# Birational geometry of Calabi-Yau pairs and $3$-dimensional Cremona
transformations

Carolina Araujo  Carolina Araujo
IMPA
Estrada Dona Castorina 110
22460-320 Rio de Janeiro
Brazil<EMAIL_ADDRESS>,  Alessio Corti  Alessio Corti
Department of Mathematics
Imperial College London
180 Queen’s Gate
London SW7 2AZ
United Kingdom<EMAIL_ADDRESS>and  Alex Massarenti  Alex Massarenti
Dipartimento di Matematica e Informatica, Università di Ferrara, Via
Machiavelli 30, 44121 Ferrara, Italy<EMAIL_ADDRESS>
###### Abstract.

In this paper we develop a framework that allows one to describe the
birational geometry of Calabi–Yau pairs $(X,D)$. After establishing some
general results for Calabi–Yau pairs $(X,D)$ with mild singularities, we focus
on the special case when $X=\mathbb{P}^{3}$ and $D\subset\mathbb{P}^{3}$ is a
quartic surface. We investigate how the appearance of increasingly worse
singularities in $D$ enriches the birational geometry of the pair
$(\mathbb{P}^{3},D)$, and lead to interesting subgroups of the Cremona group
of $\mathbb{P}^{3}$.

###### Key words and phrases:

Sarkisov program, Calabi–Yau pairs, Cremona group

###### 2020 Mathematics Subject Classification:

14E30, 14E05, 14E07

###### Contents

  1. 1 Introduction
  2. 2 The Sarkisov program for Mf CY pairs
  3. 3 Proof of Theorem A
  4. 4 Extremal contractions
  5. 5 Proof of Theorem B
  6. 6 Proof of Theorem C

##  1\. Introduction

###  1.1. Overview

In [Ogu13, Ogu12], Oguiso addressed the following question, attributed to
Gizatullin:

_Which automorphisms of a smooth quartic surface $D\subset\mathbb{P}^{3}$ are
induced by Cremona transformations of $\mathbb{P}^{3}$?_

He produced several interesting examples of smooth quartic surfaces in
$\mathbb{P}^{3}$, among which:

  1. (1)

A smooth quartic surface $D\subset\mathbb{P}^{3}$ with
$\operatorname{Aut}(D)\cong\mathbb{Z}$, and such that no nontrivial
automorphism of $D$ is induced by a Cremona transformation of
$\mathbb{P}^{3}$, see [Ogu13, Theorem 1.2] and [Ogu12, Theorem 1.8].

  2. (2)

A smooth quartic surface $D\subset\mathbb{P}^{3}$ with
$\operatorname{Aut}(D)\cong\mathbb{Z}_{2}*\mathbb{Z}_{2}*\mathbb{Z}_{2}$, and
such that every automorphism of $D$ is induced by a Cremona transformation of
$\mathbb{P}^{3}$, see [Ogu12, Theorem 1.7].

More recently, Paiva and Quedo produced examples of smooth quartic surfaces
$D\subset\mathbb{P}^{3}$ with
$\operatorname{Aut}(D)\cong\mathbb{Z}_{2}*\mathbb{Z}_{2}$, and such that no
nontrivial automorphism of $D$ is induced by a Cremona transformation of
$\mathbb{P}^{3}$, see [PQ23, Theorem 17].

The pair $(\mathbb{P}^{3},D)$, where $D\subset\mathbb{P}^{3}$ is a smooth
quartic surface, is an example of a _Calabi–Yau (CY) pair_ , that is, a pair
$(X,D)$, consisting of a normal projective variety $X$ and an effective Weil
divisor $D$ on $X$ such that $K_{X}+D\sim 0$. Oguiso’s results can be
interpreted as statements about the _birational geometry of the CY pair
$(\mathbb{P}^{3},D)$_, which is the theme of this paper.

Given a CY pair $(X,D)$, there is a rational volume form $\omega$ on $X$,
unique up to scaling by a nonzero constant, such that
$D+\operatorname{div}_{X}\omega=0$. Our goal is to understand birational self-
maps of $X$ that preserve the volume form $\omega$, up to scaling. In
particular, we are interested in the structure of the group
$\operatorname{Bir}(X,D)$ of such volume preserving birational maps. When the
pair $(X,D)$ has _canonical singularities_ , see Definition 1.2, a birational
self-map of $X$ is volume preserving if and only if it restricts to a
birational self-map of $D$ (Proposition 2.6): hence, there is a group
homomorphism

$r\colon\operatorname{Bir}(X,D)\ \to\ \operatorname{Bir}(D)\,.$

When $D\subset\mathbb{P}^{3}$ is a smooth quartic surface, the pair
$(\mathbb{P}^{3},D)$ has canonical (even terminal) singularities and
$\operatorname{Bir}(D)=\operatorname{Aut}(D)$. So Gizatullin’s question can be
re-stated as follows:

_For a smooth quartic surface $D\subset\mathbb{P}^{3}$, what is the image of
the restriction homomorphism_

$r\colon\operatorname{Bir}(\mathbb{P}^{3},D)\ \to\ \operatorname{Aut}(D)\,?$

We are also interested in describing the kernel of $r$. More generally, in
dimension $n\geq 3$, consider a CY pair $(\mathbb{P}^{n},D_{n+1})$, where
$D_{n+1}$ is a degree $n+1$ hypersurface. In this case,
$\operatorname{Bir}(\mathbb{P}^{n},D_{n+1})$ is a subgroup of the Cremona
group $\operatorname{Bir}(\mathbb{P}^{n})$. If we want to produce interesting
subgroups of $\operatorname{Bir}(\mathbb{P}^{n})$ in this way, then we must
pick a hypersurface $D_{n+1}$ with special properties. Indeed, our first main
result (Theorem A) implies that, if $D_{n+1}$ is general (very general if
$n=3$), then the group $\operatorname{Bir}(\mathbb{P}^{n},D_{n+1})$ is
trivial. More precisely, if the pair $(\mathbb{P}^{n},D_{n+1})$ has _terminal
singularities_ (see Definition 1.2), and the divisor class group of $D_{n+1}$
is generated by $\mathcal{O}_{D_{n+1}}(1)$, then
$\operatorname{Bir}(\mathbb{P}^{n},D_{n+1})=\operatorname{Aut}(\mathbb{P}^{n},D_{n+1})$.111We
denote by $\operatorname{Aut}(X,D)$ the group of automorphisms of $X$ that
stabilize $D$.

The $2$-dimensional case $(\mathbb{P}^{2},C)$ has also been considered. When
$C$ is a smooth cubic, the restriction homomorphism
$r\colon\operatorname{Bir}(\mathbb{P}^{2},C)\ \to\ \operatorname{Aut}(C)$ is
surjective, see [Ogu12, Theorem 2.2]. In this case, generators for
$\operatorname{Bir}(\mathbb{P}^{2},C)$ are described in [Pan07, Théorème
1.4.], while the kernel of $r$ is generated by its elements of degree $3$
[Bla08, Theorem 1]. When $C$ is the sum of the three coordinate lines,
generators for $\operatorname{Bir}(\mathbb{P}^{2},C)$ were given in [Bla13,
Theorem 1].

Next we turn to the setting of _singular_ quartic surfaces
$D\subset\mathbb{P}^{3}$. Our results illustrate how the appearance of
increasingly worse singularities on $D$ enriches the birational geometry of
the pair $(\mathbb{P}^{3},D)$. The simplest case to consider is when
$D\subset\mathbb{P}^{3}$ is a quartic surface having a single
$A_{1}$-singularity222In this case the pair $(\mathbb{P}^{3},D)$ has canonical
but not terminal singularities., and the class group of $D$ is generated by
$\mathcal{O}_{D}(1)$. Theorem B gives a complete description of
$\operatorname{Bir}(\mathbb{P}^{3},D)$ in this case. We then investigate what
happens when the singularities of $D$ become worse. Theorem C addresses the
birational geometry of the pair $(\mathbb{P}^{3},D)$ when
$D\subset\mathbb{P}^{3}$ is a quartic surface with a single
$A_{2}$-singularity, and the class group of $D$ is generated by
$\mathcal{O}_{D}(1)$. At the most singular end of the spectrum, we find
rational quartic surfaces. These have been proven to be Cremona equivalent to
a plane in [Mel20, Theorem 1]. Furthermore, log Calabi–Yau pairs of the form
$(\mathbb{P}^{3},D)$ and coregularity at most one have been recently
classified in [Duc22].

Our main tool in this paper is a factorization theorem for volume preserving
birational maps. We briefly explain it here, and refer to Section 2 for
details. The _Sarkisov_ program allows one to factorize any birational map
$\psi\in\operatorname{Bir}(\mathbb{P}^{n})$ as a _chain of links_ between Mori
fibre spaces:

$\mathbb{P}^{n}=X_{0}\ $$\ X_{1}\ $$\ \ \cdots\ \ $$\ X_{n-1}\ $$\
X_{n}=\mathbb{P}^{n}.$$\operatorname{Spec}\mathbb{C}$ $\ Z_{1}\ $$\ Z_{n-1}\
$$\operatorname{Spec}\mathbb{C}$
$\scriptstyle{\psi_{1}}$$\scriptstyle{\psi_{2}}$$\scriptstyle{\psi_{n-1}}$$\scriptstyle{\psi_{n}}$$\scriptstyle{\psi}$

The _Sarkisov_ program was established in dimension 3 in [Cor95], and in
higher dimensions in [HM13]. The volume preserving version of this theorem was
established in [CK16]. It states that, if the birational map $\psi$ is volume
preserving for some _log canonical_ CY pair $(\mathbb{P}^{n},D)$, then there
is a factorization as above, and divisors $D_{i}\subset X_{i}$ making each
$(X_{i},D_{i})$ a mildly singular CY pair (see Definition 1.4 for the notion
of _(t, lc)_ pair), and each $\psi_{i}$ a volume preserving link between CY
pairs (see Definition 1.1).

A mildly singular CY pair $(X,D)$ together with a Mori fibre space structure
$X\to Z$ is called a _Mori fibered (Mf) CY pair_ (see Definition 1.5). The
_pliability_ of a Mf CY pair $(X,D)$ is the set $\mathcal{P}(X,D)$ of
equivalence classes of Mf CY pairs that admit a volume preserving birational
map to $(X,D)$ (see Definition 1.6 for the precise notion of equivalence in
this setting). Our main theorems not only describe $\operatorname{Bir}(X,D)$,
but also determine the pliability $\mathcal{P}(X,D)$ of the CY pairs in
question.

Theorem A states that, if $X$ is a Fano variety with $\rho(X)=1$, $(X,D)$ is a
CY pair with terminal singularities, and the class group of $D$ is generated
by the restriction of a divisor on $X$, then
$\mathcal{P}(X,D)=\big{\\{}(X,D)\to\operatorname{Spec}\mathbb{C}\big{\\}}$ is
a set with one element, and $\operatorname{Bir}(X,D)=\operatorname{Aut}(X,D)$.

In the setting of Theorem B, let $D\subset\mathbb{P}^{3}$ be a quartic surface
with a single $A_{1}$-singularity $z\in D$, and suppose that the class group
of $D$ is generated by $\mathcal{O}_{D}(1)$. The blow-up $X\to\mathbb{P}^{3}$
of $z\in\mathbb{P}^{3}$, together with the strict transform $D_{X}\subset X$
of $D$, and the $\mathbb{P}^{1}$-bundle $X\to\mathbb{P}^{2}$ induced by the
projection from $z$, is a Mf CY pair. Moreover, the birational morphism
$(X,D_{X})\to(\mathbb{P}^{3},D)$ is volume preserving. Theorem B states that
the pliability of $(\mathbb{P}^{3},D)$ is the set consisting of the two
elements $(\mathbb{P}^{3},D)\to\operatorname{Spec}\mathbb{C}$ and
$(X,D_{X})\to\mathbb{P}^{2}$.

Our Theorem C describes the pliability of the pair $(\mathbb{P}^{3},D)$ when
$D\subset\mathbb{P}^{3}$ is a quartic surface with a single
$A_{2}$-singularity, and the class group of $D$ is generated by
$\mathcal{O}_{D}(1)$. We were surprised to discover just how large this
pliability is.

Throughout the paper we work over the field $\mathbb{C}$ of complex numbers,
or, more generally any algebraically closed field of characteristic zero.

In the remaining of the introduction, we define all the relevant notions
regarding CY pairs and state precisely our main results, Theorems A, B and C.
In Section 2, we review the Sarkisov program for Mf CY pairs. In Sections 3, 5
and 6, we prove Theorems A, B and C, respectively.

###  1.2. Calabi-Yau pairs

######  Definition 1.1.

A _Calabi–Yau (CY) pair_ is a pair $(X,D)$ consisting of a normal projective
variety $X$ and an effective integral Weil divisor $D$ on $X$ such that
$K_{X}+D$ is linearly equivalent to $0$. There exists a top degree rational
differential form $\omega=\omega_{X,D}$ on $X$, unique up to multiplication by
a nonzero constant, such that $D+\operatorname{div}_{X}\omega=0$. With a
slight abuse of language, we call $\omega$ the _volume form_ of the CY pair
$(X,D)$.

Let $(X,D_{X})$ and $(Y,D_{Y})$ be CY pairs, with associated volume forms
$\omega_{X,D_{X}}$ and $\omega_{Y,D_{Y}}$. A birational map $\varphi\colon
X\dasharrow Y$ is _volume preserving_ if there exists a nonzero constant
$\lambda\in\mathbb{C}^{\times}$ such that

$\varphi^{*}(\omega_{Y,D_{Y}})=\lambda\omega_{X,D_{X}}.$

We abuse language and say that $\varphi$ is a _birational map of CY pairs_ to
mean that it is a volume preserving map.

We denote by $\operatorname{Bir}(X,D)$ the group of volume preserving
birational self-maps of the CY pair $(X,D)$.

Volume preserving birational maps are called _crepant birational_ in [Kol13].

Next we introduce several natural classes of singularities of pairs.

######  Definition 1.2.

A pair $(X,D)$ has _terminal singularities_ if for all $z\in X$:

  1. (a)

if $\operatorname{codim}_{z}X\leq 2$, then the pair $(X,D)$ is smooth at $z$;

  2. (b)

if $\operatorname{codim}_{z}X>2$, then for all divisorial valuations $E$ of
$X$ with $\operatorname{z}_{E}X=z$,333We denote by $\operatorname{z}_{E}X\in
X$ the _centre_ of $E$ on $X$ — which, for us, is a scheme-theoretic point of
$X$. $a(E,K_{X}+D)>0$.

A pair $(X,D)$ has _canonical singularities_ if for all divisorial valuations
$E$ with small centre on $X$, $a(E,K_{X}+D)\geq 0$. Following common usage, we
say that a pair “is” terminal (respectively canonical) if it has terminal
(respectively canonical) singularities.

######  Remark 1.3.

  1. (1)

If the pair $(X,D)$ is terminal (respectively canonical) then both $X$ and $D$
are terminal (respectively canonical). In particular, $D$ is normal.

  2. (2)

If $X$ is smooth and $(X,D)$ is terminal, then for all $z\in X$ with
$\operatorname{codim}_{z}X>2$,
$\operatorname{mult}_{z}(D)<-1+\operatorname{codim}_{z}X$.

  3. (3)

In particular, if $X$ is a smooth $3$-fold, then $(X,D)$ is terminal if and
only if $D$ is smooth.

  4. (4)

If $X$ is smooth and $z\in D$ is an isolated singularity of multiplicity
$\operatorname{mult}_{z}D<-1+\dim X$ and smooth projectivized tangent cone,
then $(X,D)$ is terminal.

######  Definition 1.4.

We say that a pair $(X,D)$ is _(t, c)_ (respectively _(t, lc)_) if $X$ has
terminal singularities and the pair $(X,D)$ has canonical (respectively log
canonical) singularities. We say that a pair $(X,D)$ is _$\mathbb{Q}$
-factorial_ if $X$ is $\mathbb{Q}$-factorial.

######  Definition 1.5.

A _Mori fibered (Mf) CY pair_ is a $\mathbb{Q}$-factorial (t, lc) CY pair
$(X,D)$ together with a Mori fiber space structure on $X$, i.e., a morphism
$f\colon X\rightarrow S$ such that $f_{*}\mathcal{O}_{X}=\mathcal{O}_{S}$,
$-K_{X}$ is $f$-ample, and $\rho(X)-\rho(S)=1$.

######  Definition 1.6.

Let $(X,D_{X})\rightarrow S_{X}$ and $(Y,D_{Y})\rightarrow S_{Y}$ be Mf CY
pairs. A volume preserving birational map
$f\colon(X,D_{X})\dasharrow(Y,D_{Y})$ is _square_ if it fits into a
commutative diagram

$X$$Y$$S_{X}$$S_{Y}$$\scriptstyle{f}$$\scriptstyle{g}$

where $g$ is birational, and the induced birational map of generic fibers
$f_{L}\colon X_{L}\dasharrow Y_{L}$ is biregular. In this case, we say that
the Mf CY pairs $(X,D_{X})\rightarrow S_{X}$ and $(Y,D_{Y})\rightarrow S_{Y}$
are square equivalent.

######  Remark 1.7.

For Mf CY pairs $(X,D_{X})\rightarrow S_{X}$ and $(Y,D_{Y})\rightarrow S_{Y}$
to be square equivalent, it is necessary that $\dim S_{X}=\dim S_{Y}$.

If $S_{X}=S_{Y}=\operatorname{Spec}\mathbb{C}$, then the Mf CY pairs are
square equivalent if and only if $(X,D_{X})$ and $(Y,D_{Y})$ are isomorphic as
CY pairs.

######  Definition 1.8.

The pliability of the Mf CY pair $(X,D_{X})$ is the set

$\mathcal{P}(X,D_{X})\ =\ \frac{\Big{\\{}\mbox{Mf CY pairs
}(Y,D_{Y})\rightarrow
S_{Y}\>\big{|}\>\operatorname{Bir}\big{(}(X,D_{X}),(Y,D_{Y})\big{)}\neq\emptyset\Big{\\}}}{\mbox{square
equivalence}}.$

We say that the Mf CY pair $(X,D_{X})\rightarrow S_{X}$ is birationally rigid
if $\mathcal{P}(X,D_{X})$ consists of a single element (necessarily the
element $(X,D_{X})\to S_{X}$).

###  1.3. Main results

######  Theorem A.

Let $(X,D)\to\operatorname{Spec}\mathbb{C}$ be a Mf CY pair (so that $X$ is a
$\mathbb{Q}$-factorial terminal Fano variety with $\rho=1$) such that

  1. (i)

$(X,D)$ is terminal, and

  2. (ii)

$\operatorname{Cl}D=\mathbb{Z}\cdot\mathcal{O}_{D}(A)$, where $A$ is a divisor
on $X$.

Then $(X,D)\to\operatorname{Spec}\mathbb{C}$ is birationally rigid, and
$\operatorname{Bir}(X,D)=\operatorname{Aut}(X,D)$.

######  Remark 1.9.

  1. (1)

Let $(X,D_{X})$ and $(Y,D_{Y})$ be _(t, c)_ CY pairs. A birational map
$\varphi\colon X\dasharrow Y$ is volume preserving if and only if $\varphi$
restricts to a birational map $D_{X}\dasharrow D_{Y}$(Proposition 2.6). Thus,
in the statement of Theorem A, we can take $\operatorname{Bir}(X,D)$ naïvely
to mean the group of birational maps $\varphi\colon X\dasharrow X$ that
stabilize $D$. Similar considerations apply to the statement of Theorem B
below.

  2. (2)

Assumption (i) is needed for the statement to hold. Theorems B and C address
the cases when $X=\mathbb{P}^{3}$ and $D$ is a generic quartic with one
singular point of type $A_{1}$ and $A_{2}$, respectively. In both cases the
pair $(X,D)$ is canonical but not terminal, and
$\operatorname{Bir}(X,D)\supsetneq\operatorname{Aut}(X,D)$.

  3. (3)

Assumption (ii) is also needed for the statement to hold. This is illustrated
for instance by Oguiso’s example (2) mentioned above.

  4. (4)

Assumption (ii) is meaningful even when $n\geq 4$. If $X=\mathbb{P}^{4}$ and
$D$ is a quintic $3$-fold with ordinary quadratic singularities, then the pair
$(X,D)$ is terminal but the restriction map
$r\colon\operatorname{Cl}X\to\operatorname{Cl}D$ is not necessarily an
isomorphism. For instance if $D$ contains a plane, then in general $D$ has
$16$ ordinary quadratic singularities and $\operatorname{Cl}D$ has rank $2$.

######  Theorem B.

Let $D\subset\mathbb{P}^{3}$ be a quartic surface. Assume the following:

  1. (i)

$D$ is smooth apart from a single $A_{1}$-singularity $z\in D$,

  2. (ii)

$\operatorname{Cl}D=\mathbb{Z}\cdot\mathcal{O}_{D}(1)$.

In what follows denote by $f\colon X\to\mathbb{P}^{3}$ the blow-up of
$z\in\mathbb{P}^{3}$, $D_{X}\subset X$ the strict transform of $D$, and
$\pi\colon X\to\mathbb{P}^{2}$ the $\mathbb{P}^{1}$-bundle induced by the
projection from $z$.

