surjective,
because the equality $\psi=\psi_{\mathcal{F}_{\psi},\mathcal{L}_{\psi}}$ holds
for every dominant rational map $\psi\colon X\dashrightarrow\mathbb{P}^{1}$. ∎

By a _resolution of $\mathcal{L}$_ we mean a resolution of the associated
rational map $\psi$. Given two linearly equivalent Weil divisors $D$ and
$D^{\prime}$ on $X$ without common irreducible component, the _pencil $\langle
D,D^{\prime}\rangle$ generated by $D$ and $D^{\prime}$_ is the pencil of
divisors on $X$, unique up to an isomorphism, which has $D$ and $D^{\prime}$
among its members. Its base scheme is equal to the scheme-theoretic
intersection of $D$ and $D^{\prime}$.

###  3C. The graph of a pencil

Let $\psi\colon X\dashrightarrow\mathbb{P}^{1}$ be a dominant rational map on
a normal variety $X$. The _graph of $\psi$_ is the scheme-theoretic closure
$\Gamma\subseteq X\times\mathbb{P}^{1}$ of the graph of the restriction of
$\psi$ to its domain of definition. We let $\gamma\colon\Gamma\rightarrow X$
and $\operatorname{p}\colon\Gamma\rightarrow\mathbb{P}^{1}$ be the
restrictions of the projections from $X\times\mathbb{P}^{1}$ onto its factors.
We obtain a commutative diagram

(3.2)
$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\operatorname{p}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\mathbb{P}^{1}}$

The proper birational morphism $\gamma\colon\Gamma\rightarrow X$ provides a
natural resolution of $\psi$ such that $\psi\circ\gamma=\operatorname{p}$. The
next proposition collects properties of this resolution.

######  Proposition 3.3 (Properties of the graph resolution).

Let $X$ be a normal variety. Let $\psi\colon X\dashrightarrow\mathbb{P}^{1}$
be a dominant rational map and let $\mathcal{L}$ be the associated pencil of
divisors. Then the following hold:

  1. (a)

For every resolution $\tau\colon X^{\prime}\rightarrow X$ of $\psi$ the
birational map $\gamma^{-1}\circ\tau\colon X^{\prime}\dashrightarrow\Gamma$ is
a proper morphism.

  2. (b)

The indeterminacy locus of $\psi$ is equal to the support of the base scheme
$\operatorname{Bs}\mathcal{L}$.

  3. (c)

The morphism $\gamma$ restricts to an isomorphism over
$X\setminus\operatorname{Bs}\mathcal{L}$, and
$\gamma^{-1}(x)\cong\mathbb{P}^{1}$ for every
$x\in\operatorname{Bs}\mathcal{L}$.

  4. (d)

For every point $p\in\mathbb{P}^{1}$ the birational morphism
$\gamma_{p}\colon\operatorname{p}^{*}(p)\rightarrow\mathcal{L}_{p}$ induced by
$\gamma$ is an isomorphism.

###### Proof.

Set $U=\operatorname{dom}(\psi)$ and let $\operatorname{pr}_{X}\colon
X\times\mathbb{P}^{1}\rightarrow X$ denote the projection.

(a) Since $\psi\circ\tau$ is a morphism, we have a morphism
$\tau\times(\psi\circ\tau)\colon X^{\prime}\rightarrow X\times\mathbb{P}^{1}$.
Since $X^{\prime}$ is irreducible and $\tau$ is surjective, the image is
contained in $\Gamma$, so we may write this morphism as a composition of some
morphism $\tau^{\prime}\colon X^{\prime}\rightarrow\Gamma$ with the closed
immersion $\Gamma\hookrightarrow X\times\mathbb{P}^{1}$. Both morphisms
$\gamma$ and $\tau=\gamma\circ\tau^{\prime}$ are proper, so $\tau^{\prime}$ is
proper too.

(b) With the notation of the proof of Proposition 3.2, $\psi$ is represented
by the morphism $f\colon W\rightarrow\mathbb{P}^{1}$ associated to the
restriction of
$\mathrm{e}\colon\mathcal{L}\otimes_{\mathbb{C}}\mathcal{O}_{X}\rightarrow\mathcal{F}$
to the open subset $W=X\setminus(\operatorname{Supp}(\operatorname{Coker}\
e)\cup\mathcal{F}_{\operatorname{Sing}})$. Let $x$ be a point of $X$ which is
not contained in the support of some member $\mathcal{L}_{p}$. Then
$\mathcal{I}_{\mathcal{L}_{p},x}=\mathcal{O}_{X,x}$, and since
$\mathcal{I}_{\mathcal{L}_{p}}\cong\mathcal{F}^{\vee}$, it follows that
$\mathcal{F}^{\vee}_{x}$, and hence $\mathcal{F}_{x}$ is a free
$\mathcal{O}_{X,x}$-module. Thus
$\mathcal{F}_{\operatorname{Sing}}\subseteq\operatorname{Supp}(\operatorname{Coker}\
e)$ and we have $U=X\setminus\operatorname{Supp}(\operatorname{Coker}\
e)=X\setminus\operatorname{Supp}(\operatorname{Bs}\mathcal{L})$.

(c) By the definition of $\Gamma$, $\gamma$ restricts to an isomorphism over
$U$. Suppose that for some $x\in\operatorname{Bs}\mathcal{L}$,
$\operatorname{pr}_{X}^{-1}(x)\cong\mathbb{P}^{1}$ is not fully contained in
$\Gamma$. Since $X$ is normal and $\gamma\colon\Gamma\rightarrow X$ is proper
and birational, it follows from [Gro61, Proposition 4.4.1] that the fibers of
$\gamma$ are connected, and hence
$\gamma^{-1}(x)=\operatorname{pr}_{X}^{-1}(x)\cap\Gamma$ consists of a unique
point $y$, and that there exists an open neighborhood $V$ of $y$ such that
$\gamma|_{V}\colon V\rightarrow X$ is an open immersion. Thus the birational
map $\gamma^{-1}$ is defined at $x$, and so is
$\psi=\operatorname{p}\circ\gamma^{-1}$. But this is impossible, because
$x\in\operatorname{Bs}\mathcal{L}=X\setminus U$.

(d) The assertion is local over $X$, so we can assume without loss of
generality that $X$ is affine, say $X=\operatorname{Spec}(A)$. Then $\psi$ is
induced by some rational function $h\in\operatorname{Frac}(A)$. Given a
representative $h=f/g$, where $f,g\in A$, let $U_{(f,g)}=X\setminus V(f,g)$.
The restriction of $\psi$ to $U_{(f,g)}$ is given by $x\mapsto[f(x):g(x)]$.
For every point $p=[\lambda:\mu]\in\mathbb{P}^{1}$ the restriction of the
Cartier divisor $\psi_{U}^{*}(p)$ to $U_{(f,g)}$ is the zero scheme of the
regular function $s_{(f,g)}=\mu f-\lambda g$. Letting $\mathcal{S}=\\{(f,g)\in
A\times(A\setminus\\{0\\}),\;h=f/g\\}$ we have
$U=\bigcup_{(f,g)\in\mathcal{S}}U_{(f,g)}$. The scheme-theoretic closure
$\mathcal{L}_{p}\subseteq X$ of $\psi_{U}^{*}(p)$ is defined by the vanishing
of all functions $s_{(f,g)}$. On the other hand, the graph $\Gamma\subseteq
X\times\mathbb{P}^{1}_{[u:v]}$ is defined by the vanishing of all sections

$\widetilde{s}_{(f,g)}=fv-gu\in
H^{0}(X\times\mathbb{P}^{1},\mathcal{O}_{X\times\mathbb{P}^{1}}(1)),$

and hence $\operatorname{p}^{*}(p)\subseteq X\times\mathbb{P}^{1}$ is defined
by the vanishing of the section $\mu u-\lambda v$ and all
$\widetilde{s}_{(f,g)}$. From this, we see directly that
$\gamma_{p}\colon\operatorname{p}^{*}(p)\rightarrow X_{p}$ is an isomorphism.
∎

######  Remark 3.4.

Let $X$ be a normal variety and let $\mathcal{L}\subseteq
H^{0}(X,\mathcal{F})$ be a pencil. The defining ideal sheaf
$\mathcal{I}_{\mathcal{L}}$ of the base scheme $\operatorname{Bs}\mathcal{L}$
can be described as follows. Consider the homomorphism

(3.3)
$\operatorname{ev}\colon\mathcal{L}\otimes_{\mathbb{C}}\mathcal{F}^{\vee}\rightarrow\mathcal{O}_{X}$

obtained from the evaluation homomorphism
$\mathrm{e}\colon\mathcal{L}\otimes_{\mathbb{C}}\mathcal{O}_{X}\rightarrow\mathcal{F}$
(3.1) by tensoring it with $\mathcal{F}^{\vee}$ and composing with the
canonical homomorphism
$\mathcal{F}\otimes_{\mathcal{O}_{X}}\mathcal{F}^{\vee}\rightarrow\mathcal{O}_{X}$.
Then the ideal sheaf $\mathcal{J}=\operatorname{Im}(\operatorname{ev})$ is
generated by the ideal sheaves $\mathcal{I}_{s}$ of the zero schemes of the
sections $s$ of $\mathcal{F}$, $s\in\mathcal{L}$. On the other hand, the
members of $\mathcal{L}$ are, by definition, the Weil divisors associated to
the zero schemes of these sections, that is, the closed subschemes of $X$ with
defining ideal sheaves $\mathcal{I}_{s}^{\vee\vee}$, $s\in\mathcal{L}$. So
$\mathcal{I}_{\mathcal{L}}$ is generated by the ideal sheaves
$\mathcal{I}_{s}^{\vee\vee}$, $s\in\mathcal{L}$. It follows that
$\mathcal{J}\subseteq\mathcal{I}_{\mathcal{L}}$, with equality in case when
$\mathcal{F}$ is invertible. Indeed, if $\mathcal{F}$ is invertible then each
$\mathcal{I}_{s}$ is an invertible ideal sheaf, so
$\mathcal{I}_{s}=\mathcal{I}_{s}^{\vee\vee}$.

For a pencil $\mathcal{L}$ on a normal variety $X$, Proposition 3.3(b) says
that $\gamma\colon\Gamma\rightarrow X$ is a universal minimal resolution of
the dominant rational map $\psi\colon X\dashrightarrow\mathbb{P}^{1}$
determined by $\mathcal{L}$, in the sense that every resolution of $\psi$
factors through it. Another natural resolution of $\psi$ is given by the blow-
up $\tau\colon\operatorname{Bl}_{\operatorname{Bs}\mathcal{L}}(X)\rightarrow
X$ of the base scheme of $\mathcal{L}$, as shown in the following lemma. In
case when members of $\mathcal{L}$ are Cartier, it is a classical fact that
the induced birational morphism
$\gamma^{-1}\circ\tau\colon\operatorname{Bl}_{\operatorname{Bs}\mathcal{L}}(X)\rightarrow\Gamma$
is an isomorphism; see e.g. [Dol12, Proposition 7.12 and §7.1.3]. This is no
longer true in general for pencils whose members are not Cartier, see Example
4.2.

######  Lemma 3.5 (The blowup of $\operatorname{Bs}\mathcal{L}$ is a
resolution).

Let $X$ be a normal variety. Let $\psi\colon X\dashrightarrow\mathbb{P}^{1}$
be a dominant rational map determined by a pencil of divisors $\mathcal{L}$.
Then the blow-up
$\tau\colon\operatorname{Bl}_{\operatorname{Bs}\mathcal{L}}(X)\rightarrow X$
of the base scheme of $\mathcal{L}$ is a resolution of $\psi$.

###### Proof.

Let $\widetilde{X}=\operatorname{Bl}_{\operatorname{Bs}\mathcal{L}}(X)$. To
verify that $\psi\circ\tau\colon\widetilde{X}\dashrightarrow\mathbb{P}^{1}$ is
a morphism we can assume without loss of generality that
$X=\operatorname{Spec}(A)$ is affine and that $\psi$ is the rational map
defined by some rational function $h\in\operatorname{Frac}(A)$. With the
notation of the proof of Proposition 3.3(c), $\operatorname{Bs}\mathcal{L}$ is
the closed subscheme of $X$ whose defining ideal $I$ generated by the regular
functions $f$ and $g$ such that $(f,g)\in\mathcal{S}$. By definition of the
blow-up, the ideal sheaf
$\mathcal{J}=\tau^{-1}(I)\cdot\mathcal{O}_{\widetilde{X}}$ is invertible. It
is generated by all regular functions $\tau^{*}f$ and $\tau^{*}g$, where
$(f,g)\in\mathcal{S}$. Thus for every point $y_{0}\in\widetilde{X}$ there
exists an element $(f,g)\in\mathcal{S}$ such that the stalk
$\mathcal{J}_{y_{0}}$ is generated by $\tau^{*}f$ or $\tau^{*}g$, say
$\tau^{*}f$, the situation being symmetric for $\tau^{*}g$. It follows that
there exists an open neighborhood $V\subseteq\widetilde{X}$ of $y_{0}$ and a
regular function $\widetilde{g}$ on $V$ such that
$\tau^{*}g|_{V}=\widetilde{g}\tau^{*}f|_{V}$. On this neighborhood the
composition $\psi\circ\tau\colon\widetilde{X}\dashrightarrow\mathbb{P}^{1}$ is
given by

$y\mapsto[(\tau^{*}f)(y):(\tau^{*}g)(y)]=[1:\widetilde{g}(y)],$

so $y_{0}\in\operatorname{dom}(\psi\circ\tau)$. Thus
$\psi\circ\tau\colon\widetilde{X}\dashrightarrow\mathbb{P}^{1}$ is a morphism,
and hence $\tau\colon\widetilde{X}\rightarrow X$ is a resolution of
$\psi\colon X\dashrightarrow\mathbb{P}^{1}$. ∎

##  4\. Terminal rank one Fano pencils and associated Mori fiber spaces

Our strategy to construct Mori fiber completions of affine spaces is to use
pencils of Weil divisors on normal projective completions of
$\mathbb{A}^{n}$’s and to produce the desired Mori fiber spaces as outputs of
relative MMP’s ran from suitable resolutions of these pencils. For this
approach to work we need in particular to find properties of a variety $X$ and
of the members of the pencil whose combination guarantees that a specific open
subset is preserved under appropriate choices of resolutions and relative
MMP’s.