Then the following holds:

  1. (1)

The pliability of the pair $(\mathbb{P}^{3},D)$ is the set with two elements

$(\mathbb{P}^{3},D)\to\operatorname{Spec}\mathbb{C}\text{ {and}
}\pi\colon(X,D_{X})\to\mathbb{P}^{2}.$

  2. (2)

The restriction homomorphism $r\colon\operatorname{Bir}(\mathbb{P}^{3},D)\
\to\ \operatorname{Bir}(D)$ induces a split exact sequence of groups:

$1\to\mathbb{G}\to\operatorname{Bir}(\mathbb{P}^{3},D)\to\operatorname{Bir}D\to
1$

where $\mathbb{G}$ is the twist of $\mathbb{G}_{m}$ corresponding to the
quadratic extension $\mathbb{C}(x,y)\subset\mathbb{C}(D)$. More precisely, the
morphism $\pi\colon D_{X}\to\mathbb{P}^{2}$ is finite of degree 2 and
identifies the function field $\mathbb{C}(D_{X})$ with a quadratic extension
$\mathbb{C}(x,y)(\sqrt{a})$ where $a=a_{6}(x,y)\in\mathbb{C}[x,y]$ is a sextic
polynomial. The group $\mathbb{G}$ is the form of $\mathbb{G}_{m}$ over
$\mathbb{C}(x,y)$ whose $\mathbb{C}(x,y)$-points are the solutions of the Pell
equation

$U^{2}-aV^{2}=1.$

######  Remark 1.10.

Assumptions (i) and (ii) are satisfied for a very general quartic surface $D$
with an $A_{1}$-singularity. Indeed, consider the moduli space of lattice-
polarized K3 surfaces containing the rank-$2$ lattice $N=\mathbb{Z}^{2}$ with
quadratic form

$\begin{pmatrix}-2&0\\\ 0&4\end{pmatrix}$

in their Picard group. A very general element of this moduli space is the
minimal resolution of a quartic surface $D$ satisfying assumptions (i) and
(ii).

######  Theorem C.

Let $D\subset\mathbb{P}^{3}$ by a quartic surface. Assume the following:

  1. (i)

$D$ is smooth apart from a single $A_{2}$-singularity $z\in D$,

  2. (ii)

$\operatorname{Cl}D=\mathbb{Z}\cdot\mathcal{O}_{D}(1)$.

Then the pliability of the pair $(\mathbb{P}^{3},D)$ is the (infinite) set
with elements: object $1$, $2$, $3^{a}$, $3^{b}$, the $3$-parameter family of
objects $4$, and the $6$-parameter family of objects $5^{a}$ constructed and
displayed in Table LABEL:table:1, modulo isomorphism. The paragraph following
Remark 1.11 explains how to read Table LABEL:table:1.

######  Remark 1.11.

Assumptions (i) and (ii) are satisfied for a very general quartic surface $D$
with an $A_{2}$-singularity. Indeed, consider the moduli space of lattice-
polarized K3 surfaces containing the rank-$3$ lattice $N=\mathbb{Z}^{3}$ with
quadratic form

$\begin{pmatrix}-2&1&0\\\ 1&-2&0\\\ 0&0&4\end{pmatrix}$

in their Picard group. A very general element of this moduli space is the
minimal resolution of a quartic surface $D$ satisfying assumptions (i) and
(ii).

Table 1. Mf CY pairs $(X^{\dagger},D^{\dagger})$ birational to $(\mathbb{P}^{3},D)$ where $D\subset\mathbb{P}^{3}$ is a quartic surface with one $A_{2}$-singularity. Object |  $X^{\dagger}\subseteq$ Ambient | Ambient coords. & wts. |  Eqn. of $X^{\dagger}$ in Ambient |  Eqn. of $D^{\dagger}$ in $X^{\dagger}$
---|---|---|---|---
$1$ | $X^{\dagger}=\mathbb{P}^{3}$ |  $\begin{array}[]{cccc}x_{0}&x_{1}&x_{2}&x_{3}\\\ \hline\cr 1&1&1&1\end{array}$ | $0$ | $x_{0}x_{1}x_{3}^{2}+Bx_{3}+C$
$2$ | $X^{\dagger}=\mathbb{F}_{1}^{3}$ |  $\begin{array}[]{ccccc}x_{0}&x_{1}&x_{2}&x_{3}&x\\\ \hline\cr 1&1&1&0&-1\\\ 0&0&0&1&1\end{array}$ | $0$ | $x_{0}x_{1}x_{3}^{2}+Bx_{3}x+Cx^{2}$
$2^{a}$ | $X^{\dagger}=\mathbb{F}_{2}^{3}$ |  $\begin{array}[]{ccccc}x_{0}&x_{1}&x_{2}&x_{3}&x\\\ \hline\cr 1&1&1&0&-2\\\ 0&0&0&1&1\end{array}$ | $0$ | $x_{0}x_{3}^{2}+Bx_{3}x+x_{1}Cx^{2}$
$2^{b}$ | $X^{\dagger}=\mathbb{F}_{2}^{3}$ |  $\begin{array}[]{ccccc}x_{0}&x_{1}&x_{2}&x_{3}&x\\\ \hline\cr 1&1&1&0&-2\\\ 0&0&0&1&1\end{array}$ | $0$ | $x_{1}x_{3}^{2}+Bx_{3}x+x_{0}Cx^{2}$
$3^{a}$ | $X^{\dagger}=\mathbb{P}(1^{3},2)$ |  $\begin{array}[]{cccc}x_{0}&x_{1}&x_{2}&y\\\ \hline\cr 1&1&1&2\end{array}$ | $0$ | $x_{0}y^{2}+By+x_{1}C$
$3^{b}$ | $X^{\dagger}=\mathbb{P}(1^{3},2)$ |  $\begin{array}[]{cccc}x_{0}&x_{1}&x_{2}&y\\\ \hline\cr 1&1&1&2\end{array}$ | $0$ | $x_{1}y^{2}+By+x_{0}C$
$4$ | $X_{4}\subset\mathbb{P}(1^{3},2^{2})$ |  $\begin{array}[]{ccccc}x_{0}&x_{1}&x_{2}&y_{0}&y_{1}\\\ \hline\cr 1&1&1&2&2\end{array}$ |  $y_{0}y_{1}+C-\\\ -L(x_{0}y_{1}-x_{1}y_{0}-B)$ | $x_{0}y_{1}-x_{1}y_{0}-B$
$5^{a}$ | $X_{4}\subset\mathbb{P}(1^{4},2)$ |  $\begin{array}[]{ccccc}x_{0}&x_{1}&x_{2}&x_{3}&y\\\ \hline\cr 1&1&1&1&2\end{array}$ |  $y(y+Q)-C+\\\ +x_{3}\bigl{(}(x_{0}+x_{1})y+x_{1}Q+B\bigr{)}$ | $y+x_{1}x_{3}$
$5^{b}$ | $X_{4}\subset\mathbb{P}(1^{4},2)$ |  $\begin{array}[]{ccccc}x_{0}&x_{1}&x_{2}&x_{3}&y\\\ \hline\cr 1&1&1&1&2\end{array}$ |  $y(y-Q)-C+\\\ +x_{3}\bigl{(}(x_{0}+x_{1})y-x_{0}Q+B\bigr{)}$ | $y+x_{0}x_{3}$

#### How to read Table LABEL:table:1

The rows of Table LABEL:table:1 display pairs $(X^{\dagger},D^{\dagger})$
birational to the pair $(\mathbb{P}^{3},D)$ of Theorem C.

The first row of Table LABEL:table:1 displays the pair $(\mathbb{P}^{3},D)$
itself. We have chosen homogeneous coordinates $x_{0},\dots,x_{3}$ on
$\mathbb{P}^{3}$ such that the singular point $z\in D$ is the point
$z=[0:0:0:1]$, and the tangent cone to $D$ at $z$ is
$(x_{0}x_{1}=0)\subset\mathbb{P}^{3}$. In these coordinates, the equation of
$D$ is written as

$D=\Bigl{(}x_{0}x_{1}x_{3}^{2}+Bx_{3}+C=0\Bigr{)}\subset\mathbb{P}^{3},$

where $B=B_{3}(x_{0},x_{1},x_{2})$ and $C=C_{4}(x_{0},x_{1},x_{2})$ are
homogeneous forms of degree $3$ and $4$. The information displayed on the
first row is self-explanatory: the second column states that the pair
$(X^{\dagger},D^{\dagger})$ is contained in the ambient $\mathbb{P}^{3}$, the
third column names homogeneous coordinates on $\mathbb{P}^{3}$ all of weight
one, the fourth column displays the equation of
$X^{\dagger}\subset\mathbb{P}^{3}$ — the zero “equation”, since
$X^{\dagger}=\mathbb{P}^{3}$ in this case — and the last column displays the
equation of $D^{\dagger}$ that we just explained.

For all integers $k\geq 0$, we denote by $\mathbb{F}_{k}^{3}\to\mathbb{P}^{2}$
the $\mathbb{P}^{1}$-bundle
$\mathbb{P}\big{(}\mathcal{O}\oplus\mathcal{O}(k)\big{)}$ over
$\mathbb{P}^{2}$ with homogeneous coordinates and weights

$\begin{array}[]{ccccc}x_{0}&x_{1}&x_{2}&x_{3}&x\\\ \hline\cr 1&1&1&0&-k\\\
0&0&0&1&1\end{array}$

The second row of the table displays object $2$, that is, the pair
$(\mathbb{F}^{3}_{1},D^{\dagger})$ where the equation of $D^{\dagger}$ is as
shown in the last column.

In the row that displays object $4$, $L=L(x_{0},x_{1},x_{2})$ is a homogeneous
linear form; in the rows that display objects $5^{a}$ and $5^{b}$,
$Q=Q(x_{0},x_{1},x_{2})$ is a homogeneous quadratic form.

######  Remark 1.12.

At the end of Section 2, we exhibit explicit volume preserving birational maps
between the Mf CY pairs in Table LABEL:table:1. These maps are constructed as
Sarkisov links, and this is how they naturally appear in the proof of Theorem
C. In particular, our constructions show that object $2$ is square equivalent
to objects $2^{a}$ and $2^{b}$. Moreover, in Example 2.15 we show that
families $5^{a}$ and $5^{b}$ are isomorphic. This is why objects $2^{a}$ and
$2^{b}$, and the family $5^{b}$, appear in Table LABEL:table:1 while they are
omitted in the statement of Theorem C.

######  Remark 1.13.

In order to prove Theorem B and Theorem C, we consider a Mf CY pair
$(Y,D_{Y})/T$, and a volume preserving birational map
$\varphi\colon(\mathbb{P}^{3},D)\dasharrow(Y,D_{Y})$ that is not biregular.
The proofs proceeds by studying explicitly the links of a Sarkisov
factorization of $\varphi$. To control these links, and the divisorial
contractions, flips, flops and antiflips that constitute them, we need some
explicit classification results for divisorial contractions and analytic
neighbourhoods of curves. We develop this material further than strictly
needed for the proof of Theorem B, enough for what we need in the proof of
Theorem C. These results, contained in Section 4, are of independent interest;
they represent the first steps towards developing a technology for working
explicitly with the volume preserving birational maps of $3$-fold Mf CY pairs.

#### Acknowledgements

Carolina Araujo was partially supported by grants from CNPq, Faperj and
CAPES/COFECUB. Part of this work was developed during the authors’ visit to
ICTP, funded by Carolina Araujo’s ICTP Simons Associateship. We thank ICTP for
the great working conditions, and Simons Foundation for the financial support.

Alessio Corti was partially supported by EPSRC Programme Grant EP/N03189X/1
_Classification, computation and construction: new problems in geometry_. This
research was started during a visit of Alessio Corti to the IMPA funded by
CNPq Visiting Researcher grant.

Alex Massarenti is a member of the Gruppo Nazionale per le Strutture
Algebriche, Geometriche e le loro Applicazioni of the Istituto Nazionale di
Alta Matematica F. Severi (GNSAGA-INDAM).

##  2\. The Sarkisov program for Mf CY pairs

In this section we review the factorization theorem for volume preserving
birational maps between Mf CY pairs established in [CK16]. This is the main
tool in the proof of our results. We start by recalling a valuative
interpretation of the volume preserving condition.

######  Proposition 2.1 ([CK16, Remark 1.7]).

Let $(X,D_{X})$ and $(Y,D_{Y})$ be CY pairs, and $f\colon X\dasharrow Y$ an
arbitrary birational map. The following conditions are equivalent:

  1. (1)

The map $f\colon(X,D_{X})\dasharrow(Y,D_{Y})$ is volume preserving.

  2. (2)

For all geometric valuations $E$ with center on both $X$ and $Y$, the
discrepancies of $E$ with respect to the pairs $(X,D_{X})$ and $(Y,D_{Y})$ are
equal: $a(E,K_{X}+D_{X})=a(E,K_{Y}+D_{Y})$.

  3. (3)

Let

$W$$X$$Y$$\scriptstyle{f}$$\scriptstyle{q}$$\scriptstyle{p}$

be a common log resolution of the pairs $(X,D_{X})$ and $(Y,D_{Y})$. The
birational map $f$ induces an isomorphism
$f_{*}\colon\Omega^{n}_{k(X)/k}\rightarrow\Omega^{n}_{k(Y)/k}$, where $n$
denotes the common dimension of $X$ and $Y$. Abusing notation, we write

$p^{*}(K_{X}+D_{X})=q^{*}(K_{Y}+D_{Y})$

to mean that for all meromorphic differentials $\omega\in\Omega^{n}_{k(X)/k}$,
we have

$p^{*}\big{(}D_{X}+\operatorname{div}(\omega)\big{)}=q^{*}\big{(}D_{Y}+\operatorname{div}(f_{*}\omega)\big{)}.$

The condition is: for some (or equivalently for all) common log resolution as
above, we have $p^{*}(K_{X}+D_{X})=q^{*}(K_{Y}+D_{Y})$.

######  2.2.

A _Mori divisorial contraction_ is a divisorial contraction $f\colon Z\to X$
from a $\mathbb{Q}$-factorial terminal variety $Z$, associated to an extremal
ray $R\subset\operatorname{\overline{NE}}(Z)$ such that $K_{Z}\cdot R<0$. In
particular, $X$ also has $\mathbb{Q}$-factorial terminal singularities.

If $(Z,D_{Z})$ and $(X,D_{X})$ are (t, lc) CY pairs, then a Mori divisorial
contraction $f\colon Z\to X$ is volume preserving as a map of CY pairs if and
only if $K_{Z}+D_{Z}=f^{*}(K_{X}+D_{X})$, in the sense of Proposition 2.1(3).
In this case, we have $D_{X}=f_{*}(D_{Z})$.

A _Mori flip_ is a flip $\varphi\colon Z\dasharrow Z^{\prime}$ from a
$\mathbb{Q}$-factorial terminal variety $Z$, associated to an extremal ray
$R\subset\operatorname{\overline{NE}}(Z)$ such that $K_{Z}\cdot R<0$. In
particular, $Z^{\prime}$ also has $\mathbb{Q}$-factorial terminal
singularities. An _antiflip_ is the inverse of a Mori flip. A _Mori flop_ is a
flop $\varphi\colon Z\dasharrow Z^{\prime}$ between $\mathbb{Q}$-factorial
terminal varieties, associated to an extremal ray
$R\subset\operatorname{\overline{NE}}(Z)$ such that $K_{Z}\cdot R=0$.

Let $(Z,D_{Z})$ and $(Z^{\prime},D_{Z^{\prime}})$ be (t, lc) CY pairs, and
$\varphi\colon Z\dasharrow Z^{\prime}$ a Mori flip, flop or antiflip. Then
$\varphi\colon(Z,D_{Z})\dasharrow(Z^{\prime},D_{Z^{\prime}})$ is volume
preserving if and only if $D_{Z^{\prime}}=\varphi_{*}D_{Z}$.

######  2.3Sarkisov links.

We recall the definition of the four types of Sarkisov links from [Cor95]. In
the following diagrams, $X\to S$ and $X^{\prime}\to S^{\prime}$ always stand
for Mori fiber spaces.

  1. (I)

A _Sarkisov link of type (I)_ is a commutative diagram

$Z$$X^{\prime}$$X$$S^{\prime}$$S$

where $Z\to X$ is a Mori divisorial contraction, and $Z\dasharrow X^{\prime}$
is a sequence of Mori flips, flops and antiflips.

  2. (II)

A _Sarkisov link of type (II)_ is a commutative diagram

$Z$$Z^{\prime}$$X$$X^{\prime}$$S$$S$

where $Z\to X$ and $Z^{\prime}\to X^{\prime}$ are Mori divisorial
contractions, and $Z\dasharrow Z^{\prime}$ is a sequence of Mori flips, flops
and antiflips.

  3. (III)

A _Sarkisov link of type (III)_ is the inverse of a link of type (I).

  4. (IV)

A _Sarkisov link of type (IV)_ is a commutative diagram

$X$$X^{\prime}$$S$$S^{\prime}$$T$

where $X\dasharrow X^{\prime}$ is a sequence of Mori flips, flops and
antiflips, and $S\rightarrow T$ and $S^{\prime}\rightarrow T$ are Mori
contractions.

It is shown in [Cor95] and [HM13] that every birational map between Mori fiber
spaces is a composition of Sarkisov links. Next we explain the version of the
Sarkisov program for volume preserving birational maps between Mf CY pairs
established in [CK16].

######  Definition 2.4.

A _volume preserving Sarkisov link_ is a Sarkisov link as described in
Paragraph 2.3 above with the following additional data and property: there are
divisors $D_{X}$ on $X$, $D_{X^{\prime}}$ on $X^{\prime}$, $D_{Z}$ on $Z$, and
$D_{Z^{\prime}}$ on $Z^{\prime}$, making $(X,D_{X})$,
$(X^{\prime},D_{X^{\prime}})$, $(Z,D_{Z})$ and $(Z^{\prime},D_{Z^{\prime}})$
(t, lc) CY pairs, and all the divisorial contractions, Mori flips, flops and
antiflips that constitute the Sarkisov link are volume preserving for these CY
pairs.

At the end of this section we will construct explicit volume preserving
Sarkisov links between the Mf CY pairs displayed in Table 1 of the
introduction.

######  Theorem 2.5 ([CK16, Theorem 1.1]).

A volume preserving birational map between Mf CY pairs is a composition of
volume preserving Sarkisov links.

Theorem 2.5 provides an effective tool to investigate the group
$\operatorname{Bir}(X,D)$ when $(X,D)$ is a Mf CY pair. In this paper we
restrict ourselves to canonical pairs. This is the case, for instance, when
$X$ is smooth and $D\subset X$ is an irreducible hypersurface with canonical
singularities. In this case, the theory is greatly simplified for the
following reason.

######  Proposition 2.6.

Let $(X,D_{X})$ and $(Y,D_{Y})$ be (t, c) CY pairs, and $f\colon X\dasharrow
Y$ an arbitrary birational map. Then $f\colon(X,D_{X})\dasharrow(Y,D_{Y})$ is
volume preserving if and only if $f_{*}D_{X}=D_{Y}$ and
$f^{-1}_{*}D_{Y}=D_{X}$. (This condition is equivalent to asking that the
restriction of $f$ to each component of $D_{X}$ is a birational map to a
component of $D_{Y}$, and the same for $f^{-1}$).

In particular, if $(X,D)$ is a (t, c) CY pair with $D$ irreducible, then
$\operatorname{Bir}(X,D)$ coincides with the group of birational self-maps
$f\colon X\dasharrow X$ such that $f(D)=D$.

######  Remark 2.7.

  1. (1)

In many cases of interest, the assumption that $(X,D)$ is (t, c) implies that
$D$ is irreducible. This is the case for instance when $X$ is Fano with
$\dim(X)>1$. Indeed, if $(X,D)$ is (t, c), then $D$ is normal, and thus
irreducibility of $D$ is equivalent to connectedness of $D$. A key case when
connectedness fails is $X=\mathbb{P}^{1}$, $D=\\{0,\infty\\}$, and one feels
that all examples must in some way be related to this: see the main result of
[HH19] for a statement along these lines.

  2. (2)

The characterization of the volume preserving condition stated in Proposition
2.6 does not hold in general for birational maps between (t, lc) CY pairs. For
example, consider the divisor $D=L_{0}+L_{1}+L_{2}$ on $\mathbb{P}^{2}$, where
the $L_{i}$ are the three coordinate lines. It is (t, lc) but not (t, c). One
checks easily that the standard Cremona transformation

$\begin{array}[]{cccc}f\colon&(\mathbb{P}^{2},D)&\dasharrow&(\mathbb{P}^{2},D)\\\
&(x_{0}:x_{1}:x_{2})&\longmapsto&(x_{1}x_{2}:x_{0}x_{2}:x_{0}x_{1})\end{array}$

is volume preserving, but it does not restrict to a birational self-map of
$D$. Indeed, $f=f^{-1}$ contracts the lines $L_{i}$ to the three coordinate
points of $\mathbb{P}^{2}$.

###### Proof of Proposition 2.6.

If $(X,D_{X})$ is a (t, c) pair, then any divisor $E$ over $X$ such that
$a(E,K_{X}+D_{X})=-1$ must be a component of $D_{X}$. Therefore, if
$f\colon(X,D_{X})\dasharrow(Y,D_{Y})$ is a volume preserving birational map
between (t, c) CY pairs, then $f$ does not contract any component of $D_{X}$,
and $f^{-1}$ does not contract any component of $D_{Y}$. Hence,
$f_{*}D_{X}=D_{Y}$ and $f^{-1}_{*}D_{Y}=D_{X}$.

To prove the converse, consider a common log resolution

$W$$X$$Y$ .$\scriptstyle{f}$$\scriptstyle{q}$$\scriptstyle{p}$

Assume that $f_{*}D_{X}=D_{Y}$ and $f^{-1}_{*}D_{Y}=D_{X}$. Then the strict
transforms of $D_{X}$ via $p$ and of $D_{Y}$ via $q$ coincide. We denote them
by $D_{W}$.

Since $(X,D_{X})$ and $(Y,D_{Y})$ are canonical, we have

  * (i)

$K_{W}+D_{W}=p^{*}(K_{X}+D_{X})+\sum_{i}a_{i}E_{i}\sim\sum_{i}a_{i}E_{i}$,
where the $E_{i}$ are $p$-exceptional and $a_{i}\geq 0$;

  * (ii)

$K_{W}+D_{W}=q^{*}(K_{Y}+D_{Y})+\sum_{j}b_{j}F_{j}\sim\sum_{j}b_{j}F_{j}$,
where the $F_{j}$ are $q$-exceptional and $b_{j}\geq 0$.

Applying $q_{*}$ to (i) yields

$q_{*}(K_{W}+D_{W})=\sum_{i}a_{i}q_{*}E_{i}\sim 0.$

This shows that, whenever $a_{i}>0$, the divisor $E_{i}$ is $q$-exceptional.
Similarly, whenever $b_{j}>0$, the divisor $F_{j}$ is $p$-exceptional. So, up
to relabelling, we may assume that $E_{i}=F_{i}$, and we have

$\sum_{i}a_{i}E_{i}\sim\sum_{i}b_{i}E_{i}.$

The negativity lemma [KM98, Lemma 3.39] then implies that $a_{i}=b_{i}$, and
so $p^{*}(K_{X}+D_{X})=q^{*}(K_{Y}+D_{Y})$. ∎

The next lemma ensures that canonicity is preserved when we run a volume
preserving Sarkisov program for canonical Mf CY pairs.

######  Lemma 2.8.

Let $f\colon(X,D_{X})\dashrightarrow(Y,D_{Y})$ be a volume preserving
birational map between $(t,~{}lc)$ CY pairs. Then $(X,D_{X})$ is canonical if
and only if so is $(Y,D_{Y})$.

###### Proof.

First of all, note that if $(X,D_{X})$ is canonical, then $D_{X}$ is normal.
Write

$D_{X}=\sum D_{i},$

where the $D_{i}$ are the connected components of $D_{X}$. The $D_{i}$ are the
only divisorial valuations with $a(D_{i},K_{X}+D_{X})<0$. Since the map is
volume preserving, for every divisorial valuation $E$,
$a(E,K_{Y}+D_{Y})=a(E,K_{X}+D_{X})$. To show that the pair $(Y,D_{Y})$ is
canonical, all we need to show is that no $D_{i}$ is exceptional over $Y$.
Consider a common log resolution

$W$$X$$Y\,,$$\scriptstyle{f}$$\scriptstyle{q}$$\scriptstyle{p}$

denote by $\widetilde{D}_{i}$ the strict transform of $D_{i}$, and by
$D_{W}=\sum\widetilde{D}_{i}$ the strict transform of $D_{X}$. In order to
show that $(Y,D_{Y})$ is canonical, we need to show that no
$\widetilde{D}_{i}$ is $q$-exceptional.

Since $(X,D_{X})$ is (t, c), we have

$K_{W}+D_{W}=p^{*}(K_{X}+D_{X})+\sum a_{j}E_{j},$

with $a_{j}\geq 0$. If $a_{j}>0$, then $E_{j}$ is $p$-exceptional. The volume
preserving condition says that $p^{*}(K_{X}+D_{X})=q^{*}(K_{Y}+D_{Y})$, and
hence

$K_{W}+D_{W}=q^{*}(K_{Y}+D_{Y})+\sum a_{j}E_{j},$

and if $a_{j}>0$ then $E_{j}$ is $q$-exceptional. By [KM98, Theorem 5.48], the
support of $D_{W}$ is connected in the neighbourhood of every fibre of $q$.
Since the $\widetilde{D}_{i}$ are pairwise disjoint, connectedness implies
that, if $\widetilde{D}_{i}$ is $q$-exceptional, then it is not contained in
the support of $q^{*}D_{Y}$. Hence
$-1=a(\widetilde{D}_{i},K_{Y}+D_{Y})=a(\widetilde{D}_{i},K_{Y})$,
contradicting the assumption that $Y$ has terminal singularities. We conclude
that no $\widetilde{D}_{i}$ is $q$-exceptional, and thus $(Y,D_{Y})$ is
canonical. ∎

In the remaining part of this section, we revisit the Mf CY pairs in Table
LABEL:table:1, and construct explicit volume preserving Sarkisov links
connecting them. These links will appear crucially in the proof of Theorem C.
They are summarized in the following diagram:

obj. $1$$\scriptstyle{\epsilon_{a},\epsilon_{b}}$obj.
$2$$\scriptstyle{\sigma}$obj. $2^{a}$,
$2^{b}$$\scriptstyle{\nu^{a},\nu^{b}}$$\scriptstyle{\chi^{a},\chi^{b}}$obj.
$3^{a}$,
$3^{b}$$\scriptstyle{\phi^{a},\phi^{b}}$$\scriptstyle{\psi^{a},\widetilde{\psi}^{b}}$obj.
$4$obj. $5^{a}$

In Example 2.15 we produce an isomorphism between the family $5^{a}$ and the
family $5^{b}$.

#### Object 1

The pair $(\mathbb{P}^{3},D_{4})$, where

$D_{4}=\Bigl{(}x_{0}x_{1}x_{3}^{2}+Bx_{3}+C=0\Bigr{)}.$

Here and in what follows, $B=B_{3}(x_{0},x_{1},x_{2})$ and
$C=C_{4}(x_{0},x_{1},x_{2})$ are fixed homogeneous forms of the indicated
degree.

#### Object 2

The pair $\left(\mathbb{F}^{3}_{1},D_{\binom{2}{2}}\right)$, where
$\mathbb{F}^{3}_{1}$ is the $\mathbb{P}^{1}$-bundle
$\mathbb{P}\big{(}\mathcal{O}\oplus\mathcal{O}(1)\big{)}$ over
$\mathbb{P}^{2}$ with homogeneous coordinates and weights

$\begin{array}[]{ccccc}x_{0}&x_{1}&x_{2}&x_{3}&x\\\ \hline\cr 1&1&1&0&-1\\\
0&0&0&1&1\end{array}\,,$

$\text{ and }\ \ \
D_{\binom{2}{2}}=\Bigl{(}x_{0}x_{1}x_{3}^{2}+Bx_{3}x+Cx^{2}=0\Bigr{)}.$

######  Example 2.9 (Map from object $2$ to object $1$).

The blow-up
$\sigma\colon\left(\mathbb{F}_{1}^{3},D_{\binom{2}{2}}\right)\to(\mathbb{P}^{3},D_{4})$
of the point $[0:0:0:1]\in\mathbb{P}^{3}$ is a volume preserving Sarkisov link
of type (I). In coordinates,
$\sigma\colon{\mathbb{F}_{1}^{3}}_{(x_{0},x_{1},x_{2},x_{3},x)}\to{\mathbb{P}^{3}}_{(x_{0},x_{1},x_{2},x_{3})}$
is given by

$(x_{0},x_{1},x_{2},x_{3},x)\mapsto\left(x_{0},x_{1},x_{2},\frac{x_{3}}{x}\right).$

#### Object $2^{a}$

The pair $\left(\mathbb{F}^{3}_{2},D_{\binom{1}{2}}^{a}\right)$, where
$\mathbb{F}^{3}_{2}$ is the $\mathbb{P}^{1}$-bundle
$\mathbb{P}\big{(}\mathcal{O}\oplus\mathcal{O}(2)\big{)}$ over
$\mathbb{P}^{2}$ with homogeneous coordinates and weights

$\begin{array}[]{ccccc}x_{0}&x_{1}&x_{2}&x_{3}&x\\\ \hline\cr 1&1&1&0&-2\\\
0&0&0&1&1\end{array}\,,$

$\text{ and }\ \ \
D_{\binom{1}{2}}^{a}=\Bigl{(}x_{0}x_{3}^{2}+Bx_{3}x+x_{1}Cx^{2}=0\Bigr{)}.$

######  Example 2.10 (Map from object $2^{a}$ to object $2$).

Consider the rational map

$\nu^{a}\colon\left(\mathbb{F}_{2}^{3},D_{\binom{1}{2}}^{a}\right)\dasharrow\left(\mathbb{F}_{1}^{3},D_{\binom{2}{2}}\right)$

obtained by blowing-up the curve $(x_{1}=x_{3}=0)\subset
D_{\binom{1}{2}}^{a}$, and then blowing-down the strict transform of the
divisor $(x_{1}=0)$. In coordinates,
$\nu^{a}\colon{\mathbb{F}_{2}^{3}}_{(x_{0},x_{1},x_{2},x_{3},x)}\dasharrow{\mathbb{F}_{1}^{3}}_{(x_{0},x_{1},x_{2},x_{3},x)}$
is given by

$(x_{0},x_{1},x_{2},x_{3},x)\mapsto(x_{0},x_{1},x_{2},x_{3},xx_{1}).$

It is a volume preserving Sarkisov link of type (II).

Similarly, there is a volume preserving Sarkisov link of type (II)
$\nu^{b}\colon\text{obj.~{}$2^{b}$}\dasharrow\text{obj.~{}$2$}$.

It follows that the Mf CY pairs $2^{a}$, $2^{b}$ and $2$ are all square
equivalent.

#### Object $3^{a}$

The pair $(\mathbb{P}(1^{3},2),D_{5}^{a})$, where

$D_{5}^{a}=\Bigl{(}x_{0}y^{2}+By+x_{1}C=0\Bigr{)}.$

Note that $D_{5}^{a}\subset\mathbb{P}(1^{3},2)$ is not a general degree five
surface in $\mathbb{P}(1^{3},2)$. Indeed $D_{5}^{a}$ always contains the curve
$y=x_{1}=0$.

#### Object $3^{b}$

The pair $(\mathbb{P}(1^{3},2),D_{5}^{b})$, where

$D_{5}^{b}=\Bigl{(}x_{1}y^{2}+By+x_{0}C=0\Bigr{)}.$

Object $3^{b}$ is object $3^{a}$ with $x_{0},x_{1}$ swapped in the same way as
object $2^{b}$ is object $2^{a}$ with $x_{0},x_{1}$ swapped. However, in
general, object $3^{b}$ is not square equivalent to object $3^{a}$. Indeed,
since $X^{\dagger}=\mathbb{P}(1,1,1,2)$ is a Mori fibre space over
$\operatorname{Spec}\mathbb{C}$, the two objects are square equivalent if and
only if they are isomorphic.

######  Example 2.11 (Map from object $1$ to object $3^{b}$).

Consider the birational map
$\epsilon_{b}\colon(\mathbb{P}^{3},D_{4})\dasharrow(\mathbb{P}(1^{3},2),D_{5}^{b})$
obtained by a weighted blow-up of $[0:0:0:1]\in\mathbb{P}^{3}$ with weights
$(2,1,1)$, followed by the contraction of the strict transform of the divisor
$(x_{0}=0)$. In coordinates,
$\epsilon_{b}\colon\mathbb{P}^{3}_{(x_{0},x_{1},x_{2},x_{3})}\dasharrow\mathbb{P}(1^{3},2)_{(x_{0},x_{1},x_{2},y)}$
is given by

$(x_{0},x_{1},x_{2},x_{3})\mapsto(x_{0},x_{1},x_{2},x_{3}x_{0}).$

It is a volume preserving Sarkisov link of type (II). Similarly, there is a
volume preserving Sarkisov link of type (II),
$\epsilon_{a}\colon\text{obj.~{}$1$}\dasharrow\text{obj.~{}$3^{a}$}$.

######  Example 2.12 (Maps from object $2^{b}$ to object $3^{b}$ and from
object $2^{a}$ to object $3^{a}$).

The blow-up
$\chi^{b}\colon\left(\mathbb{F}^{3}_{2},D_{\binom{1}{2}}^{b}\right)\to(\mathbb{P}(1^{3},2),D_{5}^{b})$
of the singular point $[0:0:0:1]\in\mathbb{P}(1^{3},2)$ is a volume preserving
Sarkisov link of type (I). In coordinates,
$\chi^{b}\colon{\mathbb{F}_{2}^{3}}_{(x_{0},x_{1},x_{2},x_{3},x)}\to\mathbb{P}(1^{3},2)_{(x_{0},x_{1},x_{2},y)}$
is given by

$(x_{0},x_{1},x_{2},x_{3},x)\mapsto\left(x_{0},x_{1},x_{2},\frac{x_{3}}{x}\right).$

From the description in coordinates, it is straightforward to check that
$\chi^{b}=\epsilon_{b}\circ\sigma\circ\nu^{b}$. Similarly, there is a volume
preserving Sarkisov link of type (I),
$\chi^{a}\colon\text{obj.~{}$2^{a}$}\to\text{obj.~{}$3^{a}$}$, and
$\chi^{a}=\epsilon_{a}\circ\sigma\circ\nu^{a}$.

####  $3$-parameter family of objects 4

Let $D_{3,4}\subset\mathbb{P}(1^{3},2^{2})_{(x_{0},x_{1},x_{2},y_{0},y_{1})}$
be the complete intersection given by equations:

$D_{3,4}=\begin{cases}y_{0}y_{1}+C&=0,\\\
x_{0}y_{1}-x_{1}y_{0}-B&=0.\end{cases}$

For each fixed linear form $L=L(x_{0},x_{1},x_{2})$, we consider the pair
$(X_{4},D_{3,4})$, where $X_{4}$ is the following quartic containing
$D_{3,4}$:

$X_{4}=\Bigl{(}y_{0}y_{1}+C-L(x_{0}y_{1}-x_{1}y_{0}-B)=0\Bigr{)}.$

######  Example 2.13 (Map from object $3^{b}$ to object $4$).

For each fixed linear form $L=L(x_{0},x_{1},x_{2})$, we construct a volume
preserving Sarkisov link of type (II),
$\phi^{b}\colon(\mathbb{P}(1^{3},2),D_{5}^{b})\dasharrow(X_{4},D_{3,4})$. In
coordinates, the map
$\phi^{b}\colon\mathbb{P}(1^{3},2)_{(x_{0},x_{1},x_{2},y)}\dasharrow\mathbb{P}(1^{3},2^{2})_{(x_{0},x_{1},x_{2},y_{0},y_{1})}$
is given by

(2.1)
$(x_{0},x_{1},x_{2},y)\mapsto\left(x_{0},x_{1},x_{2},y,-x_{1}L-\frac{x_{0}x_{1}L^{2}+BL+C}{y-x_{0}L}\right).$

This map is obtained by first blowing-up the curve
$\Gamma\subset\mathbb{P}(1,1,1,2)$ defined by:

$\Gamma=\left\\{\begin{array}[]{l}Q=y-x_{0}L=0,\\\
F=x_{0}x_{1}L^{2}+BL+C=0,\end{array}\right.$

and then blowing-down the strict transform of the divisor $(y-x_{0}L=0)$.

In what follows, we describe this map in detail, and deduce expression (2.1)
above.

First note that $\Gamma\subset D_{5}^{b}$:

(2.2) $x_{1}y^{2}+By+x_{0}C=Q\left(x_{1}(y+x_{0}L)+B\right)+Fx_{0}.$

In order to describe the blow-up of $\mathbb{P}(1,1,1,2)$ along $\Gamma$,
consider the toric variety $\mathbb{F}$ with coordinates and weight matrix:

$\begin{array}[]{cccccc}z_{0}&z_{1}&z_{2}&z_{3}&u&v\\\ \hline\cr
1&1&1&2&0&-2\\\ 0&0&0&0&1&1\end{array}$

and stability condition chosen so that the nef cone of $\mathbb{F}$ is the
span $\langle z_{i},u\rangle_{+}$:

$z_{0},z_{1},z_{2}$$z_{3}$$u$$v$

This choice of stability condition gives the irrelevant ideal
$(z_{0},z_{1},z_{2},z_{3})(u,v)$ and yields a $\mathbb{P}^{1}$-bundle morphism
$\pi\colon\mathbb{F}\to\mathbb{P}(1,1,1,2)$. The other contraction of
$\mathbb{F}$ is the divisorial contraction
$\pi^{\prime}\colon\mathbb{F}\to\mathbb{P}(1,1,1,2,2)$ that maps the divisor
$v=0$ to the point $[0:0:0:0:1]\in\mathbb{P}(1,1,1,2,2)$. In coordinates,
$\pi^{\prime}\colon\mathbb{F}_{(z_{0},z_{1},z_{2},z_{3},u,v)}\to\mathbb{P}(1,1,1,2,2)_{(x_{0},x_{1},x_{2},y_{0},y_{1})}$
is given by

$(z_{0},z_{1},z_{2},z_{3},u,v)\mapsto\left(z_{0},z_{1},z_{2},z_{3},\frac{u}{v}\right).$

The blow-up $Z$ of $\mathbb{P}(1,1,1,2)$ along $\Gamma$ is cut out in
$\mathbb{F}$ by the equation

$uQ+vF=0.$

Let us describe the equation of $\pi^{\prime}(Z)\subset\mathbb{P}(1,1,1,2,2)$.
From $\frac{u}{v}=-\frac{F}{Q}$, we get $y_{1}Q+F=0$ and hence
$\pi^{\prime}(Z)$ is cut out by the equation

$y_{1}(y_{0}-x_{0}L)+x_{0}x_{1}L^{2}+BL+C=0.$

Combining with Equation (2.2), we see that the strict transform of $D_{5}^{b}$
in $\pi^{\prime}(Z)$ is cut out by the equation

$x_{0}y_{1}=x_{1}y_{0}+x_{0}x_{1}L+B.$

In coordinates, the composition
$\pi^{\prime}\circ\pi\colon\mathbb{P}(1^{3},2)_{(x_{0},x_{1},x_{2},y)}\to\pi^{\prime}(Z)\subset\mathbb{P}(1,1,1,2,2)_{(x_{0},x_{1},x_{2},y_{0},y_{1})}$
is given by

$(x_{0},x_{1},x_{2},y)\mapsto\left(x_{0},x_{1},x_{2},y,-\frac{F}{Q}\right).$

Next we compose it with the automorphism of
$\mathbb{P}(1,1,1,2,2)_{(x_{0},x_{1},x_{2},y_{0},y_{1})}$ given in coordinates
by

$(x_{0},x_{1},x_{2},y_{0},y_{1})\mapsto(x_{0},x_{1},x_{2},y_{0},y_{1}-x_{1}L).$

It is immediate to check that, in coordinates, the composed map
$\phi^{b}\colon\mathbb{P}(1^{3},2)_{(x_{0},x_{1},x_{2},y)}\dasharrow\mathbb{P}(1^{3},2^{2})_{(x_{0},x_{1},x_{2},y_{0},y_{1})}$
is given by (2.1). The image of $Z$ is given by the equation

$0=(y_{1}+x_{1}L)(y_{0}-x_{0}L)+x_{0}x_{1}L^{2}+BL+C=y_{0}y_{1}+C-L(x_{0}y_{1}-x_{1}y_{0}-B),$

which is precisely the equation of $X_{4}$. The strict transform of
$D_{5}^{b}$ is the surface of $\mathbb{P}(1^{3},2^{2})$ given by the equations

$\begin{cases}y_{0}y_{1}+C-L(x_{0}y_{1}-x_{1}y_{0}-B)&=0,\\\
x_{0}(y_{1}+x_{1}L)-x_{1}(y_{0}+x_{0}L)-B&=x_{0}y_{1}-x_{1}y_{0}-B=0,\end{cases}$

which are precisely the equations of $D_{3,4}$.

Similarly, one can construct a volume preserving Sarkisov link of type (II),
$\psi^{a}\colon\text{obj.~{}$3^{a}$}\dasharrow\text{obj.~{}$4$}$.

#### 6-parameter family of objects $5^{a}$

For each fixed quadratic form $Q=Q(x_{0},x_{1},x_{2})$, we consider the pair
$(X_{4},D_{2,4})$, where
$X_{4}\subset\mathbb{P}(1^{4},2)_{(x_{0},x_{1},x_{2},x_{3},y)}$ is the
hypersurface given by equation

$y(y+Q)-C+x_{3}\bigl{(}(x_{0}+x_{1})y+x_{1}Q+B\bigr{)}=0,$

and $D_{2,4}$ is cut out in $X_{4}$ by the equation $(y+x_{1}x_{3}=0)$.

######  Remark 2.14.

The variety $X_{4}\subset\mathbb{P}(1^{4},2)$ has a $cA_{2}$-singularity444A
singularity that is locally analytically equivalent to $uv+w^{3}+z^{3}=0$ in
$\mathbb{C}^{4}$. at the point $[0:0:0:1:0]$ and $D_{2,4}$ is isomorphic to
the original $D\subset\mathbb{P}^{3}$, as one can check by plugging in
$y=-x_{1}x_{3}$ and literally getting the equation of $D\subset\mathbb{P}^{3}$
back.

######  Example 2.15 (Isomorphism of objects $5^{a}$ and $5^{b}$).

Table 1 also lists a 6-parameter family of objects $5^{b}$. The substitution

$\widetilde{y}=-y-x_{3}(x_{0}+x_{1})$

transforms object $5^{a}$ isomorphically to object $5^{b}$.