###  4A. $\mathbb{Q}$-factorial terminal resolutions

Let $X$ be a normal variety. Let $\mathcal{L}$ be a pencil of divisors
determining a dominant rational map $\psi\colon
X\dashrightarrow\mathbb{P}^{1}$. Let $\tau\colon X^{\prime}\rightarrow X$ be a
resolution of $\mathcal{L}$, that is, a proper birational morphism from a
variety such that $\psi\circ\tau$ is a morphism. By Proposition 3.3 there
exists a unique morphism $\Gamma(\tau)\colon X^{\prime}\rightarrow\Gamma$ such
that $\tau=\gamma\circ\Gamma(\tau)$ and a commutative diagram

(4.1)
$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma(\tau)}$$\scriptstyle{\tau}$$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\operatorname{p}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\mathbb{P}^{1}}$

where $(\Gamma,\gamma,\operatorname{p})$ are as in (4.1). Let
$\nu\colon\widetilde{\Gamma}\rightarrow\Gamma$ be the normalization of
$\Gamma$. We call $\widetilde{\Gamma}$ the _normalized graph_ of $\psi$. If
$X^{\prime}$ is normal then we have
$\Gamma(\tau)=\nu\circ\widetilde{\Gamma}(\tau)$ for some unique birational
proper morphism

$\widetilde{\Gamma}(\tau)\colon X^{\prime}\rightarrow\widetilde{\Gamma}.$

######  Definition 4.1 (A thrifty resolution).

Let $X$ be a normal variety and let $\tau\colon X^{\prime}\rightarrow X$ be a
resolution of a pencil $\mathcal{L}$ on $X$.

  1. (a)

We call the image $\delta(\tau)\subseteq\mathbb{P}^{1}$ of
$\operatorname{Exc}\Gamma(\tau)$ by $\operatorname{p}\circ\ \Gamma(\tau)$ the
_discrepancy locus of $\tau$_.

  2. (b)

We say that $\tau$ is _$\mathbb{Q}$ -factorial terminal_ if $X^{\prime}$ is
$\mathbb{Q}$-factorial terminal.

  3. (c)

We say that a $\mathbb{Q}$-factorial terminal resolution $\tau$ is _thrifty_
if $\widetilde{\Gamma}(\tau)\colon X^{\prime}\rightarrow\widetilde{\Gamma}$ is
a $\mathbb{Q}$-factorial terminalization.

The discrepancy locus is a rough measure of how much a given resolution of
$\psi\colon X\dashrightarrow\mathbb{P}^{1}$ differs from the (minimal) graph
resolution.

######  Example 4.2 (The affine cone in $\mathbb{A}^{4}$).

On the affine cone $X=\\{xv-yu=0\\}\subseteq\mathbb{A}^{4}$ the Weil divisors
$\mathcal{L}_{0}=\\{u=v=0\\}$ and
$\mathcal{L}_{\infty}=\\{x=y=0\\}=\mathcal{L}_{0}+\operatorname{div}(x/u)$
generate a pencil $\mathcal{L}$, whose base scheme
$\operatorname{Bs}\mathcal{L}$ is equal to the isolated singular point
$p=(0,0,0,0)\in X$. Since the latter has codimension $3$ in $X$,
$\mathcal{L}_{0}$ is not $\mathbb{Q}$-Cartier. In particular, $X$ is not
$\mathbb{Q}$-factorial. The associated rational map is

$\psi_{\mathcal{L}}\colon
X\dashrightarrow\mathbb{P}^{1}_{[w_{0}:w_{1}]},\quad(x,y,u,v)\mapsto[u:x]=[v:y].$

Its graph $\Gamma$ is isomorphic to the sub-variety of
$X\times\mathbb{P}^{1}_{[w_{0}:w_{1}]}$ defined by the equations

$xw_{0}-uw_{1}=0\quad\textrm{and}\quad yw_{0}-vw_{1}=0.$

The morphism $\operatorname{p}\colon\Gamma\rightarrow\mathbb{P}^{1}$ is a
locally trivial $\mathbb{A}^{2}$-bundle, so $\Gamma$ is smooth. The morphism
$\gamma\colon\Gamma\rightarrow X$ is a thrifty $\mathbb{Q}$-factorial terminal
resolution of $\mathcal{L}$ with an empty discrepancy locus. It is a small
resolution of the singularity $p\in X$ with the exceptional locus consisting
of a single curve
$\gamma_{\mathcal{L}}^{-1}(p)\cong\mathbb{P}^{1}_{[w_{0}:w_{1}]}$. It can be
also described as the blow-up of the ideal sheaf of $\mathcal{L}_{0}$.

On the other hand, the blow-up
$\tau_{\mathcal{L}}\colon\widetilde{X}\rightarrow X$ of
$\operatorname{Bs}\mathcal{L}=\\{p\\}$ is a resolution of the singularity of
$X$ with exceptional divisor
$\tau_{\mathcal{L}}^{-1}(p)\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$. In
particular, it is a $\mathbb{Q}$-factorial terminal resolution of
$\mathcal{L}$. The birational proper morphism
$\tau^{\prime}:=\Gamma(\tau_{\mathcal{L}})\colon\widetilde{X}\rightarrow\Gamma$
contracts $\tau_{\mathcal{L}}^{-1}(p)$ onto $\gamma^{-1}(p)$. Since the proper
transform by $\tau^{\prime}$ of every closed fiber of
$\operatorname{p}\colon\Gamma\rightarrow\mathbb{P}^{1}$ is isomorphic to the
blow-up of the origin in $\mathbb{A}^{2}$ the discrepancy locus of
$\tau_{\mathcal{L}}$ is equal to $\mathbb{P}^{1}$.

A $\mathbb{Q}$-factorial terminal resolution $\tau\colon X^{\prime}\rightarrow
X$ with a finite discrepancy locus induces isomorphisms between general fibers
of $\operatorname{p}\colon\Gamma\rightarrow\mathbb{P}^{1}$ and their proper
transforms on $X^{\prime}$. The latter are general fibers of
$\operatorname{p}\circ\ \Gamma(\tau)\colon
X^{\prime}\rightarrow\mathbb{P}^{1}$, and since $X^{\prime}$ is terminal, they
are terminal varieties by [Kol97, Proposition 7.7]. On the other hand, by
Proposition 3.3(d) the fibers of
$\operatorname{p}\colon\Gamma\rightarrow\mathbb{P}^{1}$ are isomorphic to the
members of the pencil $\mathcal{L}$ determining $\psi\colon
X\dashrightarrow\mathbb{P}^{1}$. The terminality of general members of
$\mathcal{L}$ is thus a necessary condition for the existence of a
$\mathbb{Q}$-factorial terminal resolution of $\mathcal{L}$. This motivates
the following definition:

######  Definition 4.3 (A terminal $\mathbb{Q}$-factorial pencil).

A _terminal pencil_ (a _$\mathbb{Q}$ -factorial terminal pencil_) on a normal
variety is a pencil whose general members are terminal (respectively,
$\mathbb{Q}$-factorial terminal).

In contrast to terminality, the $\mathbb{Q}$-factoriality of general members
of $\mathcal{L}$ is not necessary for the existence of a
$\mathbb{Q}$-factorial terminal resolution of $\psi\colon
X\dashrightarrow\mathbb{P}^{1}$, as illustrated by the following example.

######  Example 4.4 ($\mathbb{Q}$-factoriality: general vs generic).

Let $X=\mathbb{P}^{4}_{[x:y:z:t:w]}$ and let $F$ and $H$ be the projective
cone over the smooth conic $\\{xy-z^{2}=0\\}\subseteq\mathbb{P}^{2}_{[x:y:z]}$
and the hyperplane $\\{t=0\\}$, respectively. Denote by $\mathcal{L}$ be the
pencil generated by $F$ and $2H$. A general member of $\mathcal{L}$ is
isomorphic to the projective cone $Z$ in $\mathbb{P}^{4}$ over the quadric
surface $\\{xy-z^{2}+t^{2}=0\\}\subseteq\mathbb{P}^{3}$. The blowup of the
vertex is smooth and the exceptional divisor has discrepancy $1$, so $Z$ is a
terminal. But it is not $\mathbb{Q}$-factorial. Indeed, its class group is
isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$ and is generated for instance by
the non-$\mathbb{Q}$-Cartier Weil divisor $\\{x=t-z=0\\}$ and the hyperplane
section $Z\cap\\{w=0\\}$. On the other hand, the graph of the rational map

$\psi\colon
X\dashrightarrow\mathbb{P}^{1},\;[x:y:z:t:w]\mapsto[xy-z^{2}:t^{2}]$

determined by $\mathcal{L}$ is isomorphic to the subvariety $\Gamma\subseteq
X\times\mathbb{P}^{1}_{[u_{0}:u_{1}]}$ with equation
$(xy-z^{2})u_{1}+t^{2}u_{0}=0$. The generic fiber of
$\operatorname{p}\colon\Gamma\rightarrow\mathbb{P}^{1}$ is isomorphic to the
projective cone $Y\subseteq\mathbb{P}^{4}_{\mathbb{C}(\lambda)}$ over the
quadric threefold in $\mathbb{P}^{3}_{\mathbb{C}(\lambda)}$ with equation
$xy-z^{2}+\lambda t^{2}=0$, where $\lambda=u_{0}/u_{1}$. As a variety over
$\mathbb{C}(\lambda)$, $Y$ is factorial, with the class group
$\operatorname{Cl}(Y)=\operatorname{Pic}(Y)\cong\mathbb{Z}$ generated by the
hyperplane section $\\{w=0\\}$. It follows that $\Gamma$ is factorial (in
particular $\mathbb{Q}$-factorial), with the class group generated by the
proper transform of $H$ and the hyperplane section $\\{w=0\\}$.

By the following criterion, on projective varieties the terminality of a
pencil is equivalent to the existence of one terminal member.

######  Lemma 4.5 (One terminal member is sufficient).

A pencil on a normal projective variety which has at least one terminal
($\mathbb{Q}$-factorial terminal) member is a terminal (respectively
$\mathbb{Q}$-factorial terminal) pencil.

###### Proof.

Since $X$ is projective and $\mathbb{P}^{1}$ is a smooth curve,
$\operatorname{p}\colon\Gamma\rightarrow\mathbb{P}^{1}$ is a flat projective
morphism. By Proposition 3.3(d), $\gamma\colon\Gamma\rightarrow X$ induces
isomorphisms between scheme-theoretic fibers of $\operatorname{p}$ and members
of $\mathcal{L}$. Assume that for some $p\in\mathbb{P}^{1}$, $\mathcal{L}_{p}$
is terminal. Then $\operatorname{p}^{*}(p)$ is terminal and so [Ish18, Theorem
9.1.14] implies that general fibers of $\operatorname{p}$, and hence general
members of $\mathcal{L}$, are terminal. As a consequence, a general fiber of
$\operatorname{p}$ has rational singularities and its singular locus has
codimension at least three. By [KM92, Theorem 12.1.10] the
$\mathbb{Q}$-factoriality of fibers of $\operatorname{p}$ is then an open
condition on the set of closed points of $\mathbb{P}^{1}$. So if
$\mathcal{L}_{p}$ is in addition $\mathbb{Q}$-factorial then general fibers of
$\operatorname{p}$, and hence the general members of $\mathcal{L}$, are
$\mathbb{Q}$-factorial terminal. ∎

We now relate properties of members of a pencil in a neighborhood of the base
locus to global properties of the graph of its associated rational map in a
neighborhood of the exceptional locus of the graph resolution.

######  Proposition 4.6 (Singularities of the graph).

Let $\mathcal{L}$ be a terminal pencil on a normal variety $X$ and let $Y$ be
a member of $\mathcal{L}$. Put $Y^{\prime}=\gamma_{*}^{-1}Y$. Then the
following hold:

  1. (a)

If $Y$ is normal then $\Gamma$ is normal in an open neighborhood of
$Y^{\prime}$.

  2. (b)

If $Y$ is klt and $K_{\Gamma}$ is $\mathbb{Q}$-Cartier in an open neighborhood
of $Y^{\prime}$ then $\Gamma$ is terminal in an open neighborhood of
$Y^{\prime}$.

  3. (c)

If $Y$ is klt, smooth in codimension $2$ and $\mathbb{Q}$-factorial then
$\Gamma$ is $\mathbb{Q}$-factorial terminal in an open neighborhood of
$Y^{\prime}$.

###### Proof.

By Proposition 3.3(d), $\gamma$ restricts to an isomorphism over each member
of $\mathcal{L}$. In particular, since $\mathcal{L}$ is a terminal pencil,
general fibers of $\operatorname{p}$ are terminal. In all three cases $Y$ is
normal and $Y^{\prime}=\operatorname{p}^{*}(\operatorname{p}(Y^{\prime}))$ is
a prime Cartier divisor on $\Gamma$. By [Gro65, Corollaire 5.12.7] there
exists a normal open neighborhood $V\subseteq\Gamma$ containing $Y^{\prime}$.
This proves (a).

(b) By Lemma 2.3 the log-pair $(V,Y^{\prime})$ is plt in a neighborhood of
$Y^{\prime}$. Let $\pi\colon V^{\prime}\rightarrow V$ be a log-resolution of
this pair and let $G$ be any exceptional prime divisor of $\pi$ whose image is
not contained in a fiber of $\operatorname{p}|_{V}$. Since $\pi$ induces a log
resolution of general fibers of $\operatorname{p}|_{V}$ and the latter are
terminal, it follows that $G$ has positive discrepancy; see e.g. [Kol97,
Proposition 7.7]. This implies by Lemma 2.1 that $V$, and hence $\Gamma$, is
terminal in an open neighborhood of $Y^{\prime}$.

(c) Since klt singularities are Cohen-Macaulay (see Section 2A), $Y^{\prime}$
satisfies Serre’s condition $S_{3}$. Since $Y^{\prime}$ is Cartier, arguing as
in the proofs of [KM92, Corollary 12.1.9, Lemma 12.1.8], we conclude that for
every Weil divisor $D$ on $V$ there exists an open neighborhood $V(D)\subseteq
V$ of $Y^{\prime}$ such that $D|_{V(D)}$ is $\mathbb{Q}$-Cartier. As in (b) we
get a terminal open neighborhood $W\subseteq V(K_{V})$ of $Y^{\prime}$. Since
terminal singularities are rational, the group
$\operatorname{Cl}(W)/\operatorname{Pic}(W)$ is finitely generated by Lemma
2.2(b). The intersection of $W$ and the open neighborhoods $V(D_{i})$, where
the $D_{i}$ range through a finite set of Weil divisors whose classes generate
$\operatorname{Cl}(W)/\operatorname{Pic}(W)$, is then a $\mathbb{Q}$-factorial
terminal open neighborhood of $Y^{\prime}$. ∎

######  Notation 4.7.

For a terminal pencil $\mathcal{L}$ on a normal variety $X$ and a finite
subset $\delta\subseteq\mathbb{P}^{1}$ we put
$\Gamma_{\delta}=\operatorname{p}^{-1}(\mathbb{P}^{1}\setminus\delta)\subseteq\Gamma$
and $X_{\delta}=X\setminus\bigcup_{p\in\delta}\mathcal{L}_{p}\subseteq X$. We
define the following property:

  1. ($\mathbf{TQ}_{\delta}$)

For every $p\in\mathbb{P}^{1}\setminus\delta$ the member $\mathcal{L}_{p}$ is
a prime divisor and on some open neighborhood of
$\operatorname{Bs}\mathcal{L}$ in $\mathcal{L}_{p}$ it is klt, smooth in
codimension $2$ and $\mathbb{Q}$-factorial.

Note that for a pencil $\mathcal{L}$ whose general members are
$\mathbb{Q}$-factorial terminal the condition ($\mathbf{TQ}_{\delta}$) holds
for the finite set $\delta\subset\mathbb{P}^{1}$ consisting of points $p$ for
which $\mathcal{L}_{p}$ is not $\mathbb{Q}$-factorial terminal.

######  Corollary 4.8 (Controlling the discrepancy locus).

Let $\mathcal{L}$ be a terminal pencil on a normal variety $X$ and let
$\delta\subset\mathbb{P}^{1}$ be a finite set. If $X_{\delta}$ is
$\mathbb{Q}$-factorial terminal and $(\mathbf{TQ}_{\delta})$ holds then
$\Gamma_{\delta}$ is $\mathbb{Q}$-factorial terminal. Consequently, the
discrepancy locus of every thrifty $\mathbb{Q}$-factorial terminal resolution
of $\mathcal{L}$ is contained in $\delta$.

###### Proof.

Let $E=\gamma^{-1}(\operatorname{Bs}\mathcal{L})_{\mathrm{red}}$ be the
exceptional locus of $\gamma$. By assumption $\Gamma_{\delta}\setminus E\cong
X_{\delta}\setminus\operatorname{Bs}\mathcal{L}$ is $\mathbb{Q}$-factorial
terminal. On the other hand, it follows from Proposition 4.6 that for every
$p\in\mathbb{P}^{1}\setminus\delta$ the open set $\Gamma_{\delta}$ is
$\mathbb{Q}$-factorial and terminal in a neighborhood of the intersection of
$E$ with the proper transform of $\mathcal{L}_{p}$. Since the union of such
neighborhoods is an open neighborhood of $E\cap\Gamma_{\delta}$ in
$\Gamma_{\delta}$, it follows that $\Gamma_{\delta}$ is $\mathbb{Q}$-factorial
terminal. The second assertion follows from Lemma 2.4. ∎

###  4B. Terminal rank one Fano pencils and relative MMPs

In this subsection we consider pencils of Weil divisors whose general members
are terminal Fano varieties of Picard rank one and the outputs of relative
MMP’s ran from their resolutions. We keep the notation of subsection 4A.

######  Definition 4.9 (A terminal rank one Fano pencil).

Let $X$ be a normal projective variety of dimension at least $2$. A _terminal
rank one Fano pencil_ on $X$ is a pencil $\mathcal{L}$ whose general members
are terminal Fano varieties of Picard rank one. The _degeneracy locus_ of
$\mathcal{L}$ is the finite set $\delta(\mathcal{L})\subset\mathbb{P}^{1}$
consisting of points $p$ such that the member $\mathcal{L}_{p}$ is either
reducible or has Picard rank strictly higher than one.

It is known that general fibers of Mori fiber spaces can have Picard rank
higher than one (see e.g. Example 6.2). But the additional assumption that the
Picard rank of general fibers is one, which we impose in Definition 4.9,
allows to control the effect of running relative MMP’s on resolutions of such
pencils more easily. Even with this restriction there is still a large natural
geometric supply of pencils that can be used to construct Mori fiber
completions of $\mathbb{A}^{n}$’s, see Section 6.

######  Definition 4.10 (A compatible thrifty resolution).

Let $\mathcal{L}$ be a terminal rank one Fano pencil on a normal projective
variety. A _compatible thrifty resolution_ of $\mathcal{L}$ is a thrifty
$\mathbb{Q}$-factorial terminal resolution of $\mathcal{L}$ (see Definition
4.1) whose discrepancy locus is contained in the degeneracy locus of
$\mathcal{L}$.

######  Example 4.11 (Simple low-dimensional examples).

  1. (a)

A terminal rank one Fano pencil on $\mathbb{P}^{2}$ consists of lines and or
conics, and the usual minimal resolution of base points of the pencil is a
compatible thrifty resolution.

  2. (b)

Since $\mathbb{P}^{2}$ is the only terminal del Pezzo surface of Picard rank
one, the only terminal rank one Fano pencils on $\mathbb{P}^{3}$ are the
pencils of planes. Clearly, the base scheme of every such pencil is a line and
its blowup gives a compatible thrifty resolution.

Let $X_{0}$ be a $\mathbb{Q}$-factorial terminal projective variety and let
$f_{0}\colon X_{0}\rightarrow\mathbb{P}^{1}$ be a surjective morphism. Recall
[KM98, 3.31, Example 2.16] that a $K_{X_{0}}$-MMP $\varphi\colon
X_{0}\dashrightarrow X_{m}=\hat{X}$ relative to $f_{0}$ consists of a finite
sequence $\varphi=\varphi_{m}\circ\cdots\circ\varphi_{1}$ of birational maps

$\textstyle{X_{k-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{k}}$$\textstyle{X_{k}}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{f_{k-1}}$$\textstyle{f_{k}}$

between $\mathbb{Q}$-factorial terminal projective varieties, where each
$\varphi_{k}$ is associated to an extremal ray of the closure
$\operatorname{NE}(X_{k-1}/\mathbb{P}^{1})$ of the relative cone of $1$-cycles
of $X_{k-1}$ over $\mathbb{P}^{1}$. The morphisms $f_{k}\colon
X_{k}\rightarrow\mathbb{P}^{1}$ are the induced surjections. Each
$\varphi_{k}$ is either a relative divisorial contraction or a flip whose
flipping and flipped curves are contained in fibers of $f_{k-1}$ and $f_{k}$.
We say that a relative MMP terminates if either $K_{\hat{X}/\mathbb{P}^{1}}$
is $f_{m}$-nef or if $f_{m}$ factors through a Mori fiber space given by a
contraction of a extremal ray in $\operatorname{NE}(X_{m}/\mathbb{P}^{1})$.

######  Proposition 4.12 (Mori Fiber completions from pencils with compatible
resolutions).

Let $\mathcal{L}$ be a terminal rank one Fano pencil on a normal projective
variety $X$ and let $\psi_{\mathcal{L}}\colon X\dashrightarrow\mathbb{P}^{1}$
be the associated dominant rational map. Assume that $\mathcal{L}$ has a
compatible thrifty resolution $\tau\colon X^{\prime}\rightarrow X$. Then there
exists a $K_{X^{\prime}}$-MMP $\varphi\colon X^{\prime}\dashrightarrow\hat{X}$
relative to $\psi_{\mathcal{L}}\circ\tau\colon
X^{\prime}\rightarrow\mathbb{P}^{1}$ which terminates. Furthermore, every
terminating $K_{X^{\prime}}$-MMP relative to $\psi_{\mathcal{L}}\circ\tau$
restricts to an isomorphism over $\mathbb{P}^{1}\setminus\delta(\mathcal{L})$
and its output is a Mori fiber space over $\mathbb{P}^{1}$.

###### Proof.

Put $X^{\prime}=X_{0}$ and $f_{0}=\psi_{\mathcal{L}}\circ\tau\colon
X_{0}\rightarrow\mathbb{P}^{1}$. Since general fibers of $f_{0}$ are Fano
varieties, their canonical divisors are not pseudo-effective, hence
$K_{X_{0}}$ is not pseudo-effective over $\mathbb{P}^{1}$. This implies that
the output $\hat{f}=f_{m}\colon\hat{X}\rightarrow\mathbb{P}^{1}$ of any
$K_{X_{0}}$-MMP $\varphi\colon X_{0}\dashrightarrow\hat{X}$ relative to
$f_{0}\colon X_{0}\rightarrow\mathbb{P}^{1}$ that terminates, factors through
a Mori fiber space $g\colon\hat{X}\rightarrow T$ over a normal projective
variety $T$ given by the contraction of some extremal ray of
$\overline{\operatorname{NE}}(\hat{X}/\mathbb{P}^{1})$. The termination of at
least one such relative MMP is guaranteed by [BCHM10, Corollary 1.3.3].