Indeed, rewrite the equation slightly and substitute:

$y\Bigl{(}y+Q\Bigr{)}-C+x_{3}\Bigl{(}(x_{0}+x_{1})y+x_{1}Q+B\Bigr{)}=\\\
=y\Bigl{(}y+x_{3}(x_{0}+x_{1})\Bigr{)}+Q\Bigl{(}y+x_{1}x_{3}\Bigr{)}-C+x_{3}B=\\\
=\Bigl{(}\widetilde{y}+x_{3}(x_{0}+x_{1})\Bigr{)}\widetilde{y}-Q\Bigl{(}\widetilde{y}+x_{0}x_{3}\Bigr{)}-C+x_{3}B=\\\
=\widetilde{y}\Bigl{(}\widetilde{y}-Q\Bigr{)}-C+x_{3}\Bigl{(}(x_{0}+x_{1})\widetilde{y}-x_{0}Q+B\Bigr{)}$

######  Example 2.16 (Maps from objects $3^{\bullet}$ to objects
$5^{\bullet}$).

For each quadratic form $Q=Q(x_{0},x_{1},x_{2})$, we construct a volume
preserving Sarkisov link of type (II),
$\psi^{a}\colon(\mathbb{P}(1^{3},2),D_{5}^{a})\dasharrow(X_{4},D_{2,4})$. In
coordinates, the map
$\psi^{a}\colon\mathbb{P}(1^{3},2)_{(x_{0},x_{1},x_{2},y)}\to\mathbb{P}(1^{4},2)_{(x_{0},x_{1},x_{2},x_{3},y)}$
is given by:

(2.3)
$(x_{0},x_{1},x_{2},y)\mapsto\left(x_{0},x_{1},x_{2},-\frac{y(y+Q)-C}{x_{0}y+B+x_{1}(y+Q)},y\right).$

This map is obtained by first blowing-up the curve
$\Gamma\subset\mathbb{P}(1^{3},2)$ defined by equations:

$\left\\{\begin{array}[]{l}F_{3}=x_{0}y+B+x_{1}(y+Q)=0,\\\
G_{4}=y(y+Q)-C=0,\end{array}\right.$

and then blowing-down the strict transform of the divisor
$\big{(}x_{0}y+B+x_{1}(y+Q)=0\big{)}$.

Let us describe this map in detail. First note that $\Gamma\subset D_{5}^{a}$:

(2.4) $x_{0}y^{2}+By+x_{1}C\ =\ x_{0}y^{2}+By+x_{1}y(y+Q)-x_{1}y(y+Q)+x_{1}C\
=\\\ =\ y(x_{0}y+B+x_{1}(y+Q))-x_{1}(y(y+Q)-C)\ =\ yF_{3}-x_{1}G_{4}.$

In order to describe the blow-up of $\mathbb{P}(1,1,1,2)$ along $\Gamma$,
consider the toric variety $\mathbb{F}$ with coordinates and weight matrix:

$\begin{array}[]{cccccc}x_{0}&x_{1}&x_{2}&y&u&v\\\ \hline\cr 1&1&1&2&0&-1\\\
0&0&0&0&1&1\end{array}\,,$

and stability condition chosen so that the nef cone of $\mathbb{F}$ is the
span $\langle x_{i},u\rangle_{+}$:

$x_{0},x_{1},x_{2}$$y$$u$$v$

This choice of stability condition gives the irrelevant ideal
$(x_{0},x_{1},x_{2},y)(u,v)$, and yields a $\mathbb{P}^{1}$-bundle morphism
$\pi\colon\mathbb{F}\to\mathbb{P}(1,1,1,2)$. The other contraction of
$\mathbb{F}$ is the divisorial contraction
$\pi^{\prime}\colon\mathbb{F}\to\mathbb{P}(1,1,1,1,2)$ that maps the divisor
$(v=0)$ to the point $[0:0:0:1:0]\in\mathbb{P}(1,1,1,1,2)$. In coordinates,
$\pi^{\prime}\colon\mathbb{F}_{(x_{0},x_{1},x_{2},y,u,v)}\to\mathbb{P}(1,1,1,1,2)_{(x_{0},x_{1},x_{2},x_{3},y)}$
is given by

(2.5)
$(x_{0},x_{1},x_{2},y,u,v)\mapsto\left(vx_{0},vx_{1},vx_{2},u,v^{2}y\right)\,.$

The blow-up $Z$ of $\mathbb{P}(1,1,1,2)$ along $\Gamma$ is cut out in
$\mathbb{F}$ by the equation

$uF_{3}+vG_{4}=0.$

Let us describe the equation of $\pi^{\prime}(Z)\subset\mathbb{P}(1,1,1,1,2)$.
From $\frac{u}{v}=-\frac{G_{4}}{F_{3}}$, we get $x_{3}F_{3}+G_{4}=0$, and
hence $\pi^{\prime}(Z)$ is cut out by the equation

$y(y+Q)-C+x_{3}(x_{0}y+B+x_{1}(y+Q))=0,$

which is precisely the equation of $X_{4}$. Combining with Equation (2.4), we
see that the strict transform of $D_{5}^{a}$ in $X_{4}$ is cut out by the
equation

$y+x_{1}x_{3}=0,$

and so it is precisely the divisor $D_{2,4}$.

In coordinates, the composed map
$\psi^{a}\colon\mathbb{P}(1^{3},2)_{(x_{0},x_{1},x_{2},y)}\to\mathbb{P}(1^{4},2)_{(x_{0},x_{1},x_{2},x_{3},y)}$
is given by:

$(x_{0},x_{1},x_{2},y)\mapsto\left(x_{0},x_{1},x_{2},-\frac{G_{4}}{F_{3}},y\right),$

which is precisely (2.3) above.

Similarly, there is a volume preserving Sarkisov link of type (II),
$\psi^{b}\colon\text{obj.~{}$3^{b}$}\dasharrow\text{obj.~{}$5^{b}$}$.

Finally we denote by
$\widetilde{\psi}^{a}\colon\text{obj.~{}$3^{a}$}\dasharrow\text{obj.~{}$5^{b}$}$
the composition of $\psi^{a}$ with the isomorphism in Example 2.15, and
similarly
$\widetilde{\psi}^{b}\colon\text{obj.~{}$3^{b}$}\dasharrow\text{obj.~{}$5^{a}$}$
the composition of $\psi^{b}$ with the inverse of the isomorphism in Example
2.15,

##  3\. Proof of Theorem A

The prototype of CY pairs adressed in Theorem A is $(\mathbb{P}^{n},D_{n+1})$,
where $D_{n+1}$ is a hypersurface of degree $n+1$ in $\mathbb{P}^{n}$, $n\geq
3$. The assumptions implies that $D_{n+1}$ is factorial, has terminal
singularities, and
$\operatorname{Pic}(D_{n+1})=\big{\langle}\mathcal{O}_{\mathbb{P}^{n}}(1)_{|D_{n+1}}\big{\rangle}$.
Let us sketch the proof of Theorem A in this case.

Suppose that $\psi\colon(\mathbb{P}^{n},D_{n+1})/\mathbb{C}\dasharrow(X,D)/T$
is a volume-preserving birational map between Mf CY pairs that is not
biregular. Then the first step of a Sarkisov factorization of $\psi$ is a
divisorial contraction $\pi\colon Y\rightarrow\mathbb{P}^{n}$ with center
$Z\subset\mathbb{P}^{n}$. In Proposition 3.1, we show that $Z\subset D_{n+1}$
and $\operatorname{codim}_{\mathbb{P}^{n}}(Z)=2$. The assumption that
$\operatorname{Pic}X\to\operatorname{Cl}D_{n+1}$ is an isomorphism implies
that $Z$ is a complete intersection $Z=D_{n+1}\cap D_{d}$, where
$D_{d}\subset\mathbb{P}^{n}$ is a hypersurface of degree $d$. Then we show in
Lemma 3.2 that the cone of effective divisors of $Y$ is

$\operatorname{Eff}(Y)\ =\ \langle E,\widetilde{D}_{b}\rangle_{+}\ ,$

where $E$ denotes the exceptional divisor of $\pi$, $b=\min\\{d,n+1\\}$, and
$\widetilde{D}_{b}$ denotes the strict transform of ${D}_{b}$ in $Y$.
Therefore, the first link in a Sarkisov factorization of $\psi$ is:

$Y$$Y^{\prime}$$\mathbb{P}^{n}$$W$
,$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\chi}$

where $\chi$ is a composition of Mori flips, flops and antiflips, and
$\pi^{\prime}\colon Y^{\prime}\rightarrow W$ is either a divisorial
contraction or a Mori fibre space that contracts the strict transform of
$\widetilde{D}_{b}$. In any case, $-K_{Y^{\prime}}$ is $\pi^{\prime}$-ample.
Since $-K_{Y^{\prime}}\sim\chi_{*}\widetilde{D}_{n+1}$, we must have
$b=d<n+1$, and $\pi^{\prime}\colon Y^{\prime}\to W$ is a divisorial
contraction with exceptional divisor $\widetilde{D}_{b}$. A straightforward
computation in the proof of Proposition 3.3 shows that in this case $W$ has
worse then terminal singularities, which is not allowed in a Sarkisov
factorization of $\psi$. This contradiction shows that the original map
$\psi\colon\mathbb{P}^{n}\dasharrow X$ is an isomorphism.

######  Proposition 3.1.

Let $(X,D_{X})$ be a (t, lc) CY pair, and
$f\colon(Y,D_{Y})\rightarrow(X,D_{X})$ a volume preserving divisorial
contraction with center $Z\subset X$. Then $Z\subset D_{X}$.

Suppose moreover that $(X,D_{X})$ is canonical, and that $D_{X}$ is terminal
at the generic point of $Z$. Then $\operatorname{codim}_{X}(Z)=2$, and $D_{Y}$
is the strict transform of $D_{X}$ in $Y$.

###### Proof.

Denote by $\widetilde{D}_{X}$ the strict transform of $D_{X}$ in $Y$, and by
$E$ the exceptional divisor of $f$. Since
$f\colon(Y,D_{Y})\rightarrow(X,D_{X})$ is volume preserving, we have
$D_{Y}=\widetilde{D}_{X}+mE$, with $m\in\\{0,1\\}$. Suppose that $Z\not\subset
D_{X}$. Then $\widetilde{D}_{X}=f^{*}D_{X}$, and the equality
$K_{Y}+D_{Y}=f^{*}(K_{X}+D_{X})$ would imply that $K_{Y}=f^{*}K_{X}-mE$,
contradicting the fact that $X$ is terminal. So we conclude that $Z\subset
D_{X}$.

Now suppose that $(X,D_{X})$ is canonical. It follows from Proposition 2.6
that $m=0$, and $D_{Y}=\widetilde{D}_{X}$. By Lemma 2.8, $(Y,D_{Y})$ is also
canonical. In particular ${D}_{Y}$ has normal support. Denote by
$\overline{f}\colon{D}_{Y}\rightarrow D_{X}$ the restriction of $f\colon Y\to
X$ to ${D}_{Y}$. By restricting to ${D}_{Y}$ the equality

$K_{Y}+D_{Y}\ =\ f^{*}(K_{X}+D_{X})$

and applying adjunction, we get that

$K_{{D}_{Y}}\ =\overline{f}^{*}K_{D_{X}}.$

If moreover $D_{X}$ is terminal at the generic point of $Z$, then this last
equality implies that $E\cap{D}_{Y}$ is not exceptional for
$\overline{f}\colon{D}_{Y}\rightarrow D_{X}$. It follows that
$\operatorname{codim}_{D_{X}}(Z)=1$ and thus $\operatorname{codim}_{X}(Z)=2$.
∎

######  Lemma 3.2.

Let $X$ be a $\mathbb{Q}$-factorial terminal Fano variety with
$\operatorname{Cl}(X)\cong\mathbb{Z}$, and denote by $H_{X}$ the ample
generator of $\operatorname{Cl}(X)$. Let $Z=H_{a}\cap H_{b}$ be an irreducible
and generically reduced complete intersection of two hypersurfaces
$H_{a}\sim_{{}_{\mathbb{Q}}}aH_{X}$ and $H_{b}\sim_{{}_{\mathbb{Q}}}bH_{X}$,
with $b\leq a$. Let $\pi\colon Y\rightarrow X$ be a terminal divisorial
contraction with center $Z$. Then the cone of effective divisors of $Y$ is

$\operatorname{Eff}(Y)\ =\ \langle E,\widetilde{H}_{b}\rangle_{+}$

where $E$ denotes the exceptional divisor of $\pi$, and $\widetilde{H}_{b}$
the strict transform of ${H}_{b}$ in $Y$.

###### Proof.

Set $H:=\pi^{*}H_{X}$, $\ell:=H^{n-1}\in N_{1}(X)$, and $\lambda=H^{n}>0$,
where $n=\dim(X)$. At the generic point of $Z$, the divisorial contraction
$\pi$ coincides with the blow-up of $Z$. In particular, $\pi_{|E}\colon
E\rightarrow Z$ is a $\mathbb{P}^{1}$-bundle over the generic point of $Z$.
Denote by $e\subset E$ a general fiber of $\pi_{|E}$. We have
$H\cdot\ell=\lambda$, $E\cdot e=-1$, and $H\cdot e=E\cdot\ell=0$. Since
$\rho(Y)=2$, the cone $\operatorname{Eff}(Y)$ has two extremal rays, one of
which is generated by $E$.

We show that $\widetilde{H}_{b}$ is dominated by a family of curves with class
in the ray

$\mathbb{R}_{\geq 0}\ [\ell-(a\lambda)e]\ \subset\ N_{1}(Y).$

Let $k$ be a positive integer such that $kH_{X}$ is very ample, and let
$C\subset H_{b}\subset X$ be a curve cut out in $H_{b}$ by $n-2$ general
members of $\big{|}kH_{X}\big{|}$. Then $C\cdot H_{a}=abk^{n-2}\lambda$ and,
inside $H_{b}$, $C$ intersects $Z$ transversally in $abk^{n-2}\lambda$ smooth
points. Therefore, the class of its strict transform $\widetilde{C}\subset Y$
in $N_{1}(Y)$ is precisely $bk^{n-2}[\ell-(a\lambda)e]$.

Let $D\subset Y$ be a prime divisor distinct from $\widetilde{H}_{b}$ and $E$,
and write $D\sim_{{}_{\mathbb{Q}}}dH-mE$. Since $D\neq E$, it has non-negative
intersection with $e$, and thus $m\geq 0$. Since $D\neq\widetilde{H}_{b}$, it
has non-negative intersection with $\ell-(a\lambda)e$, and thus $d\geq ma\geq
mb$. So we can write

$D\ \sim_{{}_{\mathbb{Q}}}\ (d-mb)H\ +\ m(bH-E)\ \sim_{{}_{\mathbb{Q}}}\
(d-mb)H\ +\ m\widetilde{H}_{b}$

with $d-mb\geq 0$. This shows that $\operatorname{Eff}(Y)=\langle
E,\widetilde{H}_{b}\rangle_{+}$. ∎

######  Proposition 3.3.

Let $(X,D)/S$ be a $\mathbb{Q}$-factorial (t, lc) Mf CY pair, where

  1. (i)

$X$ is a Fano variety with $\operatorname{Cl}(X)\cong\mathbb{Z}$; and

  2. (ii)

$D\subset X$ is a normal hypersurface such that the restriction homomorphism
$\operatorname{Cl}(X)\to\operatorname{Cl}(D)$ is an isomorphism.

Let $\psi\colon(X,D)/S\dasharrow(X^{\prime},D^{\prime})/S^{\prime}$ be a
volume preserving birational map between Mf CY pairs. Then any Sarkisov
factorization of $\psi$ starts with a divisorial contraction with center
$Z\subset D$ such that $\operatorname{codim}_{X}(Z)\geq 3$.

###### Proof.

By Theorem 2.5, $\psi\colon(X,D)\dasharrow(X^{\prime},D^{\prime})$ is a
composition of volume preserving Sarkisov links. By Proposition 3.1 the first
step of a Sarkisov factorization is a divisorial contraction $\pi\colon
Y\rightarrow X$ with center $Z\subset D$. We must rule out the possibility
that $\operatorname{codim}_{X}(Z)=2$. Suppose that this is the case. Then
$\pi$ coincides with the blow-up of $Z$ at the generic point of $Z$, and so
$(K_{Y}+D_{Y})=\pi^{*}(K_{X}+D)$, where $D_{Y}$ is the strict transform of $D$
in $Y$.

Write $H_{X}$ for the generator of $\operatorname{Cl}(X)$, and let
$\iota\in\mathbb{Z}_{>0}$ be such that
$-K_{X}\sim_{\mathbb{Q}}D\sim_{\mathbb{Q}}\iota H_{X}$. Since the restriction
homomorphism $\operatorname{Cl}(X)\to\operatorname{Cl}(D)$ is an isomorphism,
there exists an irreducible hypersurface in $H_{d}\sim_{\mathbb{Q}}\iota
H_{X}$ such that $Z=D\cap H_{d}$.555That is, the intersection $D\cap H_{d}$ is
generically reduced and irreducible, and $D\cap H_{d}=Z$ set-theoretically. We
follow the notation of Lemma 3.2 and its proof, with
$\\{a,b\\}=\\{\iota,d\\}$. Consider the first link in a Sarkisov factorization
of $\psi$:

$Y$$Y^{\prime}$$X$$W\
,$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\chi}$

where $\chi$ is a (possibly trivial) composition of Mori flips, flops and
antiflips, and $(Y^{\prime},D_{Y}^{\prime})$ is a CY pair, where
$D_{Y}^{\prime}=\chi_{*}D_{Y}$.

Next we show that $\iota=a>b=d$. By Lemma 3.2,
$\operatorname{Eff}(Y^{\prime})=\langle
E^{\prime},\widetilde{H}^{\prime}_{b}\rangle_{+}$, where $E^{\prime}$ and
$\widetilde{H}^{\prime}_{b}$ denote the strict transforms in $Y^{\prime}$ of
$E$ and $\widetilde{H}_{b}$, respectively. This implies that either the
morphism $\pi^{\prime}\colon Y^{\prime}\to W$ is a divisorial contraction with
exceptional divisor $\widetilde{H}^{\prime}_{b}$, or it is a Mori fibration
given by the linear system $\big{|}m\widetilde{H}^{\prime}_{b}\big{|}$ for
$m\gg 0$. The latter occurs if and only if $a=b$. Since
$(Y^{\prime},D_{Y}^{\prime})$ is a CY pair, and $-K_{Y^{\prime}}\sim
D_{Y}^{\prime}$ is $\pi^{\prime}$-ample, we conclude that
$D^{\prime}\neq\widetilde{H}^{\prime}_{b}$. So $\iota=a>b=d$, $D=H_{a}$, and
$\pi^{\prime}\colon Y^{\prime}\to W$ is a divisorial contraction with
exceptional divisor $\widetilde{H}^{\prime}_{b}$. Note also that $W$ is a
$\mathbb{Q}$-factorial terminal Fano variety with $\rho(W)=1$.

Since $\pi$ coincides with the blow-up of $Z$ at the generic point of $Z$, $W$
is terminal, and $\chi$ is a small birational map, we can write

$\pi^{*}K_{X}+E\ =\ K_{Y}\ =\ \chi^{*}K_{Y^{\prime}}\ =\
\chi^{*}(\pi^{\prime*}K_{W}+t\widetilde{H}^{\prime}_{b})\ =\
\chi^{*}\pi^{\prime*}K_{W}\ +\ t\widetilde{H}_{b}$

for some positive rational number $t$. Recall from the proof of Lemma 3.2 that
$\widetilde{H}_{b}$ is dominated by a family of curves with class in the ray

$\mathbb{R}_{\geq 0}\ [\ell-(a\lambda)e]\ \subset\ N_{1}(Y)\ ,$

where $\lambda=H_{X}^{\dim(X)}>0$. By intersecting the divisors above with
$[\ell-(a\lambda)e]$, we get

$\displaystyle 0\ =\ -\lambda\iota+\lambda a\ $ $\displaystyle=\
(\pi^{*}K_{X}\ +\ E)\ \cdot\ [\ell-(a\lambda)e]$ $\displaystyle=\
(\chi^{*}\pi^{\prime*}K_{W}\ +\ t\widetilde{H}_{b})\ \cdot\
[\ell-(a\lambda)e]$ $\displaystyle=\
K_{W}\cdot(\pi^{\prime}\circ\chi)_{*}[\ell-(a\lambda)e]\ +\ t\lambda(b-a)<0.$

This contradiction shows that $\operatorname{codim}_{D}(Z)\geq 2$, and hence
$\operatorname{codim}_{X}(Z)\geq 3$. ∎

###### Proof of Theorem A.

The result follows directly from Propositions 3.1 and 3.3. ∎

######  Remark 3.4.

It follows from Theorem A that a _very general_ quartic surface
$D\subset\mathbb{P}^{3}$ satisfies
$\operatorname{Bir}(\mathbb{P}^{3},D)=\operatorname{Aut}(\mathbb{P}^{3},D)$.
By very general we mean that $D$ is smooth and satisfies the Lefschetz
condition:
$\operatorname{Cl}D=\mathbb{Z}\cdot\mathcal{O}_{\mathbb{P}^{3}}(1)_{|D}$. In
this special case, a stronger result holds, and $D$ can be taken to be only
_general_. Indeed, to conclude that
$\operatorname{Bir}(\mathbb{P}^{3},D)=\operatorname{Aut}(\mathbb{P}^{3},D)$,
it is enough to assume that $D$ is smooth and any curve of degree $<16$ in $D$
is a complete intersection of $D$ with another surface
$S\subset\mathbb{P}^{3}$, see [Tak98, Theorem 2.3 and Remark 2.4] and [Ogu13,
Theorem 3.1].

##  4\. Extremal contractions

In this section we classify extremal contractions of different types in
various contexts. These results are the toolkit that we need to prove Theorems
B and C. Indeed, the proofs of these results centre on the classification of
all possible volume preserving Sarkisov links

$(X,D)/S\dasharrow(X^{\dagger},D^{\dagger})/S^{\dagger}$

that start from a known and given Mf CY pair $(X,D)/S$. The links are
themselves constituted of elementary modifications that are naturally
associated to contractions of extremal rays: volume preserving extremal
divisorial contractions; flips, flops, antiflips; and Mf CY pairs. These
elementary constituents are the topic of this section. The results are
technical and we advise the reader to skip this section on first reading and
use it as reference, coming back to it as needed.