We show by induction that for every $k\in\\{1,\ldots,m\\}$ the birational map
$\varphi_{k}\colon X_{k-1}\dashrightarrow X_{k}$ restricts to an isomorphism
over $\mathbb{P}^{1}\setminus\delta(\mathcal{L})$. Let $\sigma\colon
X_{k-1}\rightarrow Y$ be the birational extremal contraction associated to
some extremal ray in $\overline{\operatorname{NE}}(X_{k-1}/\mathbb{P}^{1})$
and let $C$ be a curve contracted by $\sigma$. Since the extremal ray lies in
$\overline{\operatorname{NE}}(X_{k-1}/\mathbb{P}^{1})$, $C$ is contained in
some fiber $f_{k-1}^{*}(p)$, $p\in\mathbb{P}^{1}$. By definition of a terminal
rank one Fano pencil and of a compatible thrifty resolution, for every
$p\in\mathbb{P}^{1}\setminus\delta(\mathcal{L})$ the fiber $f_{0}^{*}(p)$, and
hence by induction the fiber $f_{k-1}^{*}(p)$ endowed with its reduced
structure is a projective variety of Picard rank one. By Remark 2.6 the
restriction of $\sigma$ to $f_{k-1}^{*}(p)$ is an isomorphism. It follows that
the exceptional locus of $\sigma$ is contained in fibers of $f_{k-1}$ over
$\delta(\mathcal{L})$, hence that $\varphi_{k}$ restricts to an isomorphism
over $\mathbb{P}^{1}\setminus\delta(\mathcal{L})$. Finally, since general
fibers of $\hat{f}\colon\hat{X}\rightarrow\mathbb{P}^{1}$ have Picard rank
one, $\hat{f}\colon\hat{X}\rightarrow\mathbb{P}^{1}$ cannot be decomposed into
a Mori fiber space $g\colon\hat{X}\rightarrow T$ over a base
$T\rightarrow\mathbb{P}^{1}$ of positive relative dimension, hence
$\hat{f}\colon\hat{X}\rightarrow\mathbb{P}^{1}$ itself is a Mori fiber space.
∎

######  Corollary 4.13.

Let $X$ be a normal projective variety and let $\mathcal{L}$ be a terminal
rank one Fano pencil on $X$ that admits a compatible thrifty resolution. Then
$X_{\delta(\mathcal{L})}\setminus\operatorname{Bs}\mathcal{L}$ (see Notation
4.7) admits a Mori fiber completion $\pi\colon V\rightarrow\mathbb{P}^{1}$.
Furthermore if $p\in\mathbb{P}^{1}\setminus\delta(\mathcal{L})$ then the
scheme-theoretic fiber $\pi^{*}(p)$ is isomorphic to the member
$\mathcal{L}_{p}$ of $\mathcal{L}$.

###### Proof.

Let $\tau\colon X^{\prime}\rightarrow X$ be a compatible thrifty resolution of
$\mathcal{L}$. Put $\tau^{\prime}=\Gamma(\tau)$ and let $\varphi\colon
X^{\prime}\dashrightarrow V$ be a $K_{X^{\prime}}$-MMP relative to
$\operatorname{p}\circ\ \tau^{\prime}\colon
X^{\prime}\rightarrow\mathbb{P}^{1}$ which terminates. By Proposition 4.12,
$V$ has a structure of a Mori fiber space $\pi\colon
V\rightarrow\mathbb{P}^{1}$. The desired open embedding is given by the
restriction to $X_{\delta(\mathcal{L})}\setminus\operatorname{Bs}\mathcal{L}$
of the birational map $\varphi\circ(\gamma\circ\tau^{\prime})^{-1}\colon
X\dashrightarrow V$. Indeed, the birational map $\gamma^{-1}\colon
X\dashrightarrow\Gamma$ induces an isomorphism between
$X_{\delta(\mathcal{L})}\setminus\operatorname{Bs}\mathcal{L}$ and
$\Gamma_{\delta(\mathcal{L})}\setminus E$. On the other hand, since by the
definition of a compatible thrifty resolution the image of
$\operatorname{Exc}(\tau^{\prime})$ by
$\operatorname{p}\circ\tau^{\prime}\colon
X^{\prime}\rightarrow\Gamma\rightarrow\mathbb{P}^{1}$ is contained in
$\delta(\mathcal{L})$, the birational map $(\tau^{\prime})^{-1}$ restricts to
an isomorphism over
$\operatorname{p}^{-1}(\mathbb{P}^{1}\setminus\delta(\mathcal{L}))=\Gamma_{\delta(\mathcal{L})}$.
It follows in turn from Proposition 4.12 that the rational map
$\varphi\circ(\tau^{\prime})^{-1}\colon\Gamma\dashrightarrow V$ restricts to
an isomorphism over $\Gamma_{\delta(\mathcal{L})}$. The second assertion
follows from Proposition 3.3. ∎

The next example illustrates the process of taking a compatible thrifty
resolution of a terminal rank one Fano pencil and then running a relative MMP
as above. It shows in particular that different runs of the MMP may lead to
different Mori fiber completions.

######  Example 4.14 (Mori fiber completions of $\mathbb{A}^{2}$ from
pencils).

Let $\mathcal{L}$ be a pencil on
$\mathbb{P}^{2}=\operatorname{Proj}(\mathbb{C}[x,y,z])$ generated by a smooth
conic $C$ and twice a line $H$ tangent to $C$. Up to a projective equivalence
we may assume that $C=\\{xz-y^{2}=0\\}$ and $H=\\{x=0\\}$. The graph of
$\psi_{\mathcal{L}}$ is the surface
$\Gamma\subseteq\mathbb{P}^{2}\times\mathbb{P}^{1}_{[u:v]}$ defined by the bi-
homogeneous equation $(xz-y^{2})v-x^{2}u=0$. It is normal and its unique
singular point $p=([0:0:1],[1:0])$ is supported at the intersection of the
proper transform of $H$ with the exceptional divisor $E\cong\mathbb{P}^{1}$ of
$\gamma=\operatorname{pr}_{1}\colon\Gamma\rightarrow\mathbb{P}^{2}$. The
singular point is a cyclic quotient singularity of type $A_{3}$. Let
$\tau^{\prime}\colon X^{\prime}\rightarrow\Gamma$ be a minimal resolution of
the singularity of $\Gamma$. Then $\tau=\gamma\circ\tau^{\prime}$ is a
compatible thrifty resolution of $\mathcal{L}$. The exceptional locus of
$\tau^{\prime}$ is a chain of three smooth rational curves $F_{0}$, $F$,
$F_{1}$ with self-intersection numbers equal to $-2$, having $F$ as its middle
component. The proper transforms $E^{\prime}$ and $H^{\prime}$ of $E$ and $H$
in $X^{\prime}$ are smooth rational curves with self-intersection $-1$, and
they intersect $\operatorname{Exc}(\tau^{\prime})$ transversally along the
curves $F_{0}$ and $F$ respectively.

The $K_{X^{\prime}}$-MMP relative to $f^{\prime}=\operatorname{p}\circ\
\tau^{\prime}\colon X^{\prime}\rightarrow\mathbb{P}^{1}$ first contracts
$H^{\prime}$, then the image of $F$ and finally either the image of $F_{0}$ or
the image of $F_{1}$. The resulting Mori fiber space $\pi\colon
V\rightarrow\mathbb{P}^{1}$ is thus isomorphic to
$\rho_{0}=\operatorname{pr}_{2}\colon\mathbb{F}_{0}=\mathbb{P}^{1}\times\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$
in the first case and to the Hirzebruch surface
$\rho_{1}\colon\mathbb{F}_{1}\rightarrow\mathbb{P}^{1}$ in the second case.
The first case yields an open embedding of
$\mathbb{A}^{2}=\mathbb{P}^{2}\setminus H$ into $\mathbb{F}_{0}$ as the
complement of the proper transforms of $E$ and $F_{1}$, which are respectively
a section with self-intersection number $0$ and a fiber of $\rho_{0}$. In the
second case we obtain an open embedding of $\mathbb{A}^{2}$ into
$\mathbb{F}_{1}$ as the complement of the proper transforms of $E$ and
$F_{0}$, which are respectively the negative section and a fiber of
$\rho_{1}$.

##  5\. Obtaining Mori fiber completions from special pencils

We now consider a class of pencils to which the methods of Section 4 apply. A
_polarized ( $\mathbb{Q}$-factorial) pair_ $(X,H)$ is by definition a pair
consisting of a normal projective ($\mathbb{Q}$-factorial) variety $X$ of
dimension at least $2$ and an ample prime Weil divisor $H$ on $X$.

######  Definition 5.1 ($H$-special pencils).

Let $(X,H)$ be a polarized $\mathbb{Q}$-factorial pair. An _$H$ -special_
pencil on $X$ is a pencil $\mathcal{L}$ which satisfies the following
properties:

  1. (a)

$dH$ is a member of $\mathcal{L}$ for some integer $d\geq 1$.

  2. (b)

The base locus $\operatorname{Bs}\mathcal{L}$ is irreducible.

  3. (c)

If $d=1$ then the base scheme $\operatorname{Bs}\mathcal{L}$ is smooth or its
support is contained in $\operatorname{Sing}(H)$.

###  5A. Integrity of members

######  Lemma 5.2.

Let $(R,\mathfrak{m})$ be a noetherian integral local ring and let
$f\in\mathfrak{m}$ be a nonzero element such that the ring $R/(f)$ is regular.
Then $R$ is regular and for every $h\in\mathfrak{m}^{2}$ the ring $R/(f+h)$ is
regular.

###### Proof.

Let $\pi\colon R\rightarrow R/(f)$ be the quotient morphism. Since $R/(f)$ is
regular, $f\notin\mathfrak{m}^{2}$ and the maximal ideal $\pi(\mathfrak{m})$
is generated by a regular sequence $\pi(a_{1}),\ldots,\pi(a_{n})$, where
$a_{i}\in\mathfrak{m}$ and $n=\dim R/(f)$. It follows that $\mathfrak{m}$ is
generated by the regular sequence $f,a_{1},\ldots,a_{n}$, hence that $R$ is
regular. Furthermore, $\mathfrak{m}^{2}\subseteq(f^{2})+(a_{1},\ldots,a_{n})$,
so for some $v\in R$ we have $h-vf^{2}\in(a_{1},\ldots,a_{n})$. Since elements
in $1+\mathfrak{m}$ are invertible in $R$, we get
$(f+h,a_{1},\ldots,a_{n})=(f(1+vf),a_{1},\ldots,a_{n})=\mathfrak{m}$. It
follows that the images of $a_{1},\ldots,a_{n}$ under the quotient
homomorphism $R\rightarrow R/(f+h)$ form a regular sequence which and they
generate the maximal ideal of $R/(f+h)$. Thus, $R/(f+h)$ is regular. ∎

######  Lemma 5.3 (Integrity and smoothness of members).

Let $\mathcal{L}$ be an $H$-special pencil on a polarized
$\mathbb{Q}$-factorial pair $(X,H)$. Assume that some member of $\mathcal{L}$
is smooth and Cartier at some point $x\in\operatorname{Bs}\mathcal{L}$ and
that in case $H$ is a member of $\mathcal{L}$, $x\in\operatorname{Sing}H$.
Then every member of $\mathcal{L}$ not supported on $\operatorname{Supp}H$ is
a prime divisor which is smooth and Cartier at $x$.

###### Proof.

By assumption $\mathcal{L}$ has a member $dH$ for some positive integer $d$
and a member $F\neq dH$ which is smooth and Cartier at $x\in
S=(\operatorname{Bs}\mathcal{L})_{\mathrm{red}}=(F\cap H)_{\mathrm{red}}$. Any
two members of $\mathcal{L}$ differ by a principal divisor, so we infer that
all members of $\mathcal{L}$ are Cartier at $x$. Let $\mathfrak{m}$ be the
maximal ideal of the local ring $\mathcal{O}_{X,x}$ and let
$f,h\in\mathfrak{m}$ be generators of the ideals of $F$ and $dH$ in
$\mathcal{O}_{X,x}$, respectively. Since $F$ is smooth at $x$, by Lemma 5.2,
$f\in\mathfrak{m}\setminus\mathfrak{m}^{2}$ and $\mathcal{O}_{X,x}$ is
regular. If $d>1$ then $h\in\mathfrak{m}^{2}$. If $d=1$ then by assumption $x$
is a singular point of $H$, so by the Jacobian criterion in
$\mathcal{O}_{X,x}$ the residue class of $h$ in
$\mathfrak{m}/\mathfrak{m}^{2}$ is trivial, hence again
$h\in\mathfrak{m}^{2}$.

The variety $X$ is projective and the divisor $H$ is ample, so since $F$ is
$\mathbb{Q}$-Cartier, $S$ is a closed subset of $\operatorname{Supp}(H)$ of
pure codimension $1$. Let $Y$ be any member of $\mathcal{L}$ other than $dH$.
Write $Y=\sum_{i=1}^{r}D_{i}$, where $D_{i}$ are prime divisors. Each $D_{i}$
is $\mathbb{Q}$-Cartier and $H$ is ample, so $(D_{i}\cap H)_{\mathrm{red}}$
has pure codimension $1$ in $H$. But $(D_{i}\cap
H)_{\mathrm{red}}\subseteq(Y\cap H)_{\mathrm{red}}=S$ and $S$ is irreducible,
so $(D_{i}\cap H)_{\mathrm{red}}=S$ for each $i$. The ideal of $D_{i}$ in
$\mathcal{O}_{X,x}$ is thus contained in $\mathfrak{m}$ for every $1\leq i\leq
r$, and hence the ideal of $Y$ is contained in $\mathfrak{m}^{r}$. On the
other hand, the ideal of $Y$ in $\mathcal{O}_{X,x}$ is generated by $f+th$ for
some $t\in\mathbb{C}$. Since $f\in\mathfrak{m}\setminus\mathfrak{m}^{2}$ and
$h\in\mathfrak{m}^{2}$, we have
$f+th\in\mathfrak{m}\setminus\mathfrak{m}^{2}$. Thus $r=1$ and so $Y$ is a
prime divisor, which by Lemma 5.2 is smooth at $x$. ∎

In Lemma 5.3 the assumption that the base locus of $\mathcal{L}$ is
irreducible and that $\mathcal{L}$ has a member which is smooth at some point
of $\operatorname{Bs}\mathcal{L}$ are both necessary to ensure primeness of
all members of $\mathcal{L}$ other than $dH$. Similarly for the additional
assumption in case $d=1$, where $H$ is itself a member of $\mathcal{L}$, that
$\operatorname{Supp}(\operatorname{Bs}\mathcal{L})$ contains a singular point
of $H$. This is illustrated by the following examples.

######  Example 5.4 (Failure of integrity).

  1. (a)

Let $\mathcal{L}$ be a pencil on $\mathbb{P}^{2}$ generated by a smooth conic
$C$ and twice a line $H$ meeting $C$ at two distinct points. The sum of two
lines tangent to $C$ at those points is a reducible member of $\mathcal{L}$.
Similarly, the pencil in $\mathbb{P}^{3}$ generated by the projective cones
over $C$ and $2H$ has a reducible member consisting of the union of two
planes. In the latter case the base locus is reducible but connected.

  2. (b)

Let $\mathcal{L}$ be a pencil on $\mathbb{P}^{2}$ generated by a nodal cubic
curve $C$ and $3H$, where $H$ is the line tangent to one of the two branches
of $C$ at its singular point
$p=(\operatorname{Bs}\mathcal{L})_{\mathrm{red}}$. Then every member of the
pencil is singular at $p$ and $\mathcal{L}$ has a reducible member. Indeed, up
to a projective equivalence we have $C=\\{y^{2}z=x^{2}(x-z)\\}$, $H=\\{x=0\\}$
and $p=[0:0:1]$. Then the members of $\mathcal{L}$ are
$\mathcal{L}_{[a:b]}=\\{a(x^{2}+y^{2})z=bx^{3}\\}$ and $\mathcal{L}_{[1:0]}$
is the union of three distinct lines through $p$.

  3. (c)

Let $\mathcal{L}$ be a pencil on $\mathbb{P}^{2}$ generated a smooth conic $H$
and some other smooth conic meeting $H$ in one point $p$ only. The pencil has
a non-reduced member supported on the line tangent to both conics at $p$.

######  Corollary 5.5.

Let $\mathcal{L}$ be an $H$-special pencil on a polarized
$\mathbb{Q}$-factorial pair $(X,H)$. In case $H$ is a member of $\mathcal{L}$
and
$\operatorname{Supp}(\operatorname{Bs}\mathcal{L})\nsubseteq\operatorname{Sing}H$
assume additionally that $\operatorname{Cl}(X)=\mathbb{Z}\langle H\rangle$. If
$\mathcal{L}$ has a member which is smooth and Cartier on some neighborhood of
$\operatorname{Bs}\mathcal{L}$ then every member of $\mathcal{L}$ other than
$dH$ is a prime divisor which is smooth and Cartier on some neighborhood of
$\operatorname{Bs}\mathcal{L}$.

###### Proof.

By assumption $\mathcal{L}$ has $dH$ as a member for some positive integer
$d$. Put $S=(\operatorname{Bs}\mathcal{L})_{\mathrm{red}}$. We may assume that
$d=1$ and $S\nsubseteq\operatorname{Sing}H$, as otherwise the corollary
follows from Lemma 5.3. Then every member of $\mathcal{L}$ is prime, because
by assumption $\operatorname{Cl}(X)=\mathbb{Z}\langle H\rangle$ in this case.
Moreover, by the definition of an $H$-special pencil (see Definition 5.1(c)),
the base scheme $\operatorname{Bs}\mathcal{L}$ is smooth, hence in particular
reduced. Then for every member $Y\neq H$ of $\mathcal{L}$ we have
$S=\operatorname{Bs}\mathcal{L}=Y\cap H$ scheme-theoretically. Let
$\mathfrak{m}$ be the maximal ideal of the local ring $\mathcal{O}_{X,x}$ at a
point $x\in S$. Then the ring $\mathcal{O}_{S,x}$ is regular and isomorphic to
$\mathcal{O}_{X,x}/(y,h)=(\mathcal{O}_{X,x}/(y))/(h)$ where $y$ and $h$ are
generators of the ideals of $Y$ and $H$ in $\mathcal{O}_{X,x}$, respectively.
Since $Y\sim H$ is prime, the ring $\mathcal{O}_{X,x}/(y)$ is integral and its
quotient by $(h)$ is regular. So $\mathcal{O}_{X,x}/(y)$ is regular by Lemma
5.2, that is, $Y$ is smooth at $x$. ∎

###  5B. $H$-special pencils of Cartier divisors

Recall (Definition 2.5) that for a Fano variety $X$ for which
$\operatorname{Cl}(X)\cong\mathbb{Z}\langle H\rangle$, where $H$ is an ample
divisor, the _index of $X$_ is the unique integer $i_{X}$ for which
$-K_{X}\sim i_{X}H$.

######  Lemma 5.6.

Let $X$ be a Fano variety with the class group
$\operatorname{Cl}(X)\cong\mathbb{Z}\langle H\rangle$ and let $Y\sim dH$ be a
normal prime divisor on $X$. If $Y$ is Cartier in codimension $2$ and
$d<i_{X}$ then $Y$ is a Fano variety.

###### Proof.

Since $X$ is $\mathbb{Q}$-factorial, $K_{X}+Y$ is $\mathbb{Q}$-Cartier. Since
$Y$ is normal and Cartier in codimension $2$, the adjunction formula, Lemma
2.3, implies that $-K_{Y}=-(K_{X}+Y)|_{Y}$ is an anticanonical divisor on $Y$.
We have $i_{X}>d$, so the divisor $-(K_{X}+Y)\sim(i_{X}-d)H$ is ample, hence
$-K_{Y}$ is ample. ∎

It is more difficult to provide uniform conditions which ensure that a given
member of an $H$-special pencil has Picard rank one. For pencils of Cartier
divisors on mildly singular varieties we can rely on the following result for
Picard groups proven in [Gro68, Exposé XII, Corollary 3.6].

######  Lemma 5.7 (Grothendieck-Lefschetz theorem).

Let $Y$ be an ample effective Cartier divisor on a normal variety $X$. Assume
that $H^{i}(Y,\mathcal{O}_{X}(-\ell Y)|_{Y})=0$ for $i=1,2$ and every
$\ell>0$, and that $X\setminus Y$ is a local complete intersection. Then the
restriction homomorphism $\operatorname{Pic}X\rightarrow\operatorname{Pic}Y$
is an isomorphism.

######  Corollary 5.8 (Finding terminal Fano pencils of rank one).

Let $X$ be a Fano variety of dimension at least $4$ whose singularities are
rational and whose class group is generated by a prime Weil divisor $H$. Let
$d\in\\{1,2,\ldots,i_{X}-1\\}$ and let $\mathcal{L}\subseteq
H^{0}(X,\mathcal{O}_{X}(dH))$ be an $H$-special pencil of Cartier divisors
such that $X\setminus\operatorname{Bs}\mathcal{L}$ is a local complete
intersection and which has a terminal member smooth in a neighborhood of
$\operatorname{Bs}\mathcal{L}$. Then $\mathcal{L}$ is a terminal rank one Fano
pencil and every member of $\mathcal{L}$ other than $dH$ is non-degenerate.

###### Proof.

Since $H$ generates $\operatorname{Cl}(X)$, $X$ is in particular
$\mathbb{Q}$-factorial and $H$ is ample, so the pair $(X,H)$ is polarized
$\mathbb{Q}$-factorial. Since $\mathcal{L}$ has a terminal member and
$d<i_{X}$, general members of $\mathcal{L}$ are terminal Fano varieties by
Lemma 4.5 and Lemma 5.6. By Corollary 5.5 every member $Y$ of $\mathcal{L}$
other than $dH$ is prime. By assumption the divisor $Y$ is Cartier and, since
$\operatorname{Cl}(X)\cong\mathbb{Z}$, it is necessarily ample. Then
$\mathcal{O}_{X}(Y)$ is an ample invertible sheaf, so we have exact sequences

$0\rightarrow\mathcal{O}_{X}(-(\ell+1)Y)\rightarrow\mathcal{O}_{X}(-\ell
Y)\rightarrow\mathcal{O}_{X}(-\ell Y)|_{Y}\rightarrow 0\quad\textrm{for every
}\ell>0.$

Since $\mathcal{O}_{X}(Y)$ is ample, by [SS85, Corollary 7.67] (see also Lemma
2.2(c)), we have $H^{i}(X,\mathcal{O}_{X}(-\ell Y))=0$ for every $i\leq\dim
X-1$ and $\ell>0$. Since $\dim X\geq 4$, the associated long exact sequence of
cohomology gives $H^{i}(Y,\mathcal{O}_{X}(-\ell Y)|_{Y})=0$ for every $i=1,2$
and $\ell>0$. Since $X\setminus Y$ is contained in
$X\setminus\operatorname{Bs}\mathcal{L}$, Lemma 5.7 implies that
$\operatorname{Pic}(Y)\cong\operatorname{Pic}(X)\cong\mathbb{Z}$. ∎

Combining the above results we obtain the following theorem.

######  Theorem 5.9 (Mori fiber completions from $H$-special pencils).

Let $X$ be a Fano variety of dimension at least $4$ whose singularities are
rational and whose class group is generated by a prime Weil divisor $H$.
Assume that $X\setminus H$ is terminal and that for some
$d\in\\{1,2,\ldots,i_{X}-1\\}$ there exists a Cartier divisor $F\sim dH$ other
than $dH$ for which the following hold:

  1. (a)

$F$ is terminal,

  2. (b)

$F\cap H$ is irreducible and contained in $F_{\mathrm{reg}}$,

  3. (c)

$X\setminus(F\cap H)$ is a local complete intersection,

  4. (d)

If $d=1$ then $F\cap H$ is either a smooth scheme or its support is contained
in $\operatorname{Sing}(H)$.

Then $X\setminus H$ admits a Mori fiber completion $\pi\colon
V\rightarrow\mathbb{P}^{1}$ such that all members of the pencil
$\mathcal{L}=\langle F,dH\rangle$ other than $dH$ appear as fibers of $\pi$.

###### Proof.

The pair $(X,H)$ is a polarized $\mathbb{Q}$-factorial pair. Since $F$ is
Cartier, the assumptions (b) and (d) imply that $\mathcal{L}$ is an
$H$-special pencil and on some neighborhood of $\operatorname{Bs}\mathcal{L}$
the divisor $F$ is smooth, in particular $\mathbb{Q}$-factorial. Let
$\psi_{\mathcal{L}}\colon X\dashrightarrow\mathbb{P}^{1}$ be the dominant
rational map determined by $\mathcal{L}$. By Corollary 5.8, $\mathcal{L}$ is a
terminal rank one Fano pencil with degeneracy locus contained in
$\\{(\psi_{\mathcal{L}})_{*}H\\}$. Since by Corollary 5.5 every member of
$\mathcal{L}$ other than $dH$ is smooth in a neighborhood of
$\operatorname{Supp}(\operatorname{Bs}\mathcal{L})$, property
$(\mathbf{TQ}_{\delta})$ (see Notation 4.7) holds for
$\delta=\\{(\psi_{\mathcal{L}})_{*}H\\}$. Since $X\setminus H$ is
$\mathbb{Q}$-factorial and terminal by assumption, Corollary 4.8 implies that
every thrifty $\mathbb{Q}$-factorial terminal resolution of $\mathcal{L}$ is
compatible, that is, its degeneracy locus is contained in $\delta$. The
assertion then follows from Corollary 4.13. ∎

As a corollary we obtain Mori fiber completions of affine varieties of
dimension $\geq 4$ whose general fibers are completions of affine Fano
varieties in the sense of Definition 2.10.

######  Corollary 5.10 (Affine Fano fibers).

In the setting of Theorem 5.9 assume further that $F$ is
$\mathbb{Q}$-factorial, that $(F\cap H)_{\mathrm{red}}$ is klt and that $d\leq
i_{X}-2$. Let $\pi\colon V\rightarrow\mathbb{P}^{1}$ be the Mori fiber
completion of the affine variety $U=X\setminus H$ associated to the pencil
$\mathcal{L}=\langle F,dH\rangle$. Then the general fibers of $\pi|_{U}\colon
U\rightarrow\mathbb{P}^{1}$ are affine Fano varieties.

###### Proof.

Let $B=V\setminus U$. For a general point $p\in\mathbb{P}^{1}$, let
$V_{p}=\pi^{*}p$ and let $B_{p}$ denote the reduction of the restriction of
$B$ to $V_{p}$ as a Weil divisor. By Lemma 4.5, $V_{p}$ is a
$\mathbb{Q}$-factorial terminal Fano variety of Picard rank one. On the other
hand, it follows from the proof of Corollary 4.13 that the log pair
$(V_{p},B_{p})$ is isomorphic to the log pair
$(\mathcal{L}_{p},(\mathcal{L}_{p}\cap H)_{\mathrm{red}})$. Since
$(\mathcal{L}_{p}\cap H)_{\mathrm{red}}=(F\cap H)_{\mathrm{red}}$ is
irreducible and klt, the log pair $(V_{p},B_{p})$ is plt by Lemma 2.3. We have
$-(K_{\mathcal{L}_{p}}+({\mathcal{L}_{p}}\cap
H)_{\mathrm{red}})=(-i_{X}+d)H|_{\mathcal{L}_{p}}+H|_{\mathcal{L}_{p}}$ by
adjunction, so $-(K_{V_{p}}+B_{p})$ is ample.