###  4.1. Divisorial contractions.

######  Lemma 4.1.

Let $D\subset\mathbb{P}^{3}$ be a quartic surface with canonical
singularities, having exactly one singular point $z\in D$. Let
$f\colon(Y,D_{Y})\rightarrow(\mathbb{P}^{3},D)$ be a volume preserving
terminal divisorial contraction mapping a divisor $E\subset Y$ to a closed
point $x\in\mathbb{P}^{3}$. Then $x=z\in D$, and $D_{Y}$ is the strict
transform of $D$ in $Y$.

  1. (1)

If $z$ is a singularity of $D$ of type $A_{1}$, then $f$ is the blow-up of
$\mathbb{P}^{3}$ at $z$.

  2. (2)

If $z$ is a singularity of $D$ of type $A_{2}$, choose homogeneous coordinates
such that $D$ is defined by the equation

$x_{0}x_{1}x_{3}^{2}+x_{3}B+C\ =\ 0,$

where $B=B_{3}(x_{0},x_{1},x_{2})$ and $C=C_{4}(x_{0},x_{1},x_{2})$ are
homogeneous polynomials of degree $3$ and $4$, respectively. Then either $f$
is the blow-up of $\mathbb{P}^{3}$ at $z$, or $f$ is the weighted blow-up of
$\mathbb{P}^{3}$ at $z$ with weights $(2,1,1)$ or $(1,2,1)$ — with respect to
the affine coordinates $x_{0},x_{1},x_{2}$ in the open affine chart
$(x_{3}=1)$.

###### Proof.

By Proposition 3.1, $x=z$ is the singular point of $D$. By Proposition 2.6,
$D_{Y}$ is the strict transform of $D$ in $Y$. Choose homogeneous coordinates
on $\mathbb{P}^{3}$ such that $z=[0:0:0:1]$, and the equation of $D$ has the
form

$x_{3}^{2}Q+x_{3}B+C=0\ ,$

where $Q,B,C\in\mathbb{C}[x_{0},x_{1},x_{2}]$ are homogeneous of degree $2,3$
and $4$, respectively. Then $z\in D$ is an $A_{1}$-singularity if and only if
$Q$ is a quadratic form of rank $3$. If $z\in D$ is an $A_{2}$-singularity,
then $Q$ is a quadratic form of rank two, which — possibly after changing
homogeneous coordinates — we may assume to be $Q=x_{0}x_{1}$.

By [Kaw01, Theorem 1.1], in suitable _analytic_ coordinates at
$z\in\mathbb{P}^{3}$, the divisorial contraction $f\colon Y\to\mathbb{P}^{3}$
is the weighted blow-up of $z$ with weights $(1,a,b)$ where $a$ and $b$ are
coprime integers. The difficulty in using this result is that at this point we
do not know that these analytic coordinates at $z$ are induced from
homogeneous coordinates on $\mathbb{P}^{3}$. Instead of using the result
directly, we will use an equivalent statement that is coordinate-free.

In general, suppose that $z\in Z$ is a nonsingular point on a $3$-fold $Z$,
and that there are analytic coordinates at $z\in Z$ such that $f\colon
E\subset Y\to z\in Z$ is the weighted blow-up with weights $(1,a,b)$, where
$1\leq a<b$ and $\operatorname{hcf}(a,b)=1$. The toric description of the
weighted blow-up allows us to realize the valuation associated to $E$ as
follows. Construct inductively the tower of blow-ups:

$\cdots\to Z_{i}\to Z_{i-1}\to\dots\to Z_{1}\to Z_{0}=Z,$

where $Z_{i}\to Z_{i-1}$ is the blow-up of the centre
$\operatorname{z}_{i-1}=\operatorname{z}_{E}Z_{i-1}$ of the valuation $E$ on
$Z_{i-1}$. Note that $Z_{1}\to Z_{0}$ is the blow-up of
$\operatorname{z}_{0}=z$. For every $i$, we denote by $E_{i}\subset Z_{i}$ the
exceptional divisor, and for $j>i$ we denote by $E^{j}_{i}\subset Z_{j}$ the
strict transform of $E_{i}$ in $Z_{j}$. The following key properties follow
directly from the toric description of the weighted blow-up:

  1. (1)

For all $0\leq j<a$, the centre $\operatorname{z}_{j}$ is a closed point of
$Z_{j}$. If $j\geq 1$, then $\operatorname{z}_{j}\in E_{j}\subset Z_{j}$, and
if $j\geq 2$, then

$\operatorname{z}_{j}\in E_{j}\setminus E^{j}_{j-1}.$

  2. (2)

The centre $\operatorname{z}_{a}\in Z_{a}$ is the generic point of a line
$L_{a}\subset E_{a}\cong\mathbb{P}^{2}$. If $a\geq 2$, then

$L_{a}\not\subset E^{a}_{a-1}.$

  3. (3)

For all $a+1\leq j<b$, the centre $\operatorname{z}_{j}\in Z_{j}$ is a section

$L_{j}\subset E_{j}\setminus E^{j}_{j-1}$

of the projection $E_{j}\to L_{j-1}$.

  4. (4)

$E_{b}=E$ (by this we mean that the exceptional divisors $E_{b}$ and $E$
induce the same valuation on $Z$).

We go back to our volume preserving terminal divisorial contraction
$f\colon(Y,D_{Y})\to(\mathbb{P}^{3},D)$ mapping $E\subset Y$ to
$z\in\mathbb{P}^{3}$. Consider the tower above, starting with the blow-up
$\sigma_{1}\colon Z_{1}\to Z_{0}=\mathbb{P}^{3}$ of
$\operatorname{z}_{0}=z\in\mathbb{P}^{3}$, and denote by $D_{i}\subset Z_{i}$
the strict transform of $D$. We have that
$K_{Z_{1}}=\sigma_{1}^{*}(K_{Z_{0}})+2E_{1}$ and
$D_{1}\sim\sigma_{1}^{*}(D_{0})-2E_{1}$, hence

$K_{Z_{1}}+D_{1}=\sigma_{1}^{*}(K_{Z_{0}}+D_{0}).$

In other words, the birational morphism $(Z_{1},D_{1})\to(Z_{0},D_{0})$ is
volume preserving, and

$a(E,K_{Z_{1}}+D_{1})=0.$

If $z$ is a singularity of $D$ of type $A_{1}$ or $A_{2}$ then $D_{1}\subset
Z_{1}$ is a smooth surface. Since $a(E,K_{Z_{1}}+D_{1})=0$, either $E=E_{1}$ —
in which case we are done — or the centre $\operatorname{z}_{E}Z_{1}$ is the
generic point of a curve on $D_{1}\cap E_{1}$. In any case, key property (1)
above implies that $a=1$.

Suppose that $z\in D$ is an $A_{1}$-singularity. Then $D_{1}\cap E_{1}$ is a
nonsingular conic in $E_{1}\cong\mathbb{P}^{2}$. On the other hand, if $b>1$,
then the centre $\operatorname{z}_{E}Z_{1}$ is the generic point of a line on
$E_{1}\cong\mathbb{P}^{2}$ by key property (2) above. This implies that
$1=a=b$, i.e. $f$ is the blow-up of $\mathbb{P}^{3}$ at $z$.

Suppose now that $z\in D$ is an $A_{2}$-singularity. If $E=E_{1}$, then we are
done. Otherwise, by key property (2) above, the centre
$\operatorname{z}_{1}=\operatorname{z}_{E}Z_{1}$ is the generic point of one
of the two lines $(x_{0}=0)$, $(x_{1}=0)$ in $E_{1}\cong\mathbb{P}^{2}$.
Assume $\operatorname{z}_{1}$ is the generic point of the line
$L_{1}=(x_{0}=0)$ — the other case is similar. Write $\sigma_{2}\colon
Z_{2}\to Z_{1}$ for the blow-up of the line $L_{1}\subset Z_{1}$. Note that
$K_{Z_{2}}=\sigma_{2}^{*}(K_{Z_{1}})+E_{2}$, and
$D_{2}=\sigma_{2}^{*}(D_{1})-E_{2}$. Hence

$K_{Z_{2}}+D_{2}=\sigma_{2}^{*}(K_{Z_{1}}+D_{1}),$

in other words, the composed birational morphism
$(Z_{2},D_{2})\to(\mathbb{P}^{3},D)$ is volume preserving and
$a(E,K_{Z_{2}}+D_{2})=0$. If $E=E_{2}$, then $f$ is the weighted blow-up with
weights $(2,1,1)$ in the native homogeneous coordinates of $\mathbb{P}^{3}$,
and we are done. We show that the assumption that $E\neq E_{2}$ leads to a
contradiction. Consider the centre
$\operatorname{z}_{2}=\operatorname{z}_{E}Z_{2}$.

  1. (i)

By key property (3), $\operatorname{z}_{2}$ is the generic point of a section
$L_{2}\subset E_{2}$ of the projection $E_{2}\to L_{1}$ disjoint from
$E_{1}^{2}$.

  2. (ii)

On the other hand, since $D_{2}\subset Z_{2}$ is a smooth surface,
$\operatorname{z}_{2}$ is the generic point of a curve in $D_{2}\cap E_{2}$ by
Proposition 3.1. Since $\Gamma=D_{2}\cap E_{2}$ is irreducible,
$\operatorname{z}_{2}$ is the generic point of $\Gamma$.

Finally, since $D_{1}\cap E_{1}$ consists of the union of the two lines
$(x_{0}=0)$ or $(x_{1}=0)$ in $E_{1}\cong\mathbb{P}^{2}$, the curves
$\Gamma=D_{2}\cap E_{2}$ and $E_{1}^{2}\cap E_{2}$ intersect. This contradicts
(i) and concludes the proof. ∎

###  4.2. Extremal neighbourhoods

######  Definition 4.2.

An _extremal neighbourhood_ is the analytic germ around a projective curve
$\Gamma$ in a 3-fold $X$. In our situation the curve $\Gamma$ is always
contained in a given surface $S\subset X$.

######  Definition 4.3.

Consider an extremal neighbourhood $\Gamma\subset S\subset X$, where $X$ is a
3-fold with terminal singularities, $S\in|-K_{X}|$ is a surface with Du Val
singularities, and $\Gamma\cong\mathbb{P}^{1}$ is a smooth rational curve. We
say that the extremal neighbourhood is _trivial_ if it is isomorphic to the
analytic germ around $\Gamma$ in its normal bundle.

Our first result describes the extremal neighbourhood $\Gamma\subset S\subset
X$ in the case when $S$ and $X$ are nonsingular, and
$K_{X}\cdot\Gamma=-S\cdot\Gamma>0$.

######  Lemma 4.4.

Consider an extremal neighbourhood $\Gamma\subset S\subset X$, where $X$ is a
smooth 3-fold, $S\in|-K_{X}|$ is a smooth surface, and
$\Gamma\cong\mathbb{P}^{1}$ is a smooth rational curve. Suppose that
$k=K_{X}\cdot\Gamma=-S\cdot\Gamma\geq 1$. Then the following holds.

  1. (1)

The extremal neighbourhood $\Gamma\subset S\subset X$ is trivial.

  2. (2)

The antiflip $X\dasharrow X^{-}$ exists, and $X^{-}$ has terminal
singularities if and only if $k=1$.

###### Proof.

To prove (1), we use [Mor82, Lemma 3.33], which in fact goes back to [HR64,
Lemma 9]. It is enough to show that the analytic germ around $S\subset X$ is
trivial. Indeed, $\Gamma$ is a $(-2)$-curve on $S$, and $S$ itself is
isomorphic to the analytic germ of $\Gamma$ in its normal bundle on $S$. We
denote by $\mathcal{O}_{S}(1)$ the unique line bundle on $S$ such that
$\mathcal{O}_{S}(1)_{|\Gamma}=\mathcal{O}_{\Gamma}(1)$.

First we check that the second infinitesimal neighbourhood of $S$ in $X$ is
trivial. Indeed, the second infinitesimal neighbourhood is an infinitesimal
extension of $\mathcal{O}_{S}$ by $\mathcal{O}_{S}(-S)=\mathcal{O}_{S}(k)$:

$0\to\mathcal{O}_{S}(-S)\to\mathcal{O}_{2S}\to\mathcal{O}_{S}\to 0\ ,$

and these extensions are classified by

$H^{1}(S,T_{S}(-S))=H^{1}\Bigl{(}\mathcal{O}_{S}(-2+k)\oplus\mathcal{O}_{S}(2+k)\Bigr{)}=0.$

Now that we have shown that the second infinitesimal neighbourhood is trivial,
we are ready to apply [Mor82, Lemma 3.33], which states that the neighbourhood
of $S\subset X$ is formally trivial if
$H^{1}\bigl{(}S,T_{X|S}\otimes\mathcal{O}_{S}(-nS)\bigr{)}=0$ for all $n\geq
2$. The latter statement in turn follows from the exact sequence

$0\to T_{S}\to T_{X|S}\to\mathcal{O}_{S}(S)\to 0\ ,$

giving

$H^{1}\Bigl{(}S,\mathcal{O}_{S}(-2+nk)\oplus\mathcal{O}_{S}(2+nk)\Bigr{)}\to
H^{1}\Bigl{(}S,T_{X|S}\otimes\mathcal{O}_{S}(-nS)\Bigr{)}\to
H^{1}\Bigl{(}S,\mathcal{O}_{S}\bigl{(}(n-1)k\bigr{)}\Bigr{)}$

for all $n$. We have shown that the neighbourhood of $S$ in $X$ is formally
trivial. The neighbourhood is analytically trivial by the main result of
[HR64].

Now let us prove (2). It follows from (1) that the extremal neighbourhood
$\Gamma\subset S\subset X$ is isomorphic to the neighbourhood of
$\mathbb{P}^{1}$ in the total space of the bundle
$\mathcal{O}(-2)\oplus\mathcal{O}(-k)$. Equivalently, $X$ is isomorphic to the
neighbourhood of $\mathbb{P}^{1}=(y_{0}=y_{1}=0)$ in the geometric quotient
$\mathbb{C}^{4}/\\!\\!/\mathbb{C}^{\times}$ for the action given by the
weights:

$\begin{array}[]{cccc}x_{0}&x_{1}&y_{0}&y_{1}\\\ \hline\cr
1&1&-2&-k\end{array}\ ,$

with $(>0)$ stability condition. Under this identification, $S$ is given by
the equation $(y_{1}=0)$.

The antiflip $X\dasharrow X^{-}$ is obtained by changing the stability
condition to $(<0)$. If $k=1$, then $X^{-}$ has terminal singularities. If
$k>1$, then $X^{-}$ has strictly canonical (and Gorenstein) singularity of
type

$\frac{1}{k}(1,1,-2).$

∎

######  Remark 4.5.

In Lemma 4.4 above, the existence of the surface $S$ is needed for the
triviality of the neighbourhood, as shown by simple counterexamples.

Next we want to describe the extremal neighbourhood $\Gamma\subset S\subset X$
in the case when $S$ is nonsingular outside a single ordinary node, where $X$
has a quotient singularity of type $1/2(1,1,1)$, and
$K_{X}\cdot\Gamma=-S\cdot\Gamma>0$. In this case, we expect that the extremal
neighbourhood is trivial. If this is true, then $X$ is isomorphic to the
neighbourhood of $\mathbb{P}^{1}=(y_{0}=y_{1}=0)$ in the geometric quotient
$\mathbb{C}^{4}/\\!\\!/\mathbb{C}^{\times}$ for the action given by the
weights:

$\begin{array}[]{cccc}x_{0}&x_{1}&y_{0}&y_{1}\\\ \hline\cr
1&2&-3&-k\end{array}\ ,$

where $k$ is the integer such that $K_{X}\cdot\Gamma=-S\cdot\Gamma=k/2$, and
the stability condition is $(>0)$. As in the proof of Lemma 4.4 above, we
could then construct the antiflip $X\dasharrow X^{-}$ by changing the
stability condition to $(<0)$, and easily check that $X^{-}$ has terminal
singularities if and only if $k=1$. However, the presence of singularities
makes it much harder to prove triviality of the extremal neighbourhood.
Instead, we shall construct the flip by hand, and verify whether it has
terminal singularities.

######  Lemma 4.6.

Consider a $3$-dimensional extremal neighbourhood $\Gamma\subset S\subset X$,
where $\Gamma\cong\mathbb{P}^{1}$ is a smooth rational curve, and
$S\in|-K_{X}|$ is nonsingular outside a single ordinary node, where $X$ has a
quotient singularity of type $1/2(1,1,1)$. Suppose that
$K_{X}\cdot\Gamma=-S\cdot\Gamma=k/2>0$, with $k\geq 3$ an odd integer. Then
the antiflip $X\dasharrow X^{-}$ exists, and $X^{-}$ has worse than terminal
singularities.

###### Proof.

Consider the blow-up $(E\subset Y)\to(\Gamma\subset X)$ of the curve
$\Gamma\subset X$, denote by $S^{\prime}\subset Y$ the strict transform of
$S$, and set $\Gamma^{\prime}=S^{\prime}\cap E$. It is easy to compute that:

$E\cdot\Gamma^{\prime}=-\frac{2}{3},\quad\text{and}\quad
S^{\prime}\cdot\Gamma^{\prime}=-m+2,\;\text{where
$m=\frac{k}{2}+\frac{1}{2}\geq 2$.}$

We claim that the extremal neighbourhood of $\Gamma^{\prime}\subset Y$ is
analytically isomorphic to the analytic germ around the curve
$(y_{0}=y_{1}=0)$ in the geometric quotient
$\mathbb{C}^{4}/\\!\\!/\mathbb{C}^{\times}$ for the action given by the
weights:

$\begin{array}[]{cccc}x_{0}&x_{1}&y_{0}&y_{1}\\\ \hline\cr 1&2&-3&-2m+4\\\
\end{array},$

where $E=(y_{0}=0)$, $S^{\prime}=(y_{1}=0)$, and the stability condition is
$(>0)$.

To prove the claim, note that the curve $\Gamma^{\prime}$ is the complete
intersection of the surfaces $S^{\prime}$ and $E$. Denoting by
$P\in\Gamma^{\prime}\subset S^{\prime}\subset Y$ the singular point, choose a
divisor $B_{0}$ through $P$ such that $B_{0}\cdot\Gamma^{\prime}=\frac{1}{2}$,
and a divisor $B_{1}$ meeting $\Gamma^{\prime}$ transversally at some other
point $Q\in\Gamma^{\prime}$. The divisors $B_{0},B_{1},S^{\prime},E$ map the
neighbourhood of $\Gamma^{\prime}\subset Y$ isomorphically to the model toric
quotient neighbourhood given in the claim, in such a way that the divisors
$B_{0},B_{1},S^{\prime},E$ map to the divisors
$(x_{0}=0),(x_{1}=0),(y_{0}=0),(y_{1}=0)$. There are two cases to discuss:
$m=2$ and $m>2$.

If $m>2$, then, in order to perform the antiflip $Y\dasharrow Y^{-}$ of the
curve $\Gamma^{\prime}$, we change the stability condition to $(<0)$. Denote
by $S^{-}$ and $E^{-}\subset Y^{-}$ the strict transforms of $S^{\prime}$ and
$E$, respectively. Then $S^{-}$ and $E^{-}$ are disjoint in $Y^{-}$, and the
antiflip $X^{-}$ of the original $\Gamma\subset X$ is given by

$X^{-}=\operatorname{Proj}R(Y^{-},S^{-}).$

The divisor $E^{-}$ is contracted to a point in $X^{-}$, and this point is a
strictly canonical singularity.

If $m=2$, then $E^{-}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$, and $X^{-}$ is
the contraction of $E^{-}$ along the ruling other than the one contracted to
$X$. The image of $E^{-}$ is a curve of $A_{2}$-singularities, and so $X^{-}$
has worse than terminal singularities. ∎

In the proof of Theorem C, we need to understand some neighbourhoods where the
curve $\Gamma$ is not irreducible.

######  Lemma 4.7.

Consider an extremal neighbourhood $\Gamma\subset S\subset X$, where $X$ is a
smooth 3-fold, $S\in|-K_{X}|$ is a smooth surface, and
$\Gamma=\Gamma_{0}\cup\Gamma_{1}\subset S$ is a chain of two $(-2)$-curves
intersecting transversally. Suppose that
$-K_{X}\cdot\Gamma_{0}=S\cdot\Gamma_{0}=-a<0$ and
$-K_{X}\cdot\Gamma_{1}=S\cdot\Gamma_{1}=-b<0$. Then the extremal neighbourhood
is isomorphic to the analytic germ around the curve

$\Gamma_{0}\cup\Gamma_{1}=(x_{0}=x_{2}=0)\cup(x_{0}=x_{3}=0)$

in the geometric quotient $\mathbb{C}^{5}/\\!\\!/\mathbb{C}^{\times}$ for the
action given by the weights:

$\begin{array}[]{ccccc}x_{0}&x_{1}&x_{2}&x_{3}&x_{4}\\\
\hline\cr-a&1&1&0&-2\\\ -b&-2&0&1&1\\\ \end{array},$

where the stability condition is taken in the quadrant
$\langle(1,0),(0,1)\rangle_{+}$. Under this identification, $S$ is given by
the equation $(x_{0}=0)$.

###### Proof.

By Lemma 4.4, the neighbourhood is trivial around each of the two curves
$\Gamma_{0}$, $\Gamma_{1}$. It follows from this that we can find divisors
$D_{0},\dots,D_{4}$ on $X$ as follows:

  1. (i)

$D_{0}=S$;

  2. (ii)

$D_{0}\cap D_{1}=\Gamma_{1}$ scheme-theoretically. It follows from this that
$D_{1}$ intersects $\Gamma_{0}$ transversally at the point where it intersects
$\Gamma_{1}$;

  3. (iii)

$D_{2}$ intersects $\Gamma_{0}$ transversally at one point and is disjoint
from $\Gamma_{1}$;

  4. (iv)

$D_{3}$ intersects $\Gamma_{1}$ transversally at one point and is disjoint
from $\Gamma_{0}$;

  5. (v)

$D_{0}\cap D_{4}=\Gamma_{0}$ scheme-theoretically. It follows from this that
$D_{4}$ intersects $\Gamma_{1}$ transversally at the point where it intersects
$\Gamma_{0}$.