∎

##  6\. Mori fiber completions of affine spaces over $\mathbb{P}^{1}$

We now apply our results to the construction of $\mathbb{Q}$-factorial
terminal Mori fiber completions of affine spaces over $\mathbb{P}^{1}$. We
begin with a review of some known examples.

###  6A. Some known examples

######  Example 6.1 (Examples of product type).

For every $n\geq 1$ and every $\mathbb{Q}$-factorial terminal Fano variety $X$
of Picard rank one which is completion of $\mathbb{A}^{n}$, the projection
$\pi=\operatorname{pr}_{2}\colon
V=X\times\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$ is a Mori fiber completion
of $\mathbb{A}^{n}\times\mathbb{A}^{1}$. For instance, we can take
$X=\mathbb{P}^{n}$, which is the only possibility for $n=1,2$. For $n=3$,
smooth Fano threefolds of Picard rank one which are completions of
$\mathbb{A}^{3}$ have been classified by Furushima [Fur93] (see also [Pro91]).
These are: $\mathbb{P}^{3}$, the smooth quadric threefold in $\mathbb{P}^{4}$,
the quintic del Pezzo threefold in $\mathbb{P}^{6}$, and a four dimensional
family of prime Fano threefolds of genus $12$. In higher dimensions, other
examples of smooth Fano completions of $\mathbb{A}^{n}$ of Picard rank one are
the quintic del Pezzo fourfold [Pro94, Theorem 3.1] in $\mathbb{P}^{7}$ and
Fano-Mukai fourfolds of genus $10$ [PZ18].

There are several examples of Mori fiber completions of $\mathbb{A}^{3}$ which
are not of product type as in Example 6.1. For instance, for every
$d\in\\{1,2,3,4,5,6,8,9\\}$ there exists a Mori fiber completion $\pi\colon
V\rightarrow\mathbb{P}^{1}$ of $\mathbb{A}^{3}$, whose general fibers are
smooth del Pezzo surfaces of degree $d$. For $d=9$ we have only the locally
trivial $\mathbb{P}^{2}$-bundles over $\mathbb{P}^{1}$. A classical example
with $d=8$ is recalled below. An example with $d=7$ does not exist, because
the generic fiber of the corresponding del Pezzo fibration $\pi\colon
V\rightarrow\mathbb{P}^{1}$ would be a minimal smooth del Pezzo surface of
degree $7$ and such surfaces do not exist over a field of characteristic zero,
see [Man86, Theorem 29.4]. An example with $d=6$ can be found in [Pro16,
Theorem 1.2] and [Fuk19]. A construction for $d=5$ is given in Example 6.3
below. For examples with $d=1,2,3,4$ see [DK18, Theorem 2]. For $d\leq 6$
general fibers of the restriction of $\pi\colon V\rightarrow\mathbb{P}^{1}$ to
$\mathbb{A}^{3}$ are not isomorphic to $\mathbb{A}^{2}$. Indeed, otherwise the
generic fiber of $\pi\colon V\rightarrow\mathbb{P}^{1}$ would be a minimal
smooth del Pezzo surface of degree $d\leq 6$ over the function field
$\mathbb{C}(\mathbb{P}^{1})$, containing a Zariski open subset isomorphic to
$\mathbb{A}^{2}_{\mathbb{C}(\mathbb{P}^{1})}$, which is impossible by [DK18,
Proposition 13]. In particular, none of these completions is of product type.
Note also that for $d\neq 9$ general fibers of the corresponding Mori fiber
spaces are smooth del Pezzo surfaces of Picard rank higher than one, hence are
not associated to any terminal rank one Fano pencil (cf. Example 4.11).

######  Example 6.2 (A Mori fiber completion of $\mathbb{A}^{3}$ into a del
Pezzo fibration of degree $8$).

Let $Q\subseteq\mathbb{P}^{4}$ be a smooth quadric threefold and let $F$ and
$H$ be distinct hyperplane sections of $Q$ such that $F$ is smooth and the
scheme-theoretic intersection $C=F\cap H$ is irreducible. Let $\mathcal{L}$ be
the pencil on $Q$ generated by $F$ and $H$. We see that $C$ is a smooth
rational curve and for a general member $Y$ of $\mathcal{L}$ the pair $(Y,C)$
is isomorphic to the pair consisting of $\mathbb{P}^{1}\times\mathbb{P}^{1}$
embedded as a smooth quadric surface in $\mathbb{P}^{3}$ and a smooth
hyperplane section of it. The rational map $\psi_{\mathcal{L}}\colon
Q\dashrightarrow\mathbb{P}^{1}$ lifts to a del Pezzo fibration $\pi\colon
V\rightarrow\mathbb{P}^{1}$ of degree 8 on the blow-up $\tau\colon
V\rightarrow Q$ of $Q$ with center at $C$. The variety $V$ is smooth and it
contains $Q\setminus H$ as the complement of the union of the exceptional
divisor $E$ of $\tau$ and of the proper transform of $H$. The general fibers
of $\pi\colon V\rightarrow\mathbb{P}^{1}$ have Picard rank $2$. On the other
hand, the generic fiber $V_{\eta}$ of $\pi$ is a smooth quadric surface over
the field $\mathbb{C}(\mathbb{P}^{1})$ and we have
$\rho(V_{\eta})=\rho(V/\mathbb{P}^{1})=1$. The Picard group of $V_{\eta}$ is
generated by the restriction $E_{\eta}$ of $E$. We note that since
$-(K_{V_{\eta}}+E_{\eta})=E_{\eta}$ is ample, for every open subset
$U\subseteq\mathbb{P}^{1}$, the open subset $\pi^{-1}(U)\setminus E$ of $V$ is
a relative affine Fano variety over $U$. Let now $H$ be a hyperplane section
coming from a hyperplane tangent to some point of $Q$. Then
$H\cong\mathbb{P}(1,1,2)$ and $Q\setminus H\cong\mathbb{A}^{3}$. In
particular, the corresponding del Pezzo fibration $\pi\colon
V\rightarrow\mathbb{P}^{1}$ is a Mori fiber completion of $\mathbb{A}^{3}$.

This Mori fiber completion $\pi\colon V\rightarrow\mathbb{P}^{1}$ of
$\mathbb{A}^{3}$ can also be obtained from a pencil of quadrics on
$\mathbb{P}^{3}$. Namely, consider the birational map $\varphi\colon
Q\dashrightarrow\mathbb{P}^{3}$ induced by the linear projection from the
unique singular point $q$ of $H\cong\mathbb{P}(1,1,2)$. A resolution of
$\varphi$ is given by the blow-up $\alpha\colon\widetilde{Q}\rightarrow Q$ of
$q$ with exceptional divisor $E\cong\mathbb{P}^{2}$. The birational morphism
$\varphi\circ\alpha\colon\widetilde{Q}\rightarrow\mathbb{P}^{3}$ maps $E$
isomorphically onto a hyperplane of $\mathbb{P}^{3}$, which we denote by
$H_{\infty}$, and it contracts the proper transform of $H$ onto a smooth conic
$C^{\prime}\subset H_{\infty}$. Note that $\varphi$ induces an isomorphism
$Q\setminus H\cong\mathbb{P}^{3}\setminus H_{\infty}$. The birational
transform of the pencil $\mathcal{L}$ is the pencil generated by $2H_{\infty}$
and the proper transform $F^{\prime}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$
of $F$, which is a smooth quadric surface in $\mathbb{P}^{3}$ intersecting
$H_{\infty}$ along the conic $C^{\prime}$.

Recall [Fuj81] that the quintic del Pezzo fourfold
$W_{5}\subset\mathbb{P}^{7}$ is the intersection of the Grassmannian
$\operatorname{Gr}(2,5)\subset\mathbb{P}^{9}$ with a general linear subspace
of codimension $2$. It is the unique smooth Fano fourfold with
$\operatorname{Pic}(W_{5})\cong\mathbb{Z}\langle H\rangle\cong\mathbb{Z}$,
Fano index $i_{W_{5}}=3$ and $H^{4}=5$, where $H$ is a hyperplane section of
$W_{5}$. A general hyperplane section of $W_{5}$ is a smooth Fano threefold
$V_{5}$ with $\operatorname{Pic}(V_{5})\cong\mathbb{Z}\langle
H_{\infty}\rangle\cong\mathbb{Z}$, Fano index $i_{V_{5}}=2$ and
$H_{\infty}^{3}=5$, where $H_{\infty}$ is a hyperplane section of $V_{5}$,
called the quintic del Pezzo threefold. Again this is the unique Fano
threefold with these invariants, see e.g. [Isk80, Chapter II, §1.6].

######  Example 6.3 (A pencil on the quintic del Pezzo threefold $V_{5}$).

This construction was communicated to us by Masaru Nagaoka. By [Fur93, Theorem
A], there exists an open embedding $\mathbb{A}^{3}\hookrightarrow V_{5}$ whose
complement is a normal singular hyperplane section $H_{\infty}$ (with an
isolated $A_{4}$ singularity). By adjunction the base locus of the pencil
$\mathcal{L}$ on $V_{5}$ generated by $H_{\infty}$ and a general hyperplane
section $F$ of $V_{5}$ is a smooth elliptic curve $C$. The rational map
$\psi_{\mathcal{L}}\colon V_{5}\dashrightarrow\mathbb{P}^{1}$ lifts to a del
Pezzo fibration $\pi\colon V\rightarrow\mathbb{P}^{1}$ of degree $5$ on the
blow-up $\tau\colon V\rightarrow V_{5}$ with center at $C$. The variety $V$ is
smooth and contains $\mathbb{A}^{3}$ as the complement of the union of the
proper transform of $H_{\infty}$ and of the exceptional divisor $E$ of $\tau$.
Note that since $E$ intersects general fibers of $\pi$ along an elliptic
curve, general fibers of the restriction of $\pi$ to $\mathbb{A}^{3}\subset V$
are not isomorphic to $\mathbb{A}^{2}$. Indeed, the boundary divisor for any
normal completion of $\mathbb{A}^{2}$ consists of rational curves only.

######  Example 6.4 (A pencil on the quintic del Pezzo fourfold $W_{5}$).

By [Pro94, Theorem 3.1(iv)] there exists an open embedding of $\mathbb{A}^{4}$
into the quintic del Pezzo fourfold $W_{5}$ such that the complement is a
normal hyperplane section $H$ of $W_{5}$ (its singular locus consists of a
unique ordinary double point $p$). Let $\mathcal{L}$ be a pencil on $W_{5}$
generated by $H$ and by a general hyperplane section $F$. The base locus of
$\mathcal{L}$ is a smooth del Pezzo surface $S$ of degree $5$. Let $\tau\colon
V\rightarrow W_{5}$ be the blowup of $W_{5}$ with center at $S$. The rational
map $\psi_{\mathcal{L}}\colon W_{5}\dashrightarrow\mathbb{P}^{1}$ lifts to a
Mori fiber space $\pi\colon V\rightarrow\mathbb{P}^{1}$ and its general fibers
are quintic del Pezzo threefolds. The variety $V$ is smooth and it contains
$\mathbb{A}^{4}$ as the complement of the union of the proper transform of $H$
and of the exceptional divisor $E$ of $\tau$. We note that $E$ intersects a
general fiber $V_{5}$ of $\pi$ along a smooth del Pezzo surface $B$ of degree
$5$. In particular, $V_{5}\setminus B$ is an affine Fano variety. Since $B$ is
smooth, by the classification in [Fur93], $V_{5}\setminus B$ is not isomorphic
to $\mathbb{A}^{3}$.

We argue that $V_{5}\setminus B$ is not super-rigid (see Definition 2.11). Let
$\ell\subseteq B$ be a line and let $T\subset V_{5}$ be the surface swept out
by the lines in $V_{5}$ intersecting $\ell$. The projection from $\ell$
defines a birational map $\alpha\colon V_{5}\dashrightarrow Q$ to a smooth
quadric threefold $Q$ in $\mathbb{P}^{4}$, which contracts $T$ onto a rational
cubic contained in a hyperplane section $Q_{0}$ of $Q$. The image of $B$ by
$\alpha$ is a smooth hyperplane section $Q_{\infty}$ of $Q$ and $\alpha$
induces an isomorphism $V_{5}\setminus(B\cup T)\cong Q\setminus(Q_{0}\cup
Q_{\infty})$. By Example 6.2, $Q\setminus(Q_{0}\cup Q_{\infty})$ contains a
relative affine Fano variety over $\mathbb{P}^{1}\setminus\\{0,\infty\\}$. So
$V_{5}\setminus B$ contains a relative affine Fano variety over a curve, hence
is not super-rigid.