Note that the intersection multiplicities of these divisors with the curves
$\Gamma_{0}$ and $\Gamma_{1}$ are as follows:

$\begin{array}[]{cccccc}&D_{0}&D_{1}&D_{2}&D_{3}&D_{4}\\\
\hline\cr\cdot\Gamma_{0}&-a&1&1&0&-2\\\ \cdot\Gamma_{1}&-b&-2&0&1&1\\\
\end{array}$

and so the divisors $D_{0},\dots,D_{4}$ map the neighbourhood isomorphically
to the model toric quotient neighbourhood given above. ∎

###  4.3. Fano and weak Fano $\mathbb{P}^{1}$-bundles over $\mathbb{P}^{2}$

We state the classification of rank two vector bundles $\mathcal{E}$ on
$\mathbb{P}^{2}$ such that $\mathbb{P}(\mathcal{E})$ is Fano or weak Fano. We
also collect some elementary facts and formulas on projective space bundles
for later use.

We begin by clarifying our conventions regarding vector bundles. In this
section, a vector bundle on a scheme is a locally free sheaf on it. Following
Grothendieck, if $\mathcal{E}$ is a vector bundle on $Y$ we denote by

$\mathbb{P}(\mathcal{E})=\operatorname{\underline{Proj}}_{\mathcal{O}_{X}}\bigoplus_{n\in\mathbb{N}}\operatorname{\mathit{Sym}}^{n}\mathcal{E}$

the space of $1$-dimensional _quotients_ of $\mathcal{E}$, by

$\pi\colon\mathbb{P}(\mathcal{E})\to Y$

the natural projection, by $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ — or
simply $\mathcal{O}(1)$ — the tautological line bundle on
$\mathbb{P}(\mathcal{E})$, so that

$\mathcal{E}=\pi_{*}\mathcal{O}(1),$

and by $\xi=c_{1}\left(\mathcal{O}(1)\right)\in
A^{1}\left(\mathbb{P}(\mathcal{E})\right)$ its first Chern class. The Chow
ring of $\mathbb{P}(\mathcal{E})$ admits the following description:

$A^{\bullet}\left(\mathbb{P}(\mathcal{E})\right)=\frac{A^{\bullet}(Y)[\xi]}{(\xi^{r}+\sum_{i=1}^{r}(-1)^{i}\xi^{r-i}c_{i}(\mathcal{E}))}\
,$

where $r=\operatorname{rk}\mathcal{E}$, and the formula defines the $i$-th
_Chern class_ $c_{i}(\mathcal{E})\in A^{i}(Y)$. The _Chern polynomial_ of
$\mathcal{E}$ is the polynomial

$c_{t}(\mathcal{E})=1+\sum_{i=1}^{r}c_{i}(\mathcal{E})t^{i}.$

In what follows, $\mathcal{E}$ is a rank two vector bundle on
$\mathbb{P}^{2}$. We denote by $\ell\in A^{1}(\mathbb{P}^{2})$ the class of a
line and we abuse notation slightly and write the first and second Chern
classes of $\mathcal{E}$ as

$c_{1}\ell,\quad c_{2}\ell^{2},\quad\text{where $c_{1},c_{2}\in\mathbb{Z}$.}$

We say that $\mathcal{E}$ is _normalised_ if $c_{1}\in\\{0,-1\\}$. We can
always achieve this by tensoring $\mathcal{E}$ with a line bundle.

######  Theorem 4.8.

Let $\mathcal{E}$ be a normalised rank two vector bundle on $\mathbb{P}^{2}$.
Then

  1. (1)

$\mathbb{P}(\mathcal{E})$ is Fano if and only if $\mathcal{E}$ is one of the
bundles in List $1$ below [MM82, SW90].

  2. (2)

$\mathbb{P}(\mathcal{E})$ is strictly weak Fano — i.e., $-K_{\mathbb{P}(E)}$
is nef and big but not ample — if and only if $\mathcal{E}$ is one of the
bundles in List $2$ below [Yas12, Theorem B], [JPR05, Theorem 3.4].

List $1$. Rank $2$ vector bundles $\mathcal{E}$ on $\mathbb{P}^{2}$ with
$c_{1}(\mathcal{E})\in\\{0,-1\\}$ such that $\mathbb{P}(\mathcal{E})$ is Fano

  1. (1)

$\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(-1),\
c_{1}=-1,c_{2}=0$;

  2. (2)

$\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(-1),\
c_{1}=0,c_{2}=-1$;

  3. (3)

$\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}},\
c_{1}=c_{2}=0$;

  4. (4)

$T_{\mathbb{P}^{2}}(-2)),\ c_{1}=-1,c_{2}=1$;

  5. (5)

$\mathcal{E}$ is determined by the exact sequence $0\
\to\mathcal{O}_{\mathbb{P}^{2}}\ \to\mathcal{E}\ \to\mathcal{I}_{p}\ \to 0$,
where $\mathcal{I}_{p}$ is the ideal sheaf of a point $p\in\mathbb{P}^{2}$,
$c_{1}=0,c_{2}=1$ (see Remark 4.9 below);

  6. (6)

$\mathcal{E}$ is a stable bundle with $c_{1}=0,c_{2}=2$;

  7. (7)

$\mathcal{E}$ is a stable bundle with $c_{1}=0,c_{2}=3$.

List $2$. Rank $2$ vector bundles $\mathcal{E}$ on $\mathbb{P}^{2}$ with
$c_{1}(\mathcal{E})\in\\{0,-1\\}$ such that $\mathbb{P}(\mathcal{E})$ is
strictly weak Fano

  1. (8)

$\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(-2)$,
$c_{1}=-1,c_{2}=-2$;

  2. (9)

$\mathcal{E}$ is determined by the exact sequence $0\
\to\mathcal{O}_{\mathbb{P}^{2}}\ \to\mathcal{E}\ \to\mathcal{I}_{p}(-1)\ \to
0$, where $\mathcal{I}_{p}$ is the ideal sheaf of a point
$p\in\mathbb{P}^{2}$, $c_{1}=-1,c_{2}=1$ (see Remark 4.9 below);

  3. (10)

$\mathcal{E}$ is a stable bundle with $c_{1}=-1,2\leq c_{2}\leq 5$;

  4. (11)

$\mathcal{E}$ is a stable bundle with $c_{1}=0,4\leq c_{2}\leq 6$.

######  Remark 4.9.

One may compute the Chern classes of the vector bundles (5) and (9) above as
follows. For $k\leq 2$, let $\mathcal{E}$ be the unique vector bundle on
$\mathbb{P}^{2}$ that sits in the exact sequence666A simple computation shows
that  $\text{for all
$k\in\mathbb{Z}$,}\quad\operatorname{Ext}^{1}_{\mathcal{O}_{\mathbb{P}^{2}}}(\mathcal{I}_{p}(k),\mathcal{O}_{\mathbb{P}^{2}})=\begin{cases}\mathbb{C}\;&\text{if
$k\leq 2$};\\\ 0\;&\text{if $k\geq 3$};\end{cases}\ ,$ and that the nontrivial
extensions are also locally nontrivial.

$0\ \to\mathcal{O}_{\mathbb{P}^{2}}\ \to\mathcal{E}\ \to\mathcal{I}_{p}(k)\
\to 0\ .$

By multiplicativity of Chern polynomials, we have
$c_{t}(\mathcal{E})=c_{t}(\mathcal{I}_{p}(k))$. The Koszul complex twisted by
$\mathcal{O}_{\mathbb{P}^{2}}(k)$,

$0\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(-2+k)\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(-1+k)^{\oplus
2}\rightarrow\mathcal{I}_{p}(k)\rightarrow 0,$

yields that $c_{t}(\mathcal{I}_{p}(k))(1+(-2+k)t)=(1+(-1+k)t)^{2}$, and from
this we conclude that $c_{1}(\mathcal{E})=k$ and $c_{2}(\mathcal{E})=1$.

######  Lemma 4.10.

Let $\mathcal{E}$ be a rank $r+1$ vector bundle on $Y$. Then the anticanonical
class of $\mathbb{P}(\mathcal{E})$ is

$-K_{\mathbb{P}(\mathcal{E})}=(r+1)\xi-\pi^{*}\bigl{(}c_{1}(\mathcal{E})+K_{Y}\bigr{)}\
.$

###### Proof.

This follows from the Euler sequence computing the relative tangent bundle:

$0\to\mathcal{O}_{\mathbb{P}(\mathcal{E})}\to\bigl{(}\pi^{*}\mathcal{E}^{\vee}\bigr{)}\otimes\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)\to
T_{\pi}\to 0\ ,$

and the exact sequence $0\to T_{\pi}\to
T_{\mathbb{P}(\mathcal{E})}\to\pi^{*}T_{Y}\to 0$. ∎

##  5\. Proof of Theorem B

In this section we prove Theorem B. Consider a Mf CY pair
$(\mathbb{P}^{3},D)$, where $D\subset\mathbb{P}^{3}$ is a quartic surface that
is nonsingular outside a unique singular point $z\in D$ of type $A_{1}$, and
such that the class group $\operatorname{Cl}(D)\cong\mathbb{Z}$ is generated
by the class of a hyperplane section. In this case, birational rigidity fails.
The following Mf CY pair $(X,D_{X})\to\mathbb{P}^{2}$ is a nontrivial element
in $\mathcal{P}(\mathbb{P}^{3},D)$. Let $\sigma\colon
X\rightarrow\mathbb{P}^{3}$ be the blow-up of $z$, and $D_{X}\subset X$ the
strict transform of $D$. It is a smooth $K3$ surface. The projection
$\mathbb{P}^{3}\dashrightarrow\mathbb{P}^{2}$ from $z$ induces a Mori
fibration $\pi\colon X\to\mathbb{P}^{2}$. One easily computes that

$K_{X}+D_{X}\ =\ \sigma^{*}(K_{\mathbb{P}^{3}}+D).$

Theorem B states that the Mf CY pair $(X,D_{X})\to\mathbb{P}^{2}$ is the only
nontrivial element in the pliability set $\mathcal{P}(\mathbb{P}^{3},D)$.

In the course of the proof, we will need the following result.

######  Proposition 5.1.

Let $Y$ be a smooth projective variety of dimension $n$, $A$ a smooth semi-
ample divisor on $Y$, and $\varphi\colon Y\rightarrow\mathbb{P}^{N}$ the
morphism induced by the linear system $|mA|$ for $m\gg 0$. Suppose that:

  1. (i)

the generic fibre dimension of $\varphi$ is $\leq n-3$;

  2. (ii)

for all $x\in\mathbb{P}^{N}$, $\dim\big{(}\varphi^{-1}(x)\big{)}\leq n-2$.

Then the cokernel of the restriction homomorphism
$H^{2}(Y;\mathbb{Z})\rightarrow H^{2}({A};\mathbb{Z})$ is torsion-free.

###### Proof.

Set $\mathcal{U}=Y\setminus{A}$, and denote by $i\colon\mathcal{U}\rightarrow
Y$ and $j\colon A\rightarrow Y$ the inclusions. Consider the long exact
sequence

$\dots\ \rightarrow H^{2}_{c}(\mathcal{U},\mathbb{Z})\
\xrightarrow{\textit{i}_{*}}\ H^{2}(Y,\mathbb{Z})\
\xrightarrow{\textit{j}^{*}}\ H^{2}({A},\mathbb{Z})\ \rightarrow\
H^{3}_{c}(\mathcal{U},\mathbb{Z})\ \rightarrow\ \dots\ $

where $H^{i}_{c}(\mathcal{U},\mathbb{Z})$ denotes the singular cohomology with
compact support. It is enough to show that $H^{3}_{c}(\mathcal{U},\mathbb{Z})$
is torsion free. By Poincaré duality, $H^{3}_{c}(\mathcal{U},\mathbb{Z})\cong
H_{2n-3}(\mathcal{U},\mathbb{Z})$.

Let $H\subset\mathbb{P}^{N}$ be the hyperplane such that $\varphi^{*}H=m{A}$.
Then $\varphi$ restricts to a proper morphism
$\varphi_{\mathcal{U}}\colon\mathcal{U}\rightarrow\mathbb{P}^{N}\setminus H$.
For each integer $k$, we denote by $\phi(k)$ be the dimension of the set of
points $y\in\varphi(\mathcal{U})$ such that
$\dim\big{(}\varphi^{-1}(y)\big{)}=k$. If this set is empty, we set
$\phi(k)=-\infty$. By the main result of [GM88, Part II, Chap. 1, Sec.
1.1$\star$] (Homotopy Dimension with Large Fibres), $\mathcal{U}$ has the
homotopy type of a CW complex of real dimension less or equal than

$n+\sup_{k}\big{\\{}2k-n+\phi(k)+\inf\\{\phi(k),0\\}\big{\\}}\ \leq\ 2n-3.$

— where the last inequality follows from the assumptions (i) and (ii).
Finally, by [Nar67, Theorem 3], $H_{2n-3}(\mathcal{U},\mathbb{Z})$ is torsion
free. ∎

We are now ready to proceed with the proof of Theorem B. We start by
collecting a few useful facts on the geometry of $D_{X}$.

######  5.2The geometry of $D_{X}$.

The surface $D_{X}\subset X$ intersects the exceptional divisor
$E\cong\mathbb{P}^{2}$ of $\sigma\colon X\rightarrow\mathbb{P}^{3}$
transversely along a smooth conic $e$. Denote by $h$ the pull-back of a
general hyperplane under $\sigma_{|D_{X}}\colon D_{X}\to\mathbb{P}^{3}$. Then
$\operatorname{Pic}(D_{X})=\mathbb{Z}[h]\oplus\mathbb{Z}[e]$, and the
intersection matrix of $\operatorname{Pic}(D_{X})$ with respect to the basis
$\big{(}[h],[e]\big{)}$ is

(5.1) $\left(\begin{array}[]{cc}4&0\\\ 0&-2\end{array}\right).$

The condition that $\operatorname{Cl}(D)\cong\mathbb{Z}$ is generated by the
class of a hyperplane section guarantees that $D$ contains no lines, and so
the restriction $\pi_{|D_{X}}\colon D_{X}\to\mathbb{P}^{2}$ is finite of
degree $2$. It ramifies over a sextic curve. The associated involution
$\tau\colon D_{X}\to D_{X}$ maps the $(-2)$-curve $e$ to another $(-2)$-curve
$e^{\prime}$, the strict transform of the intersection of $D$ with its tangent
cone at $z$. Note that
$\operatorname{NE}(D_{X})=\left\langle[e],[e^{\prime}]\right\rangle_{+}$, and
$e+e^{\prime}\sim(\pi_{|D_{X}})^{*}\mathcal{O}_{\mathbb{P}^{2}}(2)\sim 2h-2e$.
Thus, $e^{\prime}\sim 2h-3e$.

######  Lemma 5.3.

With the notation and assumptions of the preceding discussion 5.2, we have

$\operatorname{Bir}(D_{X})=G_{D}\rtimes\langle\tau\rangle,$

where $G_{D}\subset\operatorname{PGL}_{4}(\mathbb{C})$ is the group of
projective automorphisms of $D\subset\mathbb{P}^{3}$.

###### Proof.

Since $D_{X}$ is the unique minimal model of $D$, it is follows that
$\operatorname{Bir}(D)=\operatorname{Aut}(D_{X})$. Let us denote this group by
$G$. It naturally acts on $\operatorname{Pic}(D_{X})$ respecting the quadratic
form and the Mori cone, thus we get a group homomorphism

$\rho\colon G\to\\{\pm 1\\}$

where $\rho(g)=1$ if $\rho$ fixes $e$ and $e^{\prime}$, and $\rho(g)=-1$ if
$g$ exchanges $e$ with $e^{\prime}$. The involution $\tau$ exchanges $e$ and
$e^{\prime}$, thus $G=G_{D}\rtimes\langle\tau\rangle$, where $G_{D}=Ker(\rho)$
is the subgroup of $\operatorname{Aut}(D_{X})$ of elements that act trivially
on $\operatorname{Pic}(X)$. Thus $G_{D}$ fixes the class
$h\in\operatorname{Pic}(X)$ and hence it acts on the linear system
$|\mathcal{O}_{D}(1)|$ as projective linear transformations. ∎

######  Remark 5.4.

It is easy to see that if $D$ is general then the group $G_{D}$ of projective
automorphisms of $D\subset\mathbb{P}^{3}$ is trivial.

We are now ready to prove the first conclusion of Theorem B, i.e., that the
pliability of the pair $(\mathbb{P}^{3},D)$ is the set with two elements
$\\{(\mathbb{P}^{3},D)/\operatorname{Spec}\mathbb{C},(X,D_{X})/\mathbb{P}^{2}\\}$.

###### Proof of Theorem B(1).

Let $(\mathbb{P}^{3},D)$ be a Mf CY pair, where $D\subset\mathbb{P}^{3}$ is a
quartic surface having exactly one singular point $z\in D$ of type $A_{1}$,
and such that $\operatorname{Cl}(D)\cong\mathbb{Z}$ is generated by the class
of a hyperplane section. Note that $(\mathbb{P}^{3},D)$ is canonical.

Let $(Y,D_{Y})\rightarrow S_{Y}$ be a Mf CY pair, and
$\varphi\colon(\mathbb{P}^{3},D)\dasharrow(Y,D_{Y})$ a volume preserving
birational map that is not biregular. By Theorem 2.5, the map $\varphi$
factors as the composition of volume preserving Sarkisov links. By Proposition
3.3, the first step of any Sarkisov factorization is a divisorial contraction
with zero-dimensional center. By Lemma 4.1, this divisorial contraction
$\sigma\colon X\rightarrow\mathbb{P}^{3}$ is the blow-up of $z$. Let
$\pi\colon X\to\mathbb{P}^{2}$ be the Mori fibration induced by the projection
from $z$, and assume that $\varphi\circ\sigma\colon X\dasharrow\mathbb{P}^{3}$
is not biregular. Let $D_{X}\subset X$ be the strict transform of $D$. Since
by assumption $D$ does not contain any line, the restriction
$\pi_{|D_{X}}\colon D_{X}\to\mathbb{P}^{2}$ is a finite morphism of degree
$2$.

Note that $X$ has Picard rank two, and the morphisms $\sigma$ and $\pi$ are
the two extremal contractions of $X$. Therefore, the Sarkisov factorization of
$\varphi$ must proceed with a volume preserving divisorial contraction
$g\colon(Z,D_{Z})\rightarrow(X,D_{X})$. Since $D_{X}\subset X$ is smooth,
Proposition 3.1 implies that the center of the divisorial contraction $g\colon
Z\rightarrow X$ is a curve $\mathcal{C}\subset D_{X}$, and $D_{Z}$ is the
strict transform of $D_{X}$ in $Z$. At the generic point of $\mathcal{C}$, the
morphism $g\colon Z\rightarrow X$ coincides with the blow-up of $\mathcal{C}$.
We denote by $F\subset Z$ the exceptional divisor of $g\colon Z\rightarrow X$.

In order to describe the next link in a Sarkisov factorization of $\varphi$,
we describe the classes of irreducible curves that are contracted by $\pi\circ
g$. They are either contained in fibers of $g_{|F}\colon
F\rightarrow\mathcal{C}$, or they are strict transforms of fibers of
$\pi\colon X\to\mathbb{P}^{2}$. Set
$\Gamma=\pi(\mathcal{C})\subset\mathbb{P}^{2}$, and denote by $d$ the degree
of the finite morphism $\pi_{|\mathcal{C}}\colon\mathcal{C}\to\Gamma$. We know
that $d\in\\{1,2\\}$ (as we just said, $\mathcal{C}\subset D_{X}$), but we
will soon see that $d=1$. For all $q\in\mathbb{P}^{2}\setminus\Gamma$,
$(\pi\circ g)^{-1}(q)\cong\mathbb{P}^{1}$. For a general point $q\in\Gamma$,
$(\pi\circ g)^{-1}(q)$ has $d+1$ rational components: the strict transform of
$\pi^{-1}(q)$, and $d$ fibers of $g_{|F}\colon F\rightarrow\mathcal{C}$,
corresponding to the $d$ points of intersection $\pi^{-1}(q)\cap\mathcal{C}$.

The next link in the Sarkisov factorization of $\varphi$ is either of type (I)
or of type (II). We show that it cannot be a Sarkisov link of type (I), as in
the following diagram:

$Z$$X^{\prime}$$X$$S$$\mathbb{P}^{2}$$\scriptstyle{r}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{g}$$\scriptstyle{\pi}$

Here $Z\dasharrow X^{\prime}$ is a sequence of Mori flips, flops and
antiflips, $\pi^{\prime}\colon X^{\prime}\to S$ is a Mori fiber space, and
$r\colon S\rightarrow\mathbb{P}^{2}$ is a divisorial contraction with center a
point $q\in\mathbb{P}^{2}$. Note that $(r\circ\pi^{\prime})^{-1}(q)$ is a
surface in $X^{\prime}$, and so its strict transform in $Z$ is also surface.
The commutativity of the above diagram then implies that $(\pi\circ
g)^{-1}(q)$ is a surface, which is impossible since the fibers of $\pi\circ g$
are $1$-dimensional.

We conclude that the next link in a Sarkisov factorization of $\varphi$ is of
type (II), as in the following diagram:

$Z$$Z^{\prime}$$X$$X^{\prime}$$\mathbb{P}^{2}$$\mathbb{P}^{2}$$\scriptstyle{g}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{g^{\prime}}$$\scriptstyle{\chi}$$\overline{\scriptstyle{g}}$

Here $\chi\colon Z\dashrightarrow Z^{\prime}$ is a sequence of Mori flips,
flops and antiflips, and $g^{\prime}\colon Z^{\prime}\to X^{\prime}$ is a
divisorial contraction. We denote by $D_{X^{\prime}}$ the strict transform of
$D_{X}$ in $X^{\prime}$. It is normal by Lemma 2.8. We will show that
$\pi^{\prime}\colon X^{\prime}\to\mathbb{P}^{2}$ is a $\mathbb{P}^{1}$-bundle
square equivalent to $\pi\colon X\to\mathbb{P}^{2}$, and that
$\overline{g}=g^{\prime}\circ\chi\circ g^{-1}\colon X\dasharrow X^{\prime}$
restricts to an isomorphism between $D_{X}$ and $D_{X^{\prime}}$.

From the description of the curves contracted by $\pi\circ g$ above, and the
fact that $\chi\colon Z\dashrightarrow Z^{\prime}$ is an isomorphism over the
complement of a finite subset of $\mathbb{P}^{2}$, we see that
$g^{\prime}\colon Z^{\prime}\to X^{\prime}$ contracts the strict transform of
$\pi^{-1}(\Gamma)$ in $Z^{\prime}$ onto a curve $\mathcal{C}^{\prime}\subset
X^{\prime}$, which is mapped to $\Gamma$ by $\pi^{\prime}$. Recall that
$\pi_{|\mathcal{C}}\colon\mathcal{C}\to\Gamma$ is a finite morphism of degree
$d\in\\{1,2\\}$. If $d=2$, then $D_{X^{\prime}}$ would be singular along
$\mathcal{C}^{\prime}$, and hence not normal. So we conclude that $d=1$, and
the general fiber of $\pi^{\prime}\colon X^{\prime}\to\mathbb{P}^{2}$ over
$\Gamma$ is irreducible, and thus isomorphic to $\mathbb{P}^{1}$. Thus,
$\pi^{\prime}\colon X^{\prime}\to\mathbb{P}^{2}$ is a $\mathbb{P}^{1}$-bundle
over the complement of a finite subset of $\mathbb{P}^{2}$. It follows from
[ARM14, Theorem 5] that $\pi^{\prime}\colon X^{\prime}\to\mathbb{P}^{2}$ is a
$\mathbb{P}^{1}$-bundle. It is clearly square equivalent to $\pi\colon
X\to\mathbb{P}^{2}$ via $\overline{g}$.

To show that the restricted map $\overline{g}_{|D_{X}}\colon D_{X}\dasharrow
D_{X^{\prime}}$ is an isomorphism, we first note that it does not contract any
curve. This follows from the commutativity of the diagram above, and the fact
that $D_{X}$ does not contain any fiber of $\pi$. By Zariski’s Main Theorem,
the birational inverse of $\overline{g}_{|D_{X}}$ is a morphism. Adjunction
yields that $K_{D_{X}}\sim 0$ and $K_{D_{X^{\prime}}}\sim 0$. Since $D_{X}$ is
smooth, in particular terminal, we conclude that
$(\overline{g}_{|D_{X}})^{-1}\colon D_{X^{\prime}}\to D_{X}$ is an
isomorphism.

The same argument as above shows that the next link in the Sarkisov
factorization of $\varphi$ cannot be of type (I). It also shows that, if it is
of type (II), then it ends with a $\mathbb{P}^{1}$-bundle
$\pi^{\prime\prime}\colon X^{\prime\prime}\to\mathbb{P}^{2}$, square
equivalent to $\pi^{\prime}\colon X^{\prime}\to\mathbb{P}^{2}$, and the
birational map $X^{\prime}\dasharrow X^{\prime\prime}$ restricts to an
isomorphism between $D_{X^{\prime}}$ and its strict transform
$D_{X^{\prime\prime}}$. So, after a finite number of Sarkisov links of type
(II), we reach a $\mathbb{P}^{1}$-bundle, which we keep denoting by
$\pi^{\prime}\colon X^{\prime}\to\mathbb{P}^{2}$, square equivalent to
$\pi\colon X\to\mathbb{P}^{2}$, and either the Sarkisov factorization of
$\varphi$ is finished, or it must continue with a link of type (III) or (IV).
Note moreover that the strict transform $D_{X^{\prime}}$ of $D_{X}$ in
$X^{\prime}$ is a smooth member of $|-K_{X^{\prime}}|$ isomorphic to $D_{X}$,
and it does not contain any fiber of $\pi^{\prime}$. In order to prove the
theorem, assuming that the Sarkisov factorization of $\varphi$ is not
finished, we must show that $X^{\prime}$ is isomorphic to the blow-up of
$\mathbb{P}^{3}$ at a point.

If the Sarkisov factorization of $\varphi$ is not finished, then the next link
starts with a birational map corresponding to an extremal ray
$R\subset\operatorname{NE}(X^{\prime})$. Let $\gamma\subset X^{\prime}$ be a
reduced and irreducible curve such that $R=\mathbb{R}_{\geq 0}[\gamma]$. We
shall show that $-K_{X^{\prime}}\cdot\gamma\geq 0$.

Suppose for a contradiction that $-K_{X^{\prime}}\cdot\gamma<0$. Then the
contraction of the extremal ray $R=\mathbb{R}_{\geq 0}[\gamma]$ is small, and
the Sarkisov link starts with an antiflip $X^{\prime}=Y^{+}\dasharrow Y^{-}$.
We will show that $Y^{-}$ has worse than terminal singularities, which is not
allowed in the definition of volume preserving Sarkisov link (Definition 2.4).

The assumption that $-K_{X^{\prime}}\cdot\gamma=D_{X^{\prime}}\cdot\gamma<0$
implies that $\gamma\subset D_{X^{\prime}}$. Recall from Paragraph 5.2 that
$\operatorname{NE}(D_{X^{\prime}})=\left\langle
e,e^{\prime}\right\rangle_{+}$, where $e$ and $e^{\prime}$ are $(-2)$-curves
in $D_{X^{\prime}}$. Therefore, either $\gamma=e$ or $\gamma=e^{\prime}$. Set
$k:=-D_{X^{\prime}}\cdot\gamma>0$. It follows from (5.1) that $k$ is even, and
hence $k\geq 2$. By Lemma 4.4, $Y^{-}$ has worse than terminal singularities,
as anticipated.

This contradiction proves that $-K_{X^{\prime}}\cdot\gamma\geq 0$. The other
extremal ray of $\operatorname{NE}(X^{\prime})$ is generated by the class of a
fiber of the $\mathbb{P}^{1}$-bundle $\pi^{\prime}\colon
X^{\prime}\to\mathbb{P}^{2}$, which has positive intersection with
$-K_{X^{\prime}}$. Therefore, if $-K_{X^{\prime}}\cdot\gamma>0$ then
$X^{\prime}$ is Fano. If $-K_{X^{\prime}}\cdot\gamma=0$, then the contraction
of the extremal ray $R=\mathbb{R}_{\geq 0}[\gamma]$, which is induced by the
linear system $\big{|}-mK_{X^{\prime}}\big{|}$ for $m\gg 0$, is small —
otherwise after the contraction we get a variety with strictly canonical
singularities, which is not allowed — and the Sarkisov link starts with a Mori
flop. In this case $X^{\prime}$ is weak Fano (i.e., $-K_{X^{\prime}}$ is nef
and big).

Theorem 4.8 and the two lists that accompany it show the rank $2$ vector
bundles $\mathcal{E}$ on $\mathbb{P}^{2}$ with
$c_{1}(\mathcal{E})\in\\{0,-1\\}$ such that $\mathbb{P}(\mathcal{E})$ is Fano
or weak Fano. Below we follow the conventions on vector bundles summarised in
section 4.3.

Let $\mathcal{E}$ be the rank two vector bundle on $\mathbb{P}^{2}$ with
$c_{1}(\mathcal{E})\in\\{0,-1\\}$ such that
$X^{\prime}\cong\mathbb{P}(\mathcal{E})$, and denote by $\pi^{\prime}\colon
X^{\prime}\to\mathbb{P}^{2}$ the natural projection. In order to prove the
theorem, we must show that
$\mathcal{E}\cong\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(-1)$.
To do so, we compare the lattice $\operatorname{Pic}(D_{X^{\prime}})$ with the
sublattice obtained as the image of the restriction homomorphism

$r\colon\ \operatorname{Pic}(X^{\prime})\ \to\
\operatorname{Pic}(D_{X^{\prime}}).$

The Picard group $\operatorname{Pic}({X^{\prime}})$ is generated by
$L^{\prime}=\big{[}(\pi^{\prime})^{*}\big{(}\mathcal{O}_{\mathbb{P}^{2}}(1)\big{)}\big{]}$
and $\xi=c_{1}\big{(}\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)\big{)}$. Working
in $A^{\bullet}(X^{\prime})$ and using that $D_{X^{\prime}}\ \sim\
-K_{X^{\prime}}\ \sim 2\xi+\ (-c_{1}+3)L^{\prime}$ (by Lemma 4.10), we compute
the intersection matrix of
$r\Big{(}\operatorname{Pic}\big{(}\mathbb{P}(\mathcal{E})\big{)}\Big{)}\subset\operatorname{Pic}(D_{X^{\prime}})$
in the basis $r(L^{\prime}),r(\xi)$:

(5.2) $\left(\begin{array}[]{cc}L^{\prime 2}\cdot D_{X^{\prime}}&\xi\cdot
L^{\prime}\cdot D_{X^{\prime}}\\\ \xi\cdot L^{\prime}\cdot
D_{X^{\prime}}&\xi^{2}\cdot
D_{X^{\prime}}\end{array}\right)=\left(\begin{array}[]{cc}2&c_{1}+3\\\
c_{1}+3&c_{1}^{2}+3c_{1}-2c_{2}\end{array}\right).$

Direct inspection shows that this matrix has rank two for all vector bundles
in the two lists, except for the one in (8), namely
$\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(-2)$. On
the other hand, we cannot have
$X^{\prime}\cong\mathbb{P}\big{(}\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(-2)\big{)}$
because the anti-canonical contraction of
$\mathbb{P}\big{(}\mathcal{O}_{\mathbb{P}^{2}}(1)\oplus\mathcal{O}_{\mathbb{P}^{2}}(-2)\big{)}$
is divisorial and not small. In all other cases, the restriction homomorphism
$r\colon\operatorname{Pic}(X^{\prime})\to\operatorname{Pic}(D_{X^{\prime}})$
is injective with finite cokernel. We claim that it is also surjective. Pick
$\alpha\in\operatorname{Pic}(D_{X^{\prime}})$. By Proposition 5.1, the first
Chern class $c_{1}(\alpha)\in H^{2}(D_{X^{\prime}},\mathbb{Z})$ is in the
image of $H^{2}(X^{\prime},\mathbb{Z})$. By the Lefschetz theorem on $(1,1)$
classes, $H^{2}(X^{\prime},\mathbb{Z})=\operatorname{Pic}(X^{\prime})$, hence
there is a line bundle $\widetilde{\alpha}$ on $X^{\prime}$ such that

$c_{1}\left(r(\widetilde{\alpha})\right)=c_{1}(\alpha).$

But $c_{1}\colon\operatorname{Pic}(D_{X^{\prime}})\to
H^{2}(D_{X^{\prime}},\mathbb{Z})$ is injective, hence
$r(\widetilde{\alpha})=\alpha$. It follows that the matrices (5.1) and (5.2)
must have the same determinant. One checks easily that this only happens in
case (1), i.e., when
$X^{\prime}\cong\mathbb{P}\big{(}\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(-1)\big{)}$
is the blow-up of $\mathbb{P}^{3}$ at a point. Denote by $D^{\prime}$ the
image of $D_{X^{\prime}}$ in $\mathbb{P}^{3}$. Since the blow-up
$(X^{\prime},D_{X^{\prime}})\rightarrow(\mathbb{P}^{3},D^{\prime})$ is volume
preserving, its restriction to $D_{X^{\prime}}$ contracts one of the two
$(-2)$-curves $e$ or $e^{\prime}$. So we have $D\cong D^{\prime}$, and
$(\mathbb{P}^{3},D)\cong(\mathbb{P}^{3},D^{\prime})$. ∎

Next, we prove the second conclusion of Theorem B, that is, we describe the
group $\operatorname{Bir}(\mathbb{P}^{3},D)$ of volume preserving birational
self-maps of $(\mathbb{P}^{3},D)$.

###### Proof of Theorem B(2).

By Proposition 2.6, since $(\mathbb{P}^{3},D)$ is canonical, there is a
restriction homomorphism

(5.3) $r\colon\operatorname{Bir}(X,D)\ \to\ \operatorname{Bir}(D).$

We use the same notation as above: $\sigma\colon X\rightarrow\mathbb{P}^{3}$
denotes the blow-up of the singular point $z\in D$, $D_{X}\subset X$ the
strict transform of $D$, and $\pi\colon X\to\mathbb{P}^{2}$ the fibration
induced by the projection from $z$. Recall from Paragraph 5.2 that the
restriction $\pi_{|D_{X}}\colon D_{X}\to\mathbb{P}^{2}$ is a double cover, and
denote by $\tau\colon D_{X}\to D_{X}$ the associated involution. Lemma 5.3
states that

$\operatorname{Bir}(D)=\operatorname{Aut}(D_{X})\cong\operatorname{Aut}(\mathbb{P}^{3},D)\rtimes\langle\tau\rangle.$

After a change of coordinates, we can write the equation of $D$ in
$\mathbb{P}^{3}$ as

$x_{3}^{2}A(x_{0},x_{1},x_{2})+x_{3}B(x_{0},x_{1},x_{2})+C(x_{0},x_{1},x_{2})=0,$

where $A=A(x_{0},x_{1},x_{2})$, $B=B(x_{0},x_{1},x_{2})$ and
$C=C(x_{0},x_{1},x_{2})$ are homogeneous of degree $2$, $3$ and $4$,
respectively. The singular point of $D$ has coordinates $z=[0:0:0:1]$.

######  Claim 5.5.

The restriction homomorphism (5.3) is surjective and admits a splitting

$\operatorname{Bir}(\mathbb{P}^{3},D)\ \xrightarrow{\curvearrowleft}\
\operatorname{Bir}(D)=\operatorname{Aut}(\mathbb{P}^{3},D)\rtimes\langle\tau\rangle.$

###### Proof of Claim 5.5.

To see that $r\colon\operatorname{Bir}(\mathbb{P}^{3},D)\ \to\
\operatorname{Bir}(D)=\operatorname{Aut}(\mathbb{P}^{3},D)\rtimes\langle\tau\rangle$
is surjective notice that the birational involution

(5.4) $\varphi(x_{0}:x_{1}:x_{2}:x_{3})=(Ax_{0}:Ax_{1}:Ax_{2}:-Ax_{3}-B),$

restricts to the nontrivial birational involution $\tau$.

Consider the splitting
$\operatorname{Bir}(D)\to\operatorname{Bir}(\mathbb{P}^{3},D)$ of $r$ that is
canonical on the normal subgroup $\operatorname{Aut}(\mathbb{P}^{3},D)$, and
sends $\tau$ to $\varphi$. To show that this is well-defined, we must check
that, for any automorphism $h\in\operatorname{Aut}(\mathbb{P}^{3},D)$, we have
$\varphi\circ h\circ\varphi\in\operatorname{Aut}(\mathbb{P}^{3},D)$. In order
to prove this, we first describe a volume preserving Sarkisov factorization of
$\varphi$:

$\mathbb{P}^{3}$$\mathbb{P}^{3}$$\scriptstyle{\sigma}$$Z$$Z^{\prime}$$X$$X^{\prime}$$X$$\mathbb{P}^{2}$$\mathbb{P}^{2}$$\mathbb{P}^{2}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\scriptstyle{\pi}$$\scriptstyle{\delta}$$\scriptstyle{\gamma}$$\scriptstyle{\sigma}$$\scriptstyle{\varphi}$

The factorization starts with the blow-up $\sigma\colon
X\rightarrow\mathbb{P}^{3}$ of $z$. Denote by $E\subset X$ its exceptional
divisor. The base locus of $\varphi\circ\sigma$ contains the curve $e=E\cap
D_{X}$. Note that $\pi\colon X\to\mathbb{P}^{2}$ maps $e$ isomorphically onto
the conic $\big{(}A=0\big{)}\subset\mathbb{P}^{2}$, and the cylinder
$\pi^{-1}\big{(}A=0\big{)}\subset X$ is precisely the strict transform of the
tangent cone of $D\subset\mathbb{P}^{3}$ at $z$. The next link in the Sarkisov
factorization is the composition $\beta\circ\alpha^{-1}$, where $\alpha$ and
$\beta$ are described as follows. The morphism $\alpha:Z\to X$ is the blow-up
of $e=E\cap D_{X}$. Denote by $F\subset Z$ its exceptional divisor. The base
locus of $\varphi\circ\sigma\circ\alpha$ contains the curve $e_{F}=F\cap
D_{Z}$. The morphism $\beta:Z\to
X^{\prime}\cong\mathbb{P}\big{(}\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(3)\big{)}$
contracts the rulings of the strict transform of the tangent cone of
$D\subset\mathbb{P}^{3}$ at $z$, and maps $F\subset Z$ isomorphically onto the
cylinder $F^{\prime}=(\pi^{\prime})^{*}\big{(}A=0\big{)}\subset X^{\prime}$.
The base locus of $\varphi\circ\sigma\circ\alpha\circ\beta^{-1}$ consists of
the curve $e_{F^{\prime}}=\beta(e_{F})\subset F^{\prime}\cap D_{X^{\prime}}$,
which is mapped by $\pi^{\prime}$ isomorphically onto the conic
$\big{(}A=0\big{)}\subset\mathbb{P}^{2}$. The next link in the Sarkisov
factorization is the composition $\delta\circ\gamma^{-1}$, where
$\gamma:Z^{\prime}\to X^{\prime}$ is the blow-up of $e_{F^{\prime}}$, and
$\delta:Z^{\prime}\to X$ contracts the rulings of the strict transform of the
cylinder $F^{\prime}$ on $Z^{\prime}$. The factorization then ends with the
blow-up $\sigma\colon X\rightarrow\mathbb{P}^{3}$ of $z$.

Any automorphism $h\in\operatorname{Aut}(\mathbb{P}^{3},D)$ fixes $z$ and
stabilizes the tangent cone of $D$ at $z$. So it lifts to an automorphism of
$X$ that stabilizes $E$, $D_{X}$, and the rulings over $\big{(}A=0\big{)}$.
Since $\alpha$ blows-up $E\cap D_{X}$, $h$ lifts to an automorphism of $Z$. On
the other hand, $\beta$ contracts the ruling of the strict transform of
$\pi^{-1}\big{(}A=0\big{)}$, and thus $h$ descends to an automorphism of
$X^{\prime}$ that stabilizes $D_{X^{\prime}}$, the ruling over
$\big{(}A=0\big{)}$, and each of the components of
$D_{X^{\prime}}\cap(\pi^{\prime})^{*}\big{(}A=0\big{)}$. The same argument
shows that $h$ lifts to $Z^{\prime}$, and then descends to $X$, always
stabilizing the strict transform of $E$. This shows that the birational map
$\varphi\circ h$ has the same Sarkisov decomposition of $\varphi$, and
therefore $\varphi\circ h\circ\varphi$ is biregular. ∎

It follows from Claim 5.5 that there is a split exact sequence:

(5.5) $1\to\mathbb{G}\to\operatorname{Bir}(\mathbb{P}^{3},D)\
\xrightarrow{\curvearrowleft}\operatorname{Bir}(D)\to 1,$

where $\mathbb{G}$ is the group of birational self maps of $\mathbb{P}^{3}$
fixing $D$ pointwise. We saw in the proof of Theorem B(1) that any
$\psi\in\operatorname{Bir}(\mathbb{P}^{3},D)$ preserves the star of lines
through $z$. Therefore, we can identify $\mathbb{G}$ with the group of
birational self-maps of $X$ over $\mathbb{P}^{2}$ fixing $D_{X}$ pointwise.

We view $X$ as a model of $\mathbb{P}^{1}$ over $\mathbb{C}(x,y)$ with
projective coordinates $(u:v)$. Setting $a(x,y)=A(1,x,y)$, $b(x,y)=B(1,x,y)$
and $c(x,y)=C(1,x,y)$, $\mathbb{G}$ becomes the identity component of the
subgroup $G_{Q}$ of $PGL\big{(}2,\mathbb{C}(x,y)\big{)}$ of projective
transformations preserving the quadratic form

$Q(u,v)\ =\ a(x,y)u^{2}+b(x,y)uv+c(x,y)v^{2}$

up to scaling.