###  6B. Pencils on smooth Fano varieties

Recall that the Grassmannian $\operatorname{Gr}(k,n)$, which parameterizes
$k$-dimensional linear subspaces of a complex vector space of dimension $n$,
is a smooth Fano variety of dimension $k(n-k)$ with class group isomorphic to
$\mathbb{Z}$ and Fano index $i_{\operatorname{Gr}(k,n)}=n$ (see e.g. [Dol12,
Lemma 10.1.1, p. 510]). It has a natural cover by affine open subsets
isomorphic to $\mathbb{A}^{k(n-k)}$. Namely, denoting the vector space by $V$,
we have the Plücker embedding

$\mathrm{pl}\colon\operatorname{Gr}(k,n)\hookrightarrow\mathbb{P}(\Lambda^{k}V)=\operatorname{Proj}(\mathbb{C}[\\{x_{I}\\}]),$

where $I$ ranges through the set of subsets of $k$ distinct elements in
$\\{1,\ldots,n\\}$, which associates to a closed point
$\Lambda\in\operatorname{Gr}(k,n)$, represented by a $k\times n$-matrix
$A_{\Lambda}$ of rank $k$, the collection of the $k\times k$-minors of
$A_{\Lambda}$. Then for every subset $I\subseteq\\{1,\ldots,n\\}$ of $k$
distinct elements the open subset
$\operatorname{Gr}(k,n)\setminus\\{x_{I}=0\\}$ is isomorphic to
$\mathbb{A}^{k(n-k)}$.

######  Proposition 6.5 (Mori fiber completions from pencils on
Grassmannians).

For $k(n-k)\geq 4$, let $H$ be a hyperplane section of
$\operatorname{Gr}(k,n)$ such that $\operatorname{Gr}(k,n)\setminus
H\cong\mathbb{A}^{k(n-k)}$, let $d\in\\{1,\ldots,n-1\\}$ and let
$F\subset\operatorname{Gr}(k,n)$ be an integral hypersurface such that $F\sim
dH$. Assume that $S=F\cap H$ is irreducible and contained in the smooth locus
of $F$ and that either $d\geq 2$ or $d=1$ and
$S\subseteq\operatorname{Sing}(H)$. Then
$\mathbb{A}^{k(n-k)}=\operatorname{Gr}(k,n)\setminus H$ admits a Mori fiber
completion over $\mathbb{P}^{1}$ such that all members of the pencil $\langle
F,dH\rangle$ other than $dH$ appear as fibers.

###### Proof.

Since $S=F\cap H$ is contained in the smooth locus of $F$, every member of
$\mathcal{L}$ other then $dH$ is smooth along $\operatorname{Bs}\mathcal{L}$
by Lemma 5.2. Since on the other hand $\operatorname{Gr}(k,n)$ is smooth, it
follows from Bertini’s theorem that a general member of $\mathcal{L}$ is
smooth away from $\operatorname{Bs}\mathcal{L}$, hence smooth. Since $S$ is by
assumption irreducible and contained in the smooth locus of $F$, the assertion
follows from Theorem 5.9. ∎

We now deduce Theorem 1.1, which asserts that given $n\geq 2$ and a hyperplane
in $H\subset\mathbb{P}^{n}$, for every integral hypersurface
$F\subseteq\mathbb{P}^{n}$ of degree $d\leq n$ such that $F\cap H$ is
irreducible and contained in the smooth locus $F_{\mathrm{reg}}$ of $F$ there
exists a completion of the affine $n$-space
$\mathbb{A}^{n}\cong\mathbb{P}^{n}\setminus H$ into a Mori fiber completion
over $\mathbb{P}^{1}$ such that all members of the pencil $\langle
F,dH\rangle$ other than $dH$ appear as fibers.

###### Proof of Theorem 1.1.

The case $d=1$ is obvious, so we may assume that $d\geq 2$. We have
$\operatorname{Gr}(1,n+1)=\mathbb{P}^{n}$, so for $n\geq 4$ the result follows
from Proposition 6.5. We are thus left with the three cases $(n,d)=(2,2)$,
$(3,2)$ and $(3,3)$. The case $(2,2)$ is treated in Example 4.14. In the case
$(3,2)$, $F\subset\mathbb{P}^{3}$ is an integral quadric surface such that
$F\cap H$ is irreducible and contained in $F_{\mathrm{reg}}$. By Corollary 5.5
and by Bertini’s theorem, a general member of the pencil $\mathcal{L}=\langle
F,2H\rangle$ is a smooth quadric surface, so the assertion follows from
Example 6.2. In the remaining case $(3,3)$, $F\subset\mathbb{P}^{3}$ is an
integral cubic surface such that $F\cap H$ is irreducible and contained in
$F_{\mathrm{reg}}$. Again, by Corollary 5.5 and Bertini’s theorem, a general
member of the pencil $\mathcal{L}=\langle F,3H\rangle$ is a smooth cubic
surface which intersects $H$ along a smooth elliptic curve, so the result
follows from [DK17, Theorem. (a)]. ∎

######  Example 6.6 (Families of Mori fiber completions of $\mathbb{A}^{n}$).

Let $n\geq 4$ and $d\geq 2$ be integers. Put
$\mathbb{P}^{n}=\operatorname{Proj}(\mathbb{C}[x_{0},\ldots,x_{n}])$ and
$H=\\{x_{0}=0\\}\subset\mathbb{P}^{n}$. Let
$\mathbb{C}[x_{1},\ldots,x_{n}]_{\leq d}$ denote the affine space of
polynomials of total degree at most $d$. Let $\mathcal{V}_{d}$ denote its open
subset consisting of polynomials $f$ of total degree precisely $d$ for which
the scheme-theoretic intersection of $H$ with the closure $F$ in
$\mathbb{P}^{n}$ of the zero locus of $f$ in
$\mathbb{A}^{n}=\operatorname{Spec}(\mathbb{C}[x_{1},\ldots,x_{n}])$ is
smooth. So, for every $f\in\mathcal{V}_{d}$, $F$ is an irreducible
hypersurface of degree $d$ which contains the smooth variety $F\cap H$ in its
smooth locus. Theorem 1.1 thus applies and for each $f\in\mathcal{V}_{d}$
gives the existence of a Mori fiber completion $\pi\colon
X\rightarrow\mathbb{P}^{1}$ of $\mathbb{A}^{n}=\mathbb{P}^{n}\setminus H$ such
that all members of the pencil $\langle F,dH\rangle$ other than $dH$ appear as
fibers, and for which we have a commutative diagram

$\textstyle{\mathbb{A}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathbb{A}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{1}}$

where the horizontal morphisms are open immersions.

We now present two other examples of completions of $\mathbb{A}^{4}$
constructed respectively from the quintic del Pezzo fourfold and Fano-Mukai
fourfolds of genus $10$.

######  Example 6.7 (Another pencil on the quintic del Pezzo fourfold
$W_{5}$).

As in Example 6.4 above, let $W_{5}\subset\mathbb{P}^{7}$ be the quintic del
Pezzo fourfold and let $H\subset W_{5}$ be a singular hyperplane section with
a unique ordinary double point $p$ whose complement is isomorphic to
$\mathbb{A}^{4}$. Let $\mathcal{L}$ be the pencil on $W_{5}$ generated by $2H$
and a general quadric section $F$. A general member $Y$ of $\mathcal{L}$ is a
smooth Fano threefold of Picard rank one and index one isomorphic to the
intersection of the Grassmannian $\operatorname{Gr}(2,5)$ with two hyperplanes
and a quadric (family $B_{10}$ in [Bea77, Theorem 5.3]). Furthermore,
$S=(\operatorname{Bs}\mathcal{L})_{\mathrm{red}}=H|_{Y}$ is a smooth
anticanonical divisor on $Y$, hence it is a smooth $\mathrm{K}3$ surface
(which implies that $Y\setminus S$ is not an affine Fano variety). By Theorem
5.9 the pencil $\mathcal{L}$ gives rise to a Mori fiber completion $\pi\colon
V\rightarrow\mathbb{P}^{1}$ of $\mathbb{A}^{4}=W_{5}\setminus H$, whose
general fibers are isomorphic to the general members of $\mathcal{L}$. Thus,
by [Bea77, Theorem 5.6(ii)], general fibers of $\pi$ are non-rational.

######  Example 6.8 (A pencil on a Fano-Mukai fourfold of genus $10$).

A Fano-Mukai fourfold of genus $10$ is a smooth Fano fourfold $X$ of Picard
rank one, Fano index $i_{X}=2$, and genus $g:=\frac{1}{2}(H^{4})+1=10$, where
$H$ is an ample generator of $\operatorname{Pic}X$. By [PZ18] the moduli space
of such fourfolds has dimension one and for every such fourfold $X$ there
exists an open embedding $\mathbb{A}^{4}\hookrightarrow X$ whose complement is
a generator $H_{\infty}$ of $\operatorname{Pic}X$ and whose singular locus $T$
is a surface. Let $\mathcal{L}$ be the pencil generated by $H_{\infty}$ and a
general smooth member $F$ of the complete linear system
$\big{|}H_{\infty}\big{|}$. Since the restriction homomorphism
$\operatorname{Pic}(X)\rightarrow\operatorname{Pic}(F)$ is an isomorphism by
Lemma 5.7, the base locus $\operatorname{Bs}\mathcal{L}=F\cap H_{\infty}$ of
$\mathcal{L}$ is an irreducible and reduced surface, which we denote by $S$.
The singular locus of $S$ is equal to the curve $C=T\cap S$. Since $S$ is
singular but not contained in the singular locus of $H_{\infty}$, we cannot
directly apply Theorem 5.9 to obtain from $\mathcal{L}$ a Mori fiber
completion of $\mathbb{A}^{4}=X\setminus H_{\infty}$ over $\mathbb{P}^{1}$.
Instead, we argue as follows: by Bertini’s theorem a general member
$\mathcal{L}_{\lambda}$ of $\mathcal{L}$ is smooth off the base locus $S$.
Furthermore, since the scheme-theoretic intersection
$\mathcal{L}_{\lambda}|_{H_{\infty}}=S$ is smooth off $C=T\cap S$, it follows
that $\mathcal{L}_{\lambda}$ is smooth off the curve $C$. Since $F$ is smooth
at every point $x\in C$ whereas $H_{\infty}$ is singular there, every member
of $\mathcal{L}$ other than $H_{\infty}$ is smooth at $x$ by Lemma 5.2. We
infer that a general member $\mathcal{L}_{\lambda}$ is smooth, so the base
locus $S$ of $\mathcal{L}$ is the scheme-theoretic transverse intersection of
any two smooth members of $\mathcal{L}$. The blow-up $\tau\colon V\rightarrow
X$ of $S$ is then a resolution of $\psi_{\mathcal{L}}\colon
X\dashrightarrow\mathbb{P}^{1}$ and $\psi_{\mathcal{L}}\circ\tau\colon
V\rightarrow\mathbb{P}^{1}$ is a Mori fiber space whose general fibers are
Fano threefolds of Picard rank one, index $1$ and genus $10$. The variety $V$
contains $\mathbb{A}^{4}$ as the complement of the union of the proper
transform of $H_{\infty}$ and of the exceptional locus of $\tau$.

######  Example 6.9 (Mori fiber completions of $\mathbb{A}^{n}$ with super-
rigid affine Fano general fibers).

Let $n\geq 4$ and let $\mathcal{L}$ be a pencil on $\mathbb{P}^{n}$ generated
by a general hypersurface of degree $n-1$ and by $(n-1)H$, where $H$ is a
hyperplane. By Corollary 5.10, $\mathcal{L}$ gives rise to a Mori fiber
completion $\pi\colon V\rightarrow\mathbb{P}^{1}$ of
$\mathbb{A}^{n}=\mathbb{P}^{n}\setminus H$ such that general fibers of
$\pi|_{\mathbb{A}^{n}}$ are smooth affine Fano varieties, isomorphic to the
complement of a smooth hyperplane section of a hypersurface of degree $n-1$ in
$\mathbb{P}^{n}$. If $n\geq 6$ then it is known that such affine Fano
varieties are super-rigid [CDP18, Theorem 2.8, Example 2.9]. For $n=4,5$ the
super-rigidity of complements of general hyperplane sections of respectively
smooth cubic threefolds in $\mathbb{P}^{4}$ and smooth quartic fourfolds in
$\mathbb{P}^{5}$ is an open problem.

###  6C. Pencils on weighted projective spaces

An important class of $\mathbb{Q}$-factorial rational Fano varieties consists
of weighted projective spaces. We fix notation and summarize some basic facts
(see e.g. [Dol82] and [IF00] for more). Given a non-decreasing sequence of
positive integers $\bar{a}=(a_{0},\ldots,a_{n})$, we define an
$\mathbb{N}$-grading on $\mathbb{C}[x_{0},\ldots,x_{n}]$ by putting $\deg
x_{i}=a_{i}$, and we let
$\mathbb{P}(\bar{a})=\operatorname{Proj}(\mathbb{C}[x_{0},\ldots,x_{n}])$. The
inclusion of graded rings
$\mathbb{C}[y_{0}^{a_{0}},\ldots,y_{n}^{a_{n}}]\subseteq\mathbb{C}[y_{0},\ldots,y_{n}]$
leads under the identification $x_{i}=y_{i}^{a_{i}}$ to a finite morphism

(6.1)
$\pi\colon\mathbb{P}^{n}\rightarrow\mathbb{P}(\bar{a})\cong\mathbb{P}^{n}/(\mathbb{Z}_{a_{0}}\times\cdots\times\mathbb{Z}_{a_{n}}),$

where the action is diagonal, by multiplication by an $a_{i}$-th root of unity
on the $i$-th factor, see [Dol82, §1.2.2]. The variety $\mathbb{P}(\bar{a})$
is covered by the affine open subsets $\\{x_{i}\neq
0\\}\cong\mathbb{A}^{n}/\mathbb{Z}_{a_{i}}$, where the generator
$\varepsilon$, a primitive $a_{i}$-th root of unity, acts by

$(x_{0},\ldots,\widehat{x_{i}}\ldots,x_{n})\mapsto(\varepsilon^{a_{0}}x_{0},\ldots,\widehat{\varepsilon^{a_{i}}x_{i}}\ldots,\varepsilon^{a_{n}}x_{n}).$

In particular, $\mathbb{P}(\bar{a})$ is normal and $\mathbb{Q}$-factorial
[KM98, Lemma 5.16], with finite quotient singularities. (For a criterion when
$\mathbb{P}(\bar{a})$ is klt see [Kas13, Proposition 2.3], cf. [CLS11,
Proposition 11.4.12]). Since
$\mathbb{P}(a_{0},da_{1},\ldots,da_{n})\cong\mathbb{P}(a_{0},a_{1},\ldots,a_{n})$
for every $d>0$, we can assume without loss of generality that
$\gcd(a_{0},\ldots\widehat{a}_{i},\ldots,a_{n})=1$ for every
$i\in\\{0,1,\ldots,n\\}$, in which case one says that the description of the
weighted projective space is _well-formed_. The singular locus of a well-
formed $\mathbb{P}(\bar{a})$ can be described as follows [IF00, 5.15]:

(6.2) $[x_{0}:\ldots:x_{n}]\in\operatorname{Sing}\mathbb{P}(\bar{a})\
\Leftrightarrow\gcd\\{a_{i}:x_{i}\neq 0\\}>1.$

By [Mor75, Proposition 2.3] the class group of $\mathbb{P}(\bar{a})$ is
isomorphic to $\mathbb{Z}$ and is generated by the class of the divisorial
sheaf $\mathcal{O}_{\mathbb{P}(\bar{a})}(1)$ on $\mathbb{P}(\bar{a})$.
Furthermore, the sheaf $\mathcal{O}_{\mathbb{P}(\bar{a})}(m)$, where $m$ is
the least common multiple of $a_{0},\ldots,a_{n}$, is invertible and its class
generates the Picard group of $\mathbb{P}(\bar{a})$; see also [CLS11, Exercise
4.1.5 and 4.2.11].

Put $\bar{x}=(x_{0},\ldots,x_{n})$. After fixing $\bar{a}$ such that the
description of $\mathbb{P}(\bar{a})$ is well-formed, we denote by
$\mathbb{C}[\bar{x}]_{(d)}$ the set of weighted homogeneous polynomials of
degree $d$ in the variables $\bar{x}$ with respect to the weights $\bar{a}$.
Let $f(\bar{x})\in\mathbb{C}[\bar{x}]_{(d)}$. A zero scheme $Y=Z(f(\bar{x}))$
has, by definition, _degree_ $d$. We say that $Y$ is _quasi-smooth_ if its
affine cone

$\mathcal{C}(Y)=\operatorname{Spec}(\mathbb{C}[\bar{x}]/(f(\bar{x})))\subseteq\mathbb{A}^{n+1}=\operatorname{Spec}(\mathbb{C}[\bar{x}])$

is smooth off the origin, equivalently, if $\\{\pi^{*}f=0\\}$ is a smooth
subvariety of $\mathbb{P}^{n}$, see (6.1). The singularities of a quasi-smooth
subvariety are finite quotient singularities, hence, in particular klt by
[Ish18, Theorem 7.4.9].

From now on we assume that $n\geq 4$ and that $a_{0}=1$. We put

(6.3) $\mathbb{P}=\mathbb{P}(1,a_{1},\ldots,a_{n})\quad\text{and}\quad
H=\\{x_{0}=0\\}\cong\mathbb{P}(a_{1},\cdots,a_{n}).$

Then $\mathcal{O}_{\mathbb{P}}(H)\cong\mathcal{O}_{\mathbb{P}}(1)$ generates
$\operatorname{Cl}(\mathbb{P})$ and $\mathbb{P}\setminus H$ is isomorphic to
the affine $n$-space $\mathbb{A}^{n}$ with inhomogeneous coordinates
$x_{i}/x_{0}^{a_{i}}$, where $i=1,\ldots,n$. Furthermore, since
$K_{\mathbb{P}}\sim-\sum_{i=0}^{n}\\{x_{i}=0\\}$ (see e.g. [Dol82, §2.1]), the
Fano index $i_{\mathbb{P}}$ of $\mathbb{P}$ is equal to
$1+\sum_{i=1}^{n}a_{i}$. The induced description of $H$ as
$\mathbb{P}(a_{1},\ldots,a_{n})$ is not necessarily well-formed (take for
instance $\bar{a}=(1,1,d,\ldots,d)$), but if it is, then by (6.2) we have
$\operatorname{Sing}\mathbb{P}=\operatorname{Sing}H$, so in this case the
singular locus of $\mathbb{P}$ has codimension at least $3$.

Theorem 1.4 is a consequence of the combination of the following result with
Corollary 4.13.

######  Proposition 6.10 (Mori fiber completions from pencils on
$\mathbb{P}(1,\bar{a})$).

Let $\mathbb{P}$ and $H$ be as in (6.3). Assume that $\mathbb{P}$ is smooth in
codimension $2$ (equivalently, the induced description of $H$ is well-formed)
and let $F\subseteq\mathbb{P}$ be a quasi-smooth terminal hypersurface of
degree $d\in\\{2,\ldots,\sum_{i=1}^{n}a_{i}\\}$. Then the pencil $\mathcal{L}$
generated by $F$ and $dH$ is a terminal rank one Fano pencil with quasi-smooth
general members and the associated rational map
$\psi_{\mathcal{L}}\colon\mathbb{P}\dashrightarrow\mathbb{P}^{1}$ admits a
compatible thrifty resolution with discrepancy locus contained in
$\\{(\psi_{\mathcal{L}})_{*}H\\}$.

###### Proof.

Let $f\in\mathbb{C}[\bar{x}]_{(d)}$ be the irreducible weighted homogeneous
polynomial defining the hypersurface $F$. The base locus
$\operatorname{Bs}\mathcal{L}$ is the codimension $2$ weighted complete
intersection of $\mathbb{P}$ with weighted homogeneous ideal
$(f(\bar{x}),x_{0}^{d})$. The graph $\Gamma$ of $\psi_{\mathcal{L}}$ is
isomorphic to the hypersurface in
$\mathbb{P}\times\mathbb{P}^{1}_{[u_{0}:u_{1}]}$ defined by the bi-homogeneous
equation $f(\bar{x})u_{1}+x_{0}^{d}u_{0}=0$ and the projection
$\operatorname{pr}_{\mathbb{P}}$ induces isomorphisms between closed fibers of
the restriction to $\Gamma$ of the projection
$\operatorname{p}=(\operatorname{pr}_{\mathbb{P}^{1}})|_{\Gamma}$ and members
of $\mathcal{L}$.

Since $F$ is terminal, $\mathcal{L}$ is a terminal pencil by Lemma 4.5. By the
Lefschetz hyperplane section theorem for weighted projective spaces [Mor75,
Theorem 3.7] and [Oka19, Remark 4.2], the group $\operatorname{Cl}(F)$ is
isomorphic to $\mathbb{Z}$. It is generated by the restriction $H|_{F}$ of $H$
to $F$ as a Weil divisor, for which
$\mathcal{O}_{F}(H|_{F})\cong(\mathcal{O}_{\mathbb{P}}(1)|_{F})^{\vee\vee}$.
This implies in particular that $F\cap H=S$ is irreducible, hence that the
exceptional locus $E\cong S\times\mathbb{P}^{1}$ of
$\operatorname{pr}_{\mathbb{P}}$ is a prime divisor on $\Gamma$. Since
$\mathbb{P}$ is smooth in codimension two, every Weil divisor on it is Cartier
in codimension two in $\mathbb{P}$. By assumption $F$ is terminal, so it is
smooth in codimension $2$, hence smooth and Cartier at general points of $S$.
We have $d\geq 2$, so every member of $\mathcal{L}$ other than $dH$ is prime
by Lemma 5.3, and since $2\leq d<i_{\mathbb{P}}$, every normal member of
$\mathcal{L}$ is Fano by Lemma 5.6. Since members of $\mathcal{L}$ are not
necessarily Cartier but only $\mathbb{Q}$-Cartier, the fact that all of them
have class group isomorphic to $\mathbb{Z}$, hence have Picard rank one,
follows again from the Lefschetz hyperplane section theorem for weighted
projective spaces. Thus, $\mathcal{L}$ is a terminal rank one Fano pencil with
degeneracy locus $\delta(\mathcal{L})=\\{(\psi_{\mathcal{L}})_{*}H\\}$.

The affine cone $\mathcal{C}(Y)$ over a member $Y$ of $\mathcal{L}$ other than
$dH$ is isomorphic to the hypersurface in $\mathbb{A}^{n+1}$ defined by the
equation $f(\bar{x})+tx_{0}^{d}=0$ for some $t\in\mathbb{C}$. Since
$\mathcal{C}(F)$ is smooth off the origin $\\{O\\}$, it follows from Bertini’s
theorem that a general $\mathcal{C}(Y)$ is smooth outside its intersection
with $\mathcal{C}(H)=\\{x_{0}=0\\}$.

Furthermore, since $d\geq 2$, it follows from Lemma 5.2 that
$\mathcal{C}(Y)\setminus\\{O\\}$ is smooth in a neighborhood of
$(\mathcal{C}(Y)\cap\\{x_{0}=0\\})\setminus\\{O\\}$. A general member $Y$ is
thus quasi-smooth and every member $Y$ other than $dH$ is klt in a
neighborhood of $S$.

In view of Lemma 2.4, every thrifty $\mathbb{Q}$-factorial terminal resolution
of $\mathcal{L}$ is compatible, provided that the open subset
$\Gamma_{\infty}=\Gamma\setminus H^{\prime}$ of $\Gamma$, where $H^{\prime}$
is the proper transform of $H$, is terminal and $\mathbb{Q}$-factorial. Thus
it remains to show that $\Gamma_{\infty}$ is $\mathbb{Q}$-factorial terminal.
Every member $Y$ of $\mathcal{L}$ other than $dH$ is klt in a neighborhood of
$S$, so by Proposition 4.6(a), $\Gamma_{\infty}$ is normal in a neighborhood
of $E_{\infty}=E\cap\Gamma_{\infty}$. We have $\Gamma_{\infty}\setminus
E_{\infty}\cong\mathbb{P}\setminus H\cong\mathbb{A}^{n}$, so $\Gamma_{\infty}$
is normal and its class group is generated by irreducible components of
$E_{\infty}$. Since $E_{\infty}$ is irreducible and $\mathbb{Q}$-Cartier, we
conclude that $\Gamma_{\infty}$ is normal and $\mathbb{Q}$-factorial. The
pencil $\mathcal{L}$ is terminal and its members other than $dH$ are klt in a
neighborhood of $S$. By Proposition 4.6(b), $\Gamma_{\infty}$ is terminal in a
neighborhood of $E_{\infty}$, hence it is terminal. ∎

As a corollary we obtain the following result (cf. Definition 2.8):

######  Corollary 6.11 (Mori fiber completions of $\mathbb{A}^{4}$ with
birationally rigid fibers).

There exist at least $95$ pairwise non weakly square birationally equivalent
Mori fiber completions of $\mathbb{A}^{4}$ over $\mathbb{P}^{1}$ with quasi-
smooth terminal birationally rigid general fibers.

###### Proof.

Let $\bar{a}_{j}=(a_{1}^{(j)},a_{2}^{(j)},a_{3}^{(j)},a_{4}^{(j)})$ for
$j=1,2$ be two distinct sequences in the list of 95 non-decreasing sequences
of [IF00, §13.3, Lemma 16.4 and §16.6]. Put
$\mathbb{P}_{j}=\mathbb{P}(1,\bar{a}_{j})$ and let
$d_{j}=\sum_{i=1}^{4}a_{i}^{(j)}$. Looking at the list one checks that the
hyperplane $H_{j}=\mathbb{P}(\bar{a}_{j})\subset\mathbb{P}_{j}$ is a well-
formed weighted projective space. Let $F_{j}\subseteq\mathbb{P}_{j}$ be a
general hypersurface of degree $d_{j}$. By construction $F_{j}$ is a quasi-
smooth and terminal Fano variety of index $1$ anticanonically embedded into