######  Lemma 5.6.

Let $Q(u,v)=Au^{2}+Buv+Cv^{2}$ be a quadratic form with coefficients in a
field $K$, and let $G_{Q}$ be the subgroup of $PGL(2,K)$ of projective
transformations preserving $Q$ up to scalar:

$G_{Q}:=\left\\{\phi=\left(\begin{array}[]{cc}\alpha&\beta\\\
\gamma&\delta\end{array}\right)\in PGL(2,K)\>\Big{|}\>Q(\phi(u,v))=\lambda
Q(u,v),\>\lambda\in K\setminus\\{0\\}\right\\}.$

Then $G_{Q}$ has two irreducible components, given by

(5.6) $\left\\{\begin{array}[]{l}\alpha=-\frac{B}{A}\gamma+\delta,\\\
\beta=-\frac{C}{A}\gamma\end{array}\right.\ \ \ and\ \ \ \
\left\\{\begin{array}[]{l}\alpha=-\delta,\\\
\beta=\frac{C}{A}\gamma-\frac{B}{A}\delta.\end{array}\right.$

###### Proof.

We have that

$Q(\phi(u,v))=u^{2}(\alpha^{2}A+\alpha\gamma B+\gamma^{2}C)+uv(2\alpha\beta
A+\alpha\delta B+\beta\gamma B+2\gamma\delta C)+v^{2}(\beta^{2}A+\beta\delta
B+\delta^{2}C).$

Therefore, $\phi=\left(\begin{array}[]{cc}\alpha&\beta\\\
\gamma&\delta\end{array}\right)\in G_{Q}$ if and only if

$\operatorname{rank}\left(\begin{array}[]{ccc}A&B&C\\\
\alpha^{2}A+\alpha\gamma B+\gamma^{2}C&2\alpha\beta A+\alpha\delta
B+\beta\gamma B+2\gamma\delta C&\beta^{2}A+\beta\delta
B+\delta^{2}C\end{array}\right)\leq 1,$

or, equivalently, if and only if

(5.7) $\left\\{\begin{array}[]{l}\alpha^{2}AC+\alpha\gamma
BC+\gamma^{2}C^{2}-\beta^{2}A^{2}-\beta\delta AB-\delta^{2}AC=0,\\\
\alpha^{2}AB+\alpha\gamma B^{2}+\gamma^{2}BC-2\alpha\beta A^{2}-\alpha\delta
AB-\beta\gamma AB-2\gamma\delta AC=0.\end{array}\right.$

If the $4$-tuple $(\alpha,\beta,\gamma,\delta)\in\mathbb{C}^{4}$ satisfies the
pair of equations (5.7), then it satisfies one of the following pairs of
equations:

$\left\\{\begin{array}[]{l}\alpha=-\delta,\\\
\beta=\frac{C}{A}\gamma-\frac{B}{A}\delta,\end{array}\right.\left\\{\begin{array}[]{l}\alpha=-\frac{B}{A}\gamma+\delta,\\\
\beta=-\frac{C}{A}\gamma,\end{array}\right.\left\\{\begin{array}[]{l}\alpha=\frac{-B+\varepsilon}{2A}\gamma,\\\
\beta=\frac{B(B-\varepsilon)-4AC}{2A\varepsilon}\delta,\end{array}\right.\left\\{\begin{array}[]{l}\alpha=\frac{-B-\varepsilon}{2A}\gamma,\\\
\beta=-\frac{B(B+\varepsilon)-4AC}{2A\varepsilon}\delta,\end{array}\right.$

where $\varepsilon\in\mathbb{C}$ is such that $\varepsilon^{2}=B^{2}-4AC$. On
the other hand, if $(\alpha,\beta,\gamma,\delta)\in\mathbb{C}^{4}$ satisfies
the third or the fourth pair of equations, then
$\det\left(\begin{array}[]{cc}\alpha&\beta\\\
\gamma&\delta\end{array}\right)=0$. Therefore $G_{Q}$ has two irreducible
components, given by (5.6). ∎

Lemma 5.6 shows that $G_{Q}$ has two irreducible components over
$\mathbb{C}(x,y)$:

(5.8)
$G_{Q}^{1}=\left\\{\left(\begin{array}[]{cc}\left(-\frac{b(x,y)}{a(x,y)}\alpha+\beta\right)&\left(-\frac{c(x,y)}{a(x,y)}\alpha\right)\\\
\alpha&\beta\end{array}\right)\right\\},\ \
G_{Q}^{2}=\left\\{\left(\begin{array}[]{cc}-\beta&\left(\frac{c(x,y)}{a(x,y)}\alpha-\frac{b(x,y)}{a(x,y)}\beta\right)\\\
\alpha&\beta\end{array}\right)\right\\}.$

The component $G_{Q}^{1}$ contains the identity, and is precisely the group
$\mathbb{G}$ of (5.5). In order to describe $\mathbb{G}$ as a form of
$\mathbb{G}_{m}$, set $w\ =\ 2a(x,y)u+b(x,y)v$, and compute

$4a(x,y)Q(u,v)\ =\ w^{2}-\delta(x,y)v^{2},$

where $\delta(x,y)=\Delta(1,x,y)$. Applying Lemma 5.6 with the new projective
coordinates $(v:w)$, $\mathbb{G}$ can be presented as the subgroup of
$PGL\big{(}2,\mathbb{C}(x,y)\big{)}$ of elements of the form

$\left(\begin{array}[]{cc}U&\delta(x,y)V\\\ V&U\end{array}\right)$

where $U,V\in\mathbb{C}(x,y)$ are such that $U^{2}-\delta(x,y)V^{2}=1$. This
is the form of $\mathbb{G}_{m}$ described in [PS94, Chapter 2, Example 2.3.2
(c)]. ∎

######  Remark 5.7.

We can write down all the elements of $\operatorname{Bir}(\mathbb{P}^{3},D)$
explicitly. It follows from (5.8) above that they can be written in one of the
following two forms:

$\varphi^{1}\colon\ (x_{0}:x_{1}:x_{2}:x_{3})\ \mapsto\
\big{(}A(Fx_{3}+G)x_{0}:A(Fx_{3}+G)x_{1}:A(Fx_{3}+G)x_{2}:(AG-
BF)x_{3}-CF\big{)}$

where either $F=0$ and $\deg(G)=0$, or $F,G\in\mathbb{C}[x_{0},x_{1},x_{2}]$
are homogeneous with $\deg(G)=\deg(F)+1$;

$\varphi^{2}\colon\ (x_{0}:x_{1}:x_{2}:x_{3})\ \mapsto\
\big{(}A(Fx_{3}+G)x_{0}:A(Fx_{3}+G)x_{1}:A(Fx_{3}+G)x_{2}:-AGx_{3}+CF-
BG\big{)}$

where either $F=0$ and $\deg(G)=0$, or $F,G\in\mathbb{C}[x_{0},x_{1},x_{2}]$
are homogeneous with $\deg(G)=\deg(F)+1$.

##  6\. Proof of Theorem C

In this section, we determine the pliability of Mf CY pairs of the form
$(\mathbb{P}^{3},D)$, where $D$ is a quartic surface having exactly one
singular point $z\in D$ of type $A_{2}$, and such that
$\operatorname{Cl}(D)\cong\mathbb{Z}\cdot\mathcal{O}_{D}(1)$ is generated by
the class of a hyperplane section. This last condition implies in particular
that $D$ does not contain lines. After a change of coordinates, we may assume
that the singular point is $z=[0:0:0:1]$, and write the equation of $D$ as

$D=\Bigl{(}x_{0}x_{1}x_{3}^{2}+Bx_{3}+C=0\Bigr{)}\subset\mathbb{P}^{3},$

where $B=B(x_{0},x_{1},x_{2})$ and $C=C(x_{0},x_{1},x_{2})$ are homogeneous
polynomials of degree $3$ and $4$, respectively.

At the end of Section 2, we constructed volume preserving Sarkisov links
between the Mf CY pairs from Table 1:

obj. $1$$\scriptstyle{\epsilon_{a},\epsilon_{b}}$obj.
$2$$\scriptstyle{\sigma}$obj. $2^{a}$,
$2^{b}$$\scriptstyle{\nu^{a},\nu^{b}}$$\scriptstyle{\chi^{a},\chi^{b}}$obj.
$3^{a}$,
$3^{b}$$\scriptstyle{\phi^{a},\phi^{b}}$$\scriptstyle{\psi^{a},\widetilde{\psi}^{b}}$obj.
$4$obj. $5^{a}$

In order to prove Theorem C, we will show that these are all the Sarkisov
links from these Mf CY pairs, except for chains of square equivalent Sarkisov
links from objects 2, $2^{a}$ and $2^{b}$.

###### Proof of Theorem C.

Let $(Y,D_{Y})/T$ be a Mf CY pair, and

$\Phi\colon(\mathbb{P}^{3},D)\dasharrow(Y,D_{Y})$

a volume preserving birational map. The goal is to show that $(Y,D_{Y})\to T$
is square equivalent to one of the objects in the conclusion of Theorem C:

  1. (1)

$(\mathbb{P}^{3},D)$ (object $1$);

  2. (2)

$(X,D_{X})=\left(\mathbb{F}^{3}_{1},D_{\binom{2}{2}}\right)$ (object $2$);

  3. (3)

$(\mathbb{P}(1^{3},2),D^{a}_{5})$ or $(\mathbb{P}(1^{3},2),D^{b}_{5})$ (object
$3^{a}$ or $3^{b}$);

  4. (4)

a member of the $3$-parameter family $\big{\\{}(X_{4},D_{3,4})\big{\\}}$, with
$X_{4}\subset\mathbb{P}(1^{3},2^{2})$ (object $4$);

  5. (5)

a member of the $6$-parameter family $\big{\\{}(X_{4},D_{2,4})\big{\\}}$, with
$X_{4}\subset\mathbb{P}(1^{4},2)$ (object $5^{a}$).

By the Sarkisov program for volume preserving birational maps of Mf CY pairs
(Theorem 2.5), there is a sequence of Mf CY pairs

$(\mathbb{P}^{3},D)/\operatorname{Spec}\mathbb{C}=(X_{0},D_{0})/S_{0}\ ,\
(X_{1},D_{1})/S_{1}\ ,\ \dots\ ,\ (X_{n},D_{n})/S_{n}=(Y,D_{Y})/T\ ,$

and volume preserving Sarkisov links

$\Phi_{i}\colon(X_{i-1},D_{i-1})/S_{i-1}\dasharrow(X_{i},D_{i})/S_{i},$

$i=1,\dots,n$, such that $\Phi=\Phi_{n}\circ\cdots\circ\Phi_{1}$.

We prove by increasing induction on $i$ that each $(X_{i},D_{i})/S_{i}$ is
either isomorphic to one of the objects $1$, $3^{a}$, $3^{b}$, $4$, and
$5^{a}$, or it is obtained from $(X,D_{X})/{\mathbb{P}^{2}}$ (object $2$)
after finitely many volume preserving Sarkisov links of type (II). In the
latter case, it is in particular square equivalent to object $2$. The base
case $i=0$ is clear since
$(X_{0},D_{0})/S_{0}=(\mathbb{P}^{3},D)/\operatorname{Spec}\mathbb{C}$ is
object $1$.

For $i>0$, we assume by induction that $(X_{i-1},D_{i-1})/S_{i-1}$ is either
isomorphic to one of the objects 1, $3^{a}$, $3^{b}$, $4$, and $5^{a}$, or is
obtained from $(X,D_{X})/{\mathbb{P}^{2}}$ after finitely many volume
preserving Sarkisov links of type (II). We shall prove that the same holds for
$(X_{i},D_{i})/S_{i}$. We discuss several cases depending on the nature of
$(X_{i-1},D_{i-1})/S_{i-1}$.

#### Case 1.

Suppose that
$(X_{i-1},D_{i-1})/S_{i-1}\cong(\mathbb{P}^{3},D)/\operatorname{Spec}\mathbb{C}$
(object $1$). We prove in Lemma 6.1 below that the only volume preserving
Sarkisov links

$(\mathbb{P}^{3},D)/\operatorname{Spec}\mathbb{C}\dasharrow(X^{\dagger},D^{\dagger})/S^{\dagger}$

are the maps $\sigma^{-1},\epsilon_{a},\epsilon_{b}$ to objects $2$, $3^{a}$
and $3^{b}$, respectively. It follows that $(X_{i},D_{i})/S_{i}$ is isomorphic
to one of the objects $2$, $3^{a}$ or $3^{b}$.

#### Case 2.

Suppose that $(X_{i-1},D_{i-1})/S_{i-1}$ is obtained from
$(X,D_{X})/{\mathbb{P}^{2}}$ after finitely many volume preserving Sarkisov
links of type (II). We show in Lemma 6.4 below that one of the following
holds:

  1. (a)

$\Phi_{i}$ is a Sarkisov link of type (II);

  2. (b)

$(X_{i-1},D_{i-1})/S_{i-1}$ is isomorphic to object $2$, and
$\Phi_{i}=\sigma$;

  3. (c)

$(X_{i-1},D_{i-1})/S_{i-1}$ is isomorphic to object $2^{a}$, and
$\Phi_{i}=\chi^{a}$;

  4. (d)

$(X_{i-1},D_{i-1})/S_{i-1}$ is isomorphic to object $2^{b}$, and
$\Phi_{i}=\chi^{b}$.

In case (a), it follows that $(X_{i},D_{i})/S_{i}$ is obtained from
$(X,D_{X})$ after finitely many volume preserving Sarkisov links of type (II).
In cases (b), (c) and (d), it follows that $(X_{i},D_{i})/S_{i}$ is isomorphic
to object $1$, $3^{a}$ and $3^{b}$, respectively.

#### Case 3.

Suppose that $(X_{i-1},D_{i-1})/S_{i-1}$ is isomorphic to object $3^{a}$ (case
$3^{b}$ is similar). We prove in Lemma 6.5 below that the only volume
preserving Sarkisov links

$(\mathbb{P}(1^{3},2),D^{a}_{5})/\operatorname{Spec}\mathbb{C}\dasharrow(X^{\dagger},D^{\dagger})/S^{\dagger}$

are the maps $\epsilon_{a}^{-1},(\chi^{a})^{-1},\phi^{a},\psi^{a}$ to objects
$1$, $2^{a}$, $4$ and $5^{a}$, respectively.

#### Case 4.

Suppose that $(X_{i-1},D_{i-1})/S_{i-1}$ is isomorphic to object $4$. We prove
in Lemma 6.6 below that the only volume preserving Sarkisov links

$(X_{4},D_{3,4})/\operatorname{Spec}\mathbb{C}\dasharrow(X^{\dagger},D^{\dagger})/S^{\dagger}$

are the maps $(\phi^{a})^{-1}$, $(\phi^{b})^{-1}$, and thus
$(X_{i},D_{i})/S_{i}$ is isomorphic to object $3^{a}$ or $3^{b}$.

#### Case 5.

Suppose that $(X_{i-1},D_{i-1})/S_{i-1}$ is isomorphic to object $5^{a}$. We
prove in Lemma 6.7 below that the only volume preserving Sarkisov links

$(X_{4},D_{3,4})/\operatorname{Spec}\mathbb{C}\dasharrow(X^{\dagger},D^{\dagger})/S^{\dagger}$

are the maps $(\psi^{a})^{-1}$ and $(\widetilde{\psi}^{b})^{-1}$, and hence
$(X_{i},D_{i})/S_{i}$ is isomorphic to object $3^{a}$ or $3^{b}$. ∎

######  Lemma 6.1.

The only volume preserving Sarkisov links

$\Psi\colon(\mathbb{P}^{3},D)/\operatorname{Spec}\mathbb{C}\dasharrow(X^{\dagger},D^{\dagger})/S^{\dagger}$

are the maps $\sigma^{-1},\epsilon_{a}$, and $\epsilon_{b}$ described in
Examples 2.9 and 2.11.

###### Proof.

Since $\mathbb{P}^{3}$ has Picard rank $1$, the link $\Psi$ begins with a
volume preserving divisorial contraction
$f\colon(Z,D_{Z})\to(\mathbb{P}^{3},D)$. By Proposition 3.3 and Lemma 4.1, the
contraction $f$ is either the usual blow-up of the singular point $z$, or the
weighted blow-up at $z$ with weights $(2,1,1)$ or $(1,2,1)$ with respect to
the affine coordinates $x_{0}$, $x_{1}$ and $x_{2}$ in the affine chart
$(x_{3}=1)$. If $f=\sigma$ is the blow-up of $z$, then $\Psi=\sigma^{-1}$. If
$f$ is the weighted blow-up at $z$ with weights $(2,1,1)$ (respectively
$(1,2,1)$), then $\Psi$ is the composition of $f$ with the contraction of the
strict transform of the divisor $(x_{0}=0)$ (respectively $(x_{1}=0)$), as
described in Example 2.11. Hence, $\Psi=\epsilon_{b}$ and
$(X^{\dagger},D^{\dagger})/S^{\dagger}\cong(\mathbb{P}(1^{3},2),D^{b}_{5})$
(respectively $\Psi=\epsilon_{a}$ and
$(X^{\dagger},D^{\dagger})/S^{\dagger}\cong(\mathbb{P}(1^{3},2),D^{a}_{5})$).
∎

As before, we denote by $\sigma\colon X\to\mathbb{P}^{3}$ the blow-up of the
singular point $z\in D$, by $D_{X}$ the (smooth) strict transform of $D$ in
$X$, and by $\pi\colon X\to\mathbb{P}^{2}$ the fibration induced by the
projection from $z$. In order to determine all the volume preserving Sarkisov
links from $(X,D_{X})/{\mathbb{P}^{2}}$, we need a good understanding of the
geometry of the smooth K3 surface $D_{X}$.

######  6.2The geometry of $D_{X}$.

Denote by $E\cong\mathbb{P}^{2}$ the exceptional divisor of the blow-up
$\sigma:X\to\mathbb{P}^{3}$. The intersection of $D_{X}$ with $E$ is the union
of two $(-2)$-curves $e_{0}$ and $e_{1}$. These curves are mapped
isomorphically via $\pi$ to the lines $(x_{0}=0)$ and $(x_{1}=0)$ in
$\mathbb{P}^{2}$, respectively. Denote by $h$ the pull-back of a general
hyperplane under $\sigma_{|D_{X}}\colon D_{X}\to\mathbb{P}^{3}$. Then
$\operatorname{Pic}(D_{X})=\mathbb{Z}[h]\oplus\mathbb{Z}[e_{0}]\oplus\mathbb{Z}[e_{1}]$,
and the intersection matrix of $\operatorname{Pic}(D_{X})$ with respect to the
basis $\big{(}[h],[e_{0}],[e_{1}]\big{)}$ is

$\left(\begin{array}[]{ccc}4&0&0\\\ 0&-2&1\\\ 0&1&-2\end{array}\right).$

By assumption, $D$ does not contain lines. Hence, the morphism
$\pi_{|D_{X}}\colon D_{X}\to\mathbb{P}^{2}$ is finite of degree $2$. Set
$\alpha=(\pi_{|D_{X}})^{*}\mathcal{O}_{\mathbb{P}^{2}}(1)$, and denote by
$\tau\colon D_{X}\to D_{X}$ the involution associated to $\pi_{|D_{X}}$. Then
$\tau\colon D_{X}\to D_{X}$ maps the $(-2)$-curves $e_{0}$ and $e_{1}$ to
other $(-2)$-curves $e^{\prime}_{0}$ and $e^{\prime}_{1}$, respectively. Note
that

$\alpha\ =\ (\pi_{|D_{X}})^{*}\mathcal{O}_{\mathbb{P}^{2}}(1)\ \sim\
h-e_{0}-e_{1}\ \sim\ e_{0}+e^{\prime}_{0}\ \sim\ e_{1}+e^{\prime}_{1}\ .$

Thus $e^{\prime}_{0}\sim h-2e_{0}-e_{1}$, and $e^{\prime}_{1}\sim
h-2e_{1}-e_{0}$. The intersection matrix of $\operatorname{Pic}(D_{X})$ with
respect to the basis $\big{(}[\alpha],[e_{0}],[e_{1}]\big{)}$ is

$\left(\begin{array}[]{ccc}2&1&1\\\ 1&-2&1\\\ 1&1&-2\end{array}\right).$

The following intersection numbers will be useful later on

$\left\\{\begin{array}[]{l}\alpha\cdot e_{0}=\alpha\cdot e_{1}=\alpha\cdot
e^{\prime}_{0}=\alpha\cdot e^{\prime}_{1}=1,\\\ e_{0}\cdot
e^{\prime}_{1}=e^{\prime}_{0}\cdot e_{1}=0,\\\ e_{0}\cdot
e_{0}^{\prime}=e_{1}\cdot e_{1}^{\prime}=3.\end{array}\right.$

The $(-2)$-curves $e_{0}$, $e_{1}$, $e^{\prime}_{0}$ and $e^{\prime}_{1}$
generate extremal rays of $\operatorname{NE}(D_{X})$. Next we show that these
are all:

(6.1) $\operatorname{NE}(D_{X})\ =\
\big{\langle}[e_{0}],[e_{1}],[e^{\prime}_{0}],[e^{\prime}_{1}]\big{\rangle}_{+}.$

Indeed, let $C\subset D_{X}$ be an irreducible curve different from $e_{0}$,
$e_{1}$, $e^{\prime}_{0}$ and $e^{\prime}_{1}$, and write $C\sim dh-
m_{0}e_{0}-m_{1}e_{1}$. By intersecting $C$ with $h$, $e_{0}$, $e_{1}$,
$e^{\prime}_{0}$ and $e^{\prime}_{1}$, we get that

$\left\\{\begin{array}[]{l}d>0,\\\ 0\leq m_{0}\leq\frac{4}{3}d,\\\ 0\leq
m_{1}\leq\frac{4}{3}d.\end{array}\right.$

Therefore, we may write

$C\ \equiv\ \frac{d}{2}{e^{\prime}}_{0}\ +\ \frac{d}{2}{e^{\prime}}_{1}\ +\
\left(\frac{3}{2}d-m_{0}\right)e_{0}\ +\
\left(\frac{3}{2}d-m_{1}\right)e_{1},$

with $\frac{3}{2}d-m_{0}>0$ and $\frac{3}{2}d-m_{1}>0$. This gives (6.1).

######  Remark 6.3.

Fix homogeneous coordinates $x_{0},x_{1},x_{2},x_{3},x$ on
$X\cong\mathbb{F}_{1}^{3}$ with weights: