# The Rigid Pham-Brieskorn Threefolds

Michael Chitayat and Adrien Dubouloz

(January 2023)

###### Abstract

We show that a $3$-dimensional Pham-Brieskorn hypersurface
$\\{X_{0}^{a_{0}}+X_{1}^{a_{1}}+X_{2}^{a_{2}}+X_{3}^{a_{3}}=0\\}$ in
$\mathbb{A}^{4}$ such that $\min\\{a_{0},a_{1},a_{2},a_{3}\\}\geq 2$ and at
most one element $i$ of $\\{0,1,2,3\\}$ satisfies $a_{i}=2$ does not admit a
non-trivial action of the additive group $\mathbb{G}_{a}$.

## Introduction

An affine variety $X$ defined over a field ${\rm\bf k}$ of characteristic zero
is called rigid if it does not admit a non-trivial action of the additive
group $\mathbb{G}_{a,{\rm\bf k}}$. Equivalently, if the coordinate ring
$B=\Gamma(X,\mathcal{O}_{X})$ does not admit a non-zero locally nilpotent
${\rm\bf k}$-derivation, then $B$ is called a rigid ring. In this article, we
are interested in the study of the rigidity of a class of varieties called
affine Pham-Brieskorn hypersurfaces. These varieties, denoted
$X_{a_{0},\ldots,a_{n}}=\mathrm{Spec}(B_{a_{0},\ldots,a_{n}})$ are defined by
equations of the form $X_{0}^{a_{0}}+\cdots+X_{n}^{a_{n}}=0$ in
$\mathbb{A}^{n+1}_{{\rm\bf k}}$, where $n\geq 2$ and $a_{0},\dots,a_{n}$ are
positive integers. If $a_{i}=1$ for some $i\in\\{0,\ldots,n\\}$ then
$X_{a_{0},\ldots,a_{n}}$ is isomorphic to $\mathbb{A}^{n}_{{\rm\bf k}}$ which
is clearly not rigid. If ${\rm\bf k}$ contains $\mathbb{Q}(i)$ and two of the
$a_{i}$ (say $a_{0}$ and $a_{1}$) equal $2$, then $X_{a_{0},\ldots,a_{n}}$ is
isomorphic to the hypersurface $uv+X_{2}^{a_{2}}+\cdots+X_{n}^{a_{n}}=0$ in
$\mathbb{A}^{n+1}_{{\rm\bf k}}$. This hypersurface admits non-trivial
$\mathbb{G}_{a,{\rm\bf k}}$-actions associated, for instance, to every locally
nilpotent ${\rm\bf k}[u]$-derivation $\partial_{i}=u\tfrac{\partial}{\partial
X_{i}}-a_{i}X^{a_{i}-1}\tfrac{\partial}{\partial v}$, $i=2,\ldots,n$, of its
coordinate ring111Note that the real Pham-Brieskorn surface
$X_{2,2,2}=\mathrm{Spec}(\mathbb{R}[X,Y,Z]/\langle X^{2}+Y^{2}+Z^{2}\rangle)$
is rigid (see e.g. [Fre17, Theorem 9.27]), so the condition that ${\rm\bf k}$
contains $\mathbb{Q}(i)$ cannot be relaxed.. In light of these observations,
the following conjecture was proposed in [FZ03, KZ00a]:

######  Main Conjecture.

An affine Pham-Brieskorn hypersurface $X_{a_{0},\dots,a_{n}}$ over an
algebraically closed field of characteristic zero is rigid if and only if
$\min\\{a_{0},\dots,a_{n}\\}\geq 2$ and at most one element $i$ of
$\\{0,\dots,n\\}$ satisfies $a_{i}=2$.

The conjecture was proved by Kaliman and Zaidenberg [KZ00b, Lemma 4] in the
$n=2$ case; despite several known partial results during the last decade (see
in particular [FMJ13], [DFMJ17], [CPW16c], [CPW16b], [CPW16a], [CD20], [LS22])
the conjecture remained open for every dimension $n\geq 3$ (see Conjecture
1.22 in [CPPZ21]).

Every affine Pham-Brieskorn hypersurface $X=X_{a_{0},\ldots,a_{n}}$ identifies
with the affine cone over the weighted hypersurface $\hat{X}$ defined by the
weighted homogeneous equation $X_{0}^{a_{0}}+\cdots+X_{n}^{a_{n}}=0$ in the
weighted projective space $\mathbb{P}=\mathbb{P}(w_{0},\ldots,w_{n})$, where
$w_{i}=\operatorname{{\rm lcm}}(a_{0},\dots,a_{n})/a_{i}$. We then say that
$X$ is a well-formed affine Pham-Brieskorn hypersurface if the corresponding
$\hat{X}$ is a well-formed weighted hypersurface in $\mathbb{P}$ in the usual
sense; that is, if
$\gcd(w_{0},\dots,w_{i-1},\hat{w_{i}},w_{i+1},\dots,w_{n})=1$ for every
$i=0,\dots,n$ and $\operatorname{{\rm
codim}}_{\hat{X}}(\hat{X}\cap\operatorname{{\rm Sing}}(\mathbb{P}))\geq 2$.
Proposition 2.1.4 provides a convenient arithmetic characterization of these
well-formed hypersurfaces: they are exactly those for which $a_{i}$ divides
$\operatorname{{\rm lcm}}(a_{0},\ldots,\hat{a}_{i},\ldots a_{n})$ for every
$i=0,\ldots,n$. With this notion at hand, our first main result consists of
the following reduction of the Main Conjecture:

######  Theorem.

To prove the Main Conjecture, it suffices to prove it in the special case
where $X_{a_{0},\ldots,a_{n}}$ is a well-formed affine Pham-Brieskorn
hypersurface.

Building on this result, we manage to complete the proof of the Main
Conjecture in the $n=3$ case, namely:

######  Main Theorem.

Let $X_{a_{0},a_{1},a_{2},a_{3}}=\mathrm{Spec}(B_{a_{0},a_{1},a_{2},a_{3}})$
be an affine Pham-Brieskorn threefold hypersurface. If
$\min\\{a_{0},a_{1},a_{2},a_{3}\\}\geq 2$ and at most one element $i$ of
$\\{0,1,2,3\\}$ satisfies $a_{i}=2$, then $X$ is rigid. Moreover, for every
integer $d\geq 1$, the Veronese subring $B_{a_{0},a_{1},a_{2},a_{3}}^{(d)}$ of
$B_{a_{0},a_{1},a_{2},a_{3}}$ is a rigid ring.

Let us briefly explain the scheme of the proof of the Main Theorem. Reducing
to well-formed affine Pham-Brieskorn hypersurfaces $X=X_{a_{0},\ldots,a_{3}}$
with associated weighted hypersurfaces $\hat{X}$ in
$\mathbb{P}(w_{0},\ldots,w_{3})$ leads us to consider the following two sub-
classes:

  1. (1)

those $\hat{X}$ with pseudo-effective canonical divisor $K_{\hat{X}}$;

  2. (2)

a finite list of cases for which $\hat{X}$ is a del Pezzo surface with cyclic
quotient singularities.

The rigidity of well-formed $3$-dimensional affine Pham-Brieskorn
hypersurfaces $X$ for which $K_{\hat{X}}$ is pseudo-effective can be inferred
by nowadays standard techniques. In the second case, which contains for
instance the affine cone $X_{3,3,3,3}$ over the Fermat cubic surface in
$\mathbb{P}^{3}$, it is known (by a general principle established by
Kishimoto-Prokhorov-Zaidenberg [KPZ13]) that the rigidity of $X$ is equivalent
to the non-existence of an anti-canonical polar cylinder in the del Pezzo
surface $\hat{X}$ (see Subsection 1.6 for the definition). Several cases in
(2) turn out to be covered by a series of results on the classification of
anti-canonical polar cylinders in del Pezzo surfaces with Du Val singularities
due to Cheltsov-Park-Won [CPW16b]. We are eventually left with the study of
the two cases $(a_{0},a_{1},a_{2},a_{3})\in\\{(2,3,4,12),(2,3,5,30)\\}$ for
which the corresponding del Pezzo surfaces have cyclic quotient singularities
that are not Du Val. In each of these two cases, we are able to extract from
the specific geometry of the del Pezzo surface at hand the non-existence of
anti-canonical polar cylinders. These two particular examples raise and
motivate the general problem of finding methods to expand the classification
given in[CPW16b] to include del Pezzo surfaces with klt singularities. In this
direction, we note that interestingly, we were unable to find a common method
to exclude the existence of anti-canonical polar cylinders in these two
examples.

The article is organized as follows. Section 1 gathers the algebraic and
geometric prelimary results required in Sections 3 and 4. In particular, we
include facts about locally nilpotent derivations, $\mathbb{Q}$-divisors,
weighted hypersurfaces in weighted projective spaces, cylinders and polar
cylinders. We also include some necessary results about singularities of log
pairs and the birational geometry of singular del Pezzo surfaces. In Section
2, we establish the theorem reducing the Main Conjecture to the case of well-
formed hypersurfaces and review the known relationship between the rigidity of
the ring $B_{a_{0},\dots,a_{n}}$ and the non-existence of canonical and anti-
canonical polar cylinders in $\operatorname{{\rm
Proj}}(B_{a_{0},\dots,a_{n}})$. Section 3 resolves the $n=3$ case of the Main
Conjecture for all but the $X_{2,3,4,12}$ and $X_{2,3,5,30}$ cases, which are
treated separately in Section 4. Finally, Section 5 gives some remarks on the
Main Conjecture in higher dimensions.

Acknowledgements. This project was partially funded by the Mitacs Globalink
Research Award number IT33113 which helped enable this collaboration. The
first author would like to thank l’Institut de Mathématiques de Bourgogne for
hosting and partially funding his visit to Dijon, and the University of Ottawa
for facilitating his visit. The second author received partial support from
the French ANR Project ANR-18-CE40-0003-01.

## Assumptions, Conventions and Notation

We assume the following throughout this article.

  * •

All rings are commutative, associative, unital and have characteristic zero.

  * •

We use the symbols $\mathbb{N},\mathbb{N}^{+},\mathbb{Z}$ to denote the sets
of natural numbers, positive integers and integers respectively.

  * •

We use the notation $``\subseteq"$ and $``\supseteq"$ for inclusion and
containment and $``\subset"$ and $``\supset"$ for proper inclusion and proper
containment.

  * •

Given an $\mathbb{N}$-graded ring $B=\bigoplus_{i\in\mathbb{N}}B_{i}$, we
denote the irrelevant ideal by $B_{+}$.

  * •

All varieties and schemes are assumed to be defined over an algebraically
closed field ${\rm\bf k}$ of characteristic zero which we fix throughout, a
variety being by convention an integral separated scheme of finite type. A
curve is a one-dimensional variety. A surface is a two-dimensional variety. In
particular, curves and surfaces are irreducible and reduced.

##  1 Preliminaries

###  1.1 Locally Nilpotent Derivations

######  1.1.1.

Let $B$ be a ring. A derivation $D:B\to B$ is locally nilpotent if for every
$b\in B$ there exists some $n\in\mathbb{N}$ such that $D^{n}(b)=0$. If $B$ is
a ring, the set of locally nilpotent derivations $D:B\to B$ is denoted
$\operatorname{{\rm LND}}(B)$. If $\operatorname{{\rm LND}}(B)=\\{0\\}$, we
say that $B$ is rigid. Given $D\in\operatorname{{\rm LND}}(B)$, an element
$t\in B$ is called a local slice of $D$ if $D(t)\neq 0$ and $D^{2}(t)=0$. In
particular, if $D(t)=1$, then $t$ is a slice of $D$. A derivation $D:B\to B$
is reducible if there exists some $b\in B$ such that $D(B)\subseteq\langle
b\rangle\neq B$. If no such $b$ exists, then $D$ is irreducible.

Let $A\subseteq B$ be an inclusion of integral domains. Then, $A$ is
factorially closed in $B$ if for all $x,y\in B\setminus\\{0\\}$, $xy\in A$
implies that $x,y\in A$.

We say that a ring $B$ is ${\rm\bf k}$-affine if it is finitely generated as a
${\rm\bf k}$-algebra. We call $B$ a ${\rm\bf k}$-domain if it is a ${\rm\bf
k}$-affine integral domain.

######  Definitions 1.1.2.

Let $B$ be a ${\rm\bf k}$-domain. For each $b\in B\setminus\\{0\\}$, define
$\deg_{D}(b)=\max\\{n\in\mathbb{N}:D^{n}(b)\neq 0\\}$; define
$\deg_{D}(0)=-\infty$. It is well-known that the map
$\deg_{D}:B\to\mathbb{N}\cup\\{-\infty\\}$ is a degree function.

If $B$ is not rigid, then given $f\in B$, the absolute degree of $f$ is
defined by

$|f|_{B}=\min\big{\\{}\,\deg_{D}(f)\,:\,D\in\operatorname{{\rm
LND}}(B)\setminus\\{0\\}\,\big{\\}}.$

If $B$ is rigid, define $|f|_{B}=-\infty$ if $f=0$, and $|f|_{B}=\infty$
otherwise. See Section 2.3 of [FMJ13] for further details.

######  Theorem 1.1.3.

Let $B$ be an integral domain, let $D:B\to B$ be a derivation, and let
$A=\ker(D)$. The following facts are well known. (Refer to [Fre17] for
instance.)

  1. (a)

If $D$ is locally nilpotent, then $A$ is a factorially closed subring of $B$.
Consequently, if $D$ is locally nilpotent and ${\rm\bf k}$ is a field included
in $B$, then $D$ is a ${\rm\bf k}$-derivation.

  2. (b)

Assume that $\mathbb{Q}\subseteq B$. If $D\neq 0$ is locally nilpotent then
$D$ has a local slice $t\in B$. For any such $t$, if we define $s=D(t)$ then
$B_{s}=A_{s}[t]$ is a polynomial ring in one variable over $A_{s}$.
Consequently, $\operatorname{{\rm trdeg}}(B/A)=1$ and $\operatorname{{\rm
Frac}}(B)\supset\operatorname{{\rm Frac}}(A)$ is purely transcendental of
transcendence degree one.

  3. (c)

If $b\in B\setminus\\{0\\}$, then the derivation $bD:B\to B$ is locally
nilpotent if and only if $D$ is locally nilpotent and $b\in A$.

  4. (d)

If $D\neq 0$ is locally nilpotent and $B$ satisfies the ascending chain
condition for principal ideals, then there exists an irreducible locally
nilpotent derivation $\delta:B\to B$ such that $D=a\delta$ for some $a\in A$.

######  1.1.4.

Let $B=\bigoplus_{n\in\mathbb{Z}}B_{n}$ be a $\mathbb{Z}$-graded ring. A
derivation $D:B\to B$ is homogeneous if there exists an $h\in\mathbb{Z}$ such
that $D(B_{g})\subseteq B_{g+h}$ for all $g\in\mathbb{Z}$. If $D$ is
homogeneous and $D\neq 0$, then $h$ is unique and we say that $D$ is
homogeneous of degree $h$. The zero derivation is said to be homogeneous of
degree $-\infty$. The set of homogeneous locally nilpotent derivations of $B$
is denoted $\operatorname{{\rm HLND}}(B)$.

A graded subring of $B$ is a subring $A$ of $B$ satisfying
$A=\bigoplus_{n\in\mathbb{Z}}(A\cap B_{n})$. If $A$ is a graded subring of $B$
then $A$ too is a $\mathbb{Z}$-graded ring. In particular, if
$D\in\operatorname{{\rm HLND}}(B)$ then $\ker(D)$ is a $\mathbb{Z}$-graded
subring of $B$. In the special case where $B$ is a finitely generated ${\rm\bf
k}$-algebra, there exists a set map $\operatorname{{\rm
LND}}(B)\to\operatorname{{\rm HLND}}(B)$ ($D\mapsto\tilde{D})$ satisfying the
condition:

> $D=0$ if and only if $\tilde{D}=0$.

The derivation $\tilde{D}$ is called the homogenization of $D$. (For further
details on this topic, see [Dai12].) In particular, we obtain the following:

######  Proposition 1.1.5.

If $B$ is a $\mathbb{Z}$-graded affine ${\rm\bf k}$-domain, then
$\operatorname{{\rm LND}}(B)\neq\\{0\\}$ if and only if $\operatorname{{\rm
HLND}}(B)\neq\\{0\\}$.

###  1.2 Type and Cotype

The following definitions appear in [CD20]. In order to keep consistent
notation throughout this article, our indices run from $0$ to $n$ (as opposed
to from $1$ to $n$).

######  Definitions 1.2.1.

Let $n\geq 2$ and $S=(b_{0},\dots,b_{n})\in\mathbb{Z}^{n+1}$.

  * •

Define222By convention, $\gcd(S)\geq 0$ and $\operatorname{{\rm lcm}}(S)\geq
0$. $\gcd(S)=\gcd(b_{0},\dots,b_{n})$ and $\operatorname{{\rm
lcm}}(S)=\operatorname{{\rm lcm}}(b_{0},\dots,b_{n})$.

  * •

If $\gcd(S)=1$, we say that $S$ is normal. If $S\neq(0,\dots,0)$ and
$d=\gcd(S)$, then the tuple
$S^{\prime}=(\frac{b_{0}}{d},\dots,\frac{b_{n}}{d})$ is normal, and is called
the normalization of $S$.

  * •

For each $j\in\\{0,\dots,n\\}$, define
$S_{j}=(b_{0},\dots,\widehat{b_{j}},\dots,b_{n})$.

  * •

We define $\operatorname{{\rm
cotype}}(S)=|\big{\\{}\,i\in\\{0,\dots,n\\}\,:\,\operatorname{{\rm
lcm}}(S_{i})\neq\operatorname{{\rm
lcm}}(S)\,\big{\\}}|=|\big{\\{}\,i\in\\{0,\dots,n\\}\,:\,b_{i}\nmid\operatorname{{\rm
lcm}}(S_{i})\,\big{\\}}|$.

Note that $\operatorname{{\rm cotype}}(S)\in\\{0,1,\dots,n+1\\}$ and that, if
$S^{\prime}$ is the normalization of $S$, then $\operatorname{{\rm
cotype}}(S)=\operatorname{{\rm cotype}}(S^{\prime})$.

######  Definition 1.2.2.

Let $n\geq 2$.

  * •

Given $S=(a_{0},\dots,a_{n})\in(\mathbb{N}^{+})^{n+1}$ and
$i\in\\{0,\dots,n\\}$, define $g_{i}(S)=\gcd(a_{i},\operatorname{{\rm
lcm}}(S_{i}))$.

  * •

Let $S=(a_{0},\dots,a_{n})$ and
$S^{\prime}=(a_{0}^{\prime},\dots,a_{n}^{\prime})$ be elements of
$(\mathbb{N}^{+})^{n+1}$ and let $i\in\\{0,\dots,n\\}$. We write
$S\leq^{i}S^{\prime}$ if and only if

$S_{i}=S^{\prime}_{i}\quad\text{and}\quad g_{i}(S^{\prime})\mid a_{i}\mid
a_{i}^{\prime}.$

We write $S<^{i}S^{\prime}$ if and only if $S\leq^{i}S^{\prime}$ and $S\neq
S^{\prime}$. Note that the relation $\leq^{i}$ is a partial order on
$(\mathbb{N}^{+})^{n+1}$.

###  1.3 Divisors and $\mathbb{Q}$-divisors

This section establishes the notation and collects some results we will use
when discussing $\mathbb{Q}$-divisors. Throughout the article, whenever we
discuss divisors on a variety $X$, we assume implicitly that $X$ is normal.

######  Definitions 1.3.1.

A prime divisor of $X$ is an irreducible closed subset of codimension one. The
group of divisors, denoted $\operatorname{{\rm Div}}(X)$, is the free abelian
group on the set of all prime divisors of $X$. A divisor $D$ is an element of
the group of divisors and is written as $D=\sum_{i\in I}a_{i}Y_{i}$ where each
$Y_{i}$ is a prime divisor and each $a_{i}\in\mathbb{Z}$. The function field
of $X$ is denoted by $K(X)$. Given $f\in K(X)^{*}$, we write
$\operatorname{{\rm div}}_{X}(f)$ (or simply $\operatorname{{\rm div}}(f)$
when $X$ is understood) for the divisor of $f$. A divisor
$D\in\operatorname{{\rm Div}}(X)$ is principal if there exists $f\in K(X)^{*}$
such that $D=\operatorname{{\rm div}}_{X}(f)$. Two divisors $D$ and
$D^{\prime}$ are linearly equivalent (we write $D\sim D^{\prime}$) if
$D-D^{\prime}$ is a principal divisor. The set of principal divisors is a
subgroup of $\operatorname{{\rm Div}}(X)$ and is denoted $\operatorname{{\rm
Princ}}(X)$. The divisor class group of $X$ is the quotient group
$\operatorname{{\rm Div}}(X)/\operatorname{{\rm Princ}}(X)$ and is denoted
$\operatorname{{\rm Cl}}(X)$.

The group of $\mathbb{Q}$-divisors, denoted $\operatorname{{\rm
Div}}(X,\mathbb{Q})$, is the group $\operatorname{{\rm
Div}}(X)\otimes_{\mathbb{Z}}\mathbb{Q}$. A $\mathbb{Q}$-divisor $D$ is written
as $D=\sum_{i\in I}a_{i}Y_{i}$ where each $Y_{i}$ is a prime divisor and
$a_{i}\in\mathbb{Q}$. If $D=\sum_{i\in I}a_{i}Y_{i}$ is a
$\mathbb{Q}$-divisor, then $\lfloor D\rfloor=\sum_{i\in I}\lfloor a_{i}\rfloor
Y_{i}$. Two $\mathbb{Q}$-divisors $D$ and $D^{\prime}$ are linearly equivalent
(we write $D\sim D^{\prime}$) if there exists $f\in K(X)^{*}$ such that
$D-D^{\prime}$ is a principal divisor. Two $\mathbb{Q}$-divisors are
$\mathbb{Q}$-linearly equivalent (we write $D\sim_{\mathbb{Q}}D^{\prime}$) if
there exists some $n\in\mathbb{N}^{+}$ such that $nD-nD^{\prime}$ is a
principal divisor. A $\mathbb{Q}$-divisor $D=\sum_{i\in I}a_{i}Y_{i}$ is
effective if $a_{i}\geq 0$ for all $i$. It is reduced if for all $i\in I$,
either $a_{i}=1$ or $a_{i}=0$. The support of a $\mathbb{Q}$-divisor
$D=\underset{i\in I}{\sum}a_{i}Y_{i}$ is $\operatorname{{\rm
Supp}}(D)=\underset{a_{i}\neq 0}{\bigcup}Y_{i}$.

A divisor $D$ is Cartier (or locally principal) if there exists an open cover
$X=\bigcup_{i\in I}U_{i}$ of $X$ such that, for each $i\in I$, $D|_{U_{i}}$ is
a principal divisor of $U_{i}$. We write $\operatorname{{\rm
CaDiv}}(X)=\big{\\{}\,D\in\operatorname{{\rm Div}}(X)\,:\,\text{$D$ is
Cartier}\,\big{\\}}$ and note that $\operatorname{{\rm CaDiv}}(X)$ is a
subgroup of $\operatorname{{\rm Div}}(X)$. Since every principal divisor is
locally principal, $\operatorname{{\rm Princ}}(X)$ is a subgroup of
$\operatorname{{\rm CaDiv}}(X)$. The quotient group $\operatorname{{\rm
CaDiv}}(X)/\operatorname{{\rm Princ}}(X)$ is called the Cartier class group of
$X$ and is denoted $\operatorname{{\rm CaCl}}(X)$. Recall that if the variety
$X$ is regular, the natural map $\operatorname{{\rm
CaCl}}(X)\to\operatorname{{\rm Cl}}(X)$ is an isomorphism and that
$\operatorname{{\rm CaCl}}(X)$ is canonically isomorphic to the Picard group
$\mathrm{Pic}(X)=H^{1}(X,\operatorname{\mathcal{O}}_{X}^{*})$.

A $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor is a $\mathbb{Q}$-divisor
$D\in\operatorname{{\rm Div}}(X,\mathbb{Q})$ such that $nD$ is a Cartier
divisor for some $n\in\mathbb{N}^{+}$. A $\mathbb{Q}$-Cartier divisor is an
integral $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor. A variety $X$ if called
$\mathbb{Q}$-factorial if every integral divisor on $X$ is
$\mathbb{Q}$-Cartier.

######  Definition 1.3.2.

Given $D\in\operatorname{{\rm Div}}(X,\mathbb{Q})$, the sheaf
$\operatorname{\mathcal{O}}_{X}(D)$ of
$\operatorname{\mathcal{O}}_{X}$-modules is defined by stipulating that if $U$
is a non-empty open subset of $X$ then

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

Note that if $D\in\operatorname{{\rm Div}}(X,\mathbb{Q})$, then by definition
$\operatorname{\mathcal{O}}_{X}(D)=\operatorname{\mathcal{O}}_{X}(\lfloor
D\rfloor)$.

######  Notation 1.3.3.

A canonical divisor on a normal variety $X$ is any integral divisor $K_{X}$ on
$X$ such that the restriction of the sheaf
$\operatorname{\mathcal{O}}_{X}(K_{X})$ to the regular locus
$X_{\mathrm{reg}}$ of $X$ is isomorphic to the canonical sheaf
$\omega_{X_{\mathrm{reg}}}=\bigwedge^{n}\Omega_{X_{\mathrm{reg}}/\mathbf{k}}$
of $X_{\mathrm{reg}}$ where $n=\dim X_{\mathrm{reg}}$. In the case where $X$
is is normal and Cohen-Macaulay, the dualizing and canonical sheaves of $X$
(denoted $\omega_{X}^{o}$ and $\omega_{X}$ respectively) are isomorphic and
are also isomorphic to $\operatorname{\mathcal{O}}_{X}(K_{X})$ where $K_{X}$
is any canonical divisor of $X$. (See Remark 5.2 in [Kov12].)

######  Definitions 1.3.4.

A Cartier divisor $D$ on a normal variety $X$ is called very ample if
$\operatorname{\mathcal{O}}_{X}(D)$ is a very ample invertible sheaf; a
$\mathbb{Q}$-divisor $D$ on $X$ is ample if there exists some
$m\in\mathbb{N}^{+}$ such that $mD$ is a very ample Cartier divisor. Assume in
addition that $X$ is projective. A $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor
$D$ is called nef if $D\cdot C\geq 0$ for every irreducible curve $C$ in $X$.

######  Definitions 1.3.5.

Let $D$ be a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $D$ on a normal
projective variety $X$. We say that $D$ is big if $\operatorname{{\rm
Vol}}_{X}(D)=\limsup_{m\to\infty}\frac{h^{0}(X,\operatorname{\mathcal{O}}_{X}(mD))}{m^{d}/d!}>0$
and that $D$ is pseudo-effective if $D+A$ is big for every ample
$\mathbb{Q}$-Cartier divisor $A$ on $X$. Note that an ample divisor is pseudo-
effective.

Recall the Nakai-Moishezon criterion which we will use repeatedly in Section
3.

######  Theorem 1.3.6.

A $\mathbb{Q}$-Cartier divisor $D\in\operatorname{{\rm Div}}(X)$ on a normal
projective surface $X$ is ample if and only if $D\cdot C>0$ for every
irreducible curve $C\in\operatorname{{\rm Div}}(X)$.

The following is a special case of Corollary 2.3.22 of [Laz04].

######  Proposition 1.3.7.

If $D$ is a nef Cartier divisor on a smooth projective surface $X$, then
$\operatorname{{\rm Vol}}_{X}(D)=D^{2}$. In particular, if $D$ is nef, then
$D$ is big if and only if $D^{2}>0$.

###  1.4 Quasismooth Weighted Complete Intersections

We collect known results on quasismooth weighted complete intersections in
weighted projective spaces.

######  1.4.1.

Let $S={\rm\bf k}_{w_{0},\dots,w_{n}}[X_{0},\dots,X_{n}]$ denote the graded
polynomial ring where $n\geq 1$ and $\deg(X_{i})=w_{i}$ for each
$i=0,\dots,n$. The weighted projective space
$\mathbb{P}=\mathbb{P}(w_{0},\dots,w_{n})=\operatorname{{\rm Proj}}(S)$ is
said to be well-formed if for each $i=0,\dots,n$, we have
$\gcd(w_{0},\dots,w_{i-1},\hat{w_{i}},w_{i+1},\dots,w_{n})=1$. Every weighted
projective space is a projective variety and is isomorphic to a well-formed
weighted projective space. Every weighted projective space is normal and
Cohen-Macaulay (see Theorem 3.1A(c) in [BR86]).

A weighted projective variety $X$ is a closed subvariety of a weighted
projective space. Whenever we write “the variety $X\subseteq\mathbb{P}$”, we
mean that $X$ is a closed subvariety of $\mathbb{P}$. Since
$\mathbb{P}(w_{0},\dots,w_{n})$ is a projective variety, every weighted
projective variety is also a projective variety.

Let $I$ be a homogeneous prime ideal of the graded ring $S={\rm\bf
k}_{w_{0},\dots,w_{n}}[X_{0},\dots,X_{n}]$ and consider the closed subvariety
$X=V_{+}(I)$ of $\mathbb{P}=\mathbb{P}(w_{0},\dots,w_{n})=\operatorname{{\rm
Proj}}(S)$. Note that $X$ is isomorphic to $\operatorname{{\rm Proj}}(S/I)$.
The closed subset $C_{X}=V(I)\subseteq\mathbb{A}^{n+1}$ is called the affine
cone over $X$; note that $C_{X}$ passes through the origin of
$\mathbb{A}^{n+1}$, and that $C_{X}\cong\operatorname{{\rm Spec}}(S/I)$ is an
integral affine scheme. The variety $X$ is called quasismooth if $C_{X}$ is
nonsingular away from the origin.

Let $I$ be a homogeneous prime ideal of the graded ring $S={\rm\bf
k}_{w_{0},\dots,w_{n}}[X_{0},\dots,X_{n}]$ and define
$X_{I}=\operatorname{{\rm Proj}}(S/I)\subseteq\mathbb{P}(w_{0},\dots,w_{n})$.
If $f_{1},\dots,f_{k}$ are homogeneous elements of $S$, we abbreviate
$X_{\langle f_{1},\dots,f_{k}\rangle}$ by $X_{f_{1},\dots,f_{k}}$. If $I$ is
generated by a regular sequence $(f_{1},\dots,f_{k})$ of homogeneous elements
of $S$ of respective degrees $d_{i}=\deg(f_{i})$ where $i=1,\dots,k$, then
$X_{I}$ is called a weighted complete intersection of multidegree
$(d_{1},\dots,d_{k})$. In particular, if $k=1$, $f_{1}=f$ and $d_{1}=d$, we
will say that $X_{I}=X_{f}$ is a weighted hypersurface of degree $d$.

######  Remark 1.4.2.

Note that $\mathbb{P}(w_{0},\dots,w_{n})$ is well-formed if and only if
$S=(w_{0},\dots,w_{n})$ is normal and $\operatorname{{\rm type}}(S)=0$. (See
Definitions 1.2.1.)

######  Definition 1.4.3.

A weighted subvariety $X$ of $\mathbb{P}$ is called well-formed if the
following conditions hold:

  * •

$\mathbb{P}$ is well-formed,

  * •

$\operatorname{{\rm codim}}_{X}(X\cap\operatorname{{\rm
Sing}}(\mathbb{P}))\geq 2$.

######  Remark 1.4.4.

Note that we will cite results due to Dimca, some of which can be found in
[Dim86]. Instead of writing that $X$ is well-formed, Dimca writes “a
quasismooth complete intersection $X$ is in general position relative to
$\operatorname{{\rm Sing}}(\mathbb{P})$ if $\operatorname{{\rm
codim}}_{X}(X\cap\operatorname{{\rm Sing}}(\mathbb{P}))\geq 2$”.

Propositions 1.4.5 and 1.4.6 appear in [Dim86], the former on page 185, the
latter as Proposition 8.

######  Proposition 1.4.5.

Let $\mathbb{P}=\mathbb{P}(w_{0},\dots,w_{n})$ be a well-formed weighted
projective space and let $\textbf{x}=[x_{0}:\ldots:x_{n}]\in\mathbb{P}$.
Define $I(\textbf{x})=\big{\\{}\,j\,:\,x_{j}\neq 0\,\big{\\}}$. For any prime
$p$, let

$\mathbb{P}_{p}=\big{\\{}\,\textbf{x}\in\mathbb{P}\,:\,p|w_{i}\text{ for every
}i\in I(\textbf{x})\,\big{\\}}.$

Then,

$\operatorname{{\rm Sing}}\mathbb{P}=\bigcup_{p\text{ prime}}\mathbb{P}_{p}.$

######  Proposition 1.4.6.

Let $\mathbb{P}=\mathbb{P}(w_{0},\dots,w_{n})$ and let $X\subseteq\mathbb{P}$
be a well-formed quasismooth weighted complete intersection of dimension at
least 2. Then $\operatorname{{\rm Sing}}(X)=X\cap\operatorname{{\rm
Sing}}(\mathbb{P})$.

######  Assumption 1.4.7.

From this point onward, whenever we consider a quasismooth weighted complete
intersection $X\subset\mathbb{P}$, we assume that $X$ is not contained in a
hyperplane of $\mathbb{P}$.

The following proposition collects some known results on singularities well-
formed quasismooth weighted complete intersections. Recall [KM98, $\S 5$] that
a variety $X$ is said to have rational singularities if for every resolution
of singularities $f:\tilde{X}\to X$ one has
$R^{i}f_{*}\operatorname{\mathcal{O}}_{\tilde{X}}=0$ for all $i>0$. This
condition is known to be equivalent to the property that $X$ is Cohen-Macaulay
and $f_{*}\omega_{\tilde{X}}=\omega_{X}$.

######  Proposition 1.4.8.

Let $\mathbb{P}=\mathbb{P}(w_{0},\dots,w_{n})$ be a well-formed weighted
projective space. If $X\subseteq\mathbb{P}$ is a well-formed quasismooth
weighted complete intersection then the following hold:

  1. (a)

$X$ is normal;

  2. (b)

$X$ has at most cyclic quotient singularities;

  3. (c)

$X$ has rational singularities;

  4. (d)

$X$ is Cohen-Macaulay;

  5. (e)

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

###### Proof.

We provide some references for the above assertions. Part (b) is stated on
page 109 of [IF00]. Part (a) follows from (b) together with the fact that
quotient of a normal variety is again normal. Part (c) is given by Corollary
7.4.10 in [Ish14], and part (d) follows from (c). Finally, Proposition 5.15 in
[KM98] states that a variety with quotient singularities is
$\mathbb{Q}$-factorial, and so (e) follows from (b). ∎

We recall that if $X\subseteq\mathbb{P}(w_{0},\dots,w_{n})$ is a weighted
complete intersection of multidegree $(d_{1},\dots,d_{k})$, then the amplitude
of $X$ is denoted by $\alpha=\sum_{i=1}^{k}d_{i}-\sum_{i=0}^{n}w_{n}$. A key
property of well-formed quasismooth complete intersections is that the
adjunction formula holds for them, in the following form:

######  Theorem 1.4.9.

Let $X\subseteq\mathbb{P}(w_{0},\dots,w_{n})$ be a well-formed quasismooth
weighted complete intersection. Then
$\omega_{X}\cong\omega_{X}^{\circ}\cong\operatorname{\mathcal{O}}_{X}(\alpha)$
where $\alpha$ is the amplitude of $X$.

###### Proof.

By Proposition 1.4.8, $X$ is normal and Cohen-Macaulay so $\omega_{X}$ and
$\omega_{X}^{\circ}$ exist and are isomorphic;
$\omega_{X}^{\circ}\cong\operatorname{\mathcal{O}}_{X}(\alpha)$ by Theorem
3.3.4 of [Dol82]. ∎

We state Lemma 7.1 of [IF00]. (We fix a small typo. The last line should read
$A_{-(k-\alpha)}$ instead of $A_{-k-\alpha}$ as in [IF00].)

######  Lemma 1.4.10.

Let $X_{f_{1},\dots,f_{r}}\subset\mathbb{P}(w_{0},\dots,w_{n})$ be a well-
formed quasismooth weighted complete intersection. Let $A$ be the graded ring
${\rm\bf k}_{w_{0},\dots,w_{n}}[X_{0},\dots,X_{n}]/\langle
f_{1},\dots,f_{r}\rangle$. Then,

$H^{i}(X,\operatorname{\mathcal{O}}_{X}(k))=\begin{cases}A_{k}&\text{if
}i=0\\\ 0&\text{if }1\leq i<\dim X\\\ A_{-(k-\alpha)}&\text{if }i=\dim X\\\
\end{cases}$

for all $k\in\mathbb{Z}$.

###  1.5 Singularities of surfaces, log pairs and singular del Pezzo surfaces

We briefly recall some basic facts about singularities of surfaces log pairs.
We refer the reader to [KM98, Chapter 2] for details. A surface _log pair_
$(S,D)$ consists of a normal surface $S$ and an effective Weil
$\mathbb{Q}$-divisor $D=\sum_{i=1}^{r}a_{i}C_{i}$ such that the divisor
$K_{S}+D$ is $\mathbb{Q}$-Cartier. A log-resolution of a surface log pair
$(S,D)$ is a proper birational morphism $\pi:\tilde{S}\to S$ such that
$\tilde{S}$ is smooth and the total transform $\pi^{*}D$ of $D$ is the support
of an SNC divisor on $\tilde{S}$. Denoting by $\tilde{C}_{i}$ the proper
transform of $C_{i}$ and by $E_{1},\dots,E_{n}$ the $\pi$-exceptional curves,
we have

$K_{\tilde{S}}+\sum_{i=1}^{r}a_{i}\tilde{C}_{i}\sim_{\mathbb{Q}}\pi^{*}(K_{S}+D)+\sum_{j=1}^{n}b_{j}E_{j}$

for some rational numbers $b_{1},\dots,b_{n}$.

######  Definitions 1.5.1.

A surface log pair $(S,D)$ is called canonical, (resp. purely log-terminal),
(resp. log-canonical) at a point $p$ if $a_{i}\leq 1$ for every $i$ such that
$p\in C_{i}$ and there exists a log-resolution $\pi:\tilde{S}\to S$ such that
$b_{j}\geq 0$ (resp. $b_{j}>-1$), (resp. $b_{j}\geq-1$) for all $j$ such that
$\pi(E_{j})=p$.

The pair $(S,D)$ is call canonical, (resp. purely log-terminal), (resp. log-
canonical) if it is so at every point $p\in S$. A pair $(S,D)$ is called
Kawamata log terminal (klt) if it is purely log-terminal and $\lfloor
D\rfloor=0$. A normal surface $S$ is said to have canonical (resp. klt)
singularities if if the log pair $(S,0)$ is canonical (resp. klt). These
notions are known to be independent the choice of log resolution and can
therefore be verified on the minimal log resolution of the surface log pair
$(S,D)$.

It is known that klt singularities of surfaces are finite quotient
singularities and that among these, canonical singularities are exactly the Du
Val A-D-E singularities, that is, the isolated singularities $s\in S$ such
that the formal completion $\hat{\mathcal{O}}_{S,s}$ of the local ring
$\mathcal{O}_{S,s}$ at $s$ is isomorphic to that of the local ring of the
quotient of $(\mathbb{A}^{2},0)$ by the action of small finite subgroups of
$\mathrm{SL}_{2}$ at the image of the origin.

######  Example 1.5.2.

[HTU17, p.286] Let $S$ be a normal projective surface with $k$ singular points
$P_{1},\dots,P_{k}$. Assume each $P_{i}$ is a cycle quotient singularity of
type $\frac{1}{n_{i}}(1,1)$ where $n_{i}\geq 2$ and let $\pi:\tilde{S}\to S$
be the minimal resolution of singularities of $S$. Then
$E_{i}=\pi^{-1}(P_{i})$ is isomorphic to $\mathbb{P}^{1}$ and has self-
intersection number $-n_{i}$ in $\tilde{S}$. Moreover, in the ramification
formula

$K_{\tilde{S}}=\pi^{*}(K_{S})+\sum_{i=1}^{k}b_{i}E_{i}$

for $\pi$, one has $b_{i}=-1+\frac{2}{n_{i}}$ for each $i=1,\dots,k$. In
particular, $P_{i}$ is a klt singularity which is canonical if and only if
$n_{i}=2$.

The following is Lemma A.3. of [CPPZ21].

######  Lemma 1.5.3.

Let $S$ be a normal surface with at most quotient singularities and let $D$ be
an effective non-zero $\mathbb{Q}$-divisor on $S$. Let $p$ be a regular point
of $S$. If $(S,D)$ is not log-canonical at $p$, then $\operatorname{{\rm
mult}}_{p}(D)>1$.

We record the following useful formula for computing intersection numbers of
Cartier divisors on complete normal surfaces. See Remark 2.5.10 in [Arc93] for
instance.

######  Lemma 1.5.4.

Let $D_{1},D_{2}$ be Cartier divisors on a complete normal surface over
${\rm\bf k}$. Then

$(D_{1}\cdot
D_{2})=\chi(\operatorname{\mathcal{O}}_{X})-\chi(\operatorname{\mathcal{O}}_{X}(D_{1}))-\chi(\operatorname{\mathcal{O}}_{X}(D_{2}))+\chi(\operatorname{\mathcal{O}}_{X}(D_{1})\otimes_{\operatorname{\mathcal{O}}_{X}}\operatorname{\mathcal{O}}_{X}(D_{2})).$

######  Definitions 1.5.5.

A del Pezzo surface (resp. weak del Pezzo surface) $X$ is a normal projective
surface over ${\rm\bf k}$ with at most quotient singularities such that
$-K_{X}$ is an ample (resp. nef and big) $\mathbb{Q}$-Cartier divisor. The
degree of a (weak) del Pezzo surface $X$ is the self-intersection number of
the canonical divisor $K_{X}$.

######  Remarks 1.5.6.

  * •

Any curve on a smooth del Pezzo surface has self intersection number at least
-1.

  * •

The minimal resolution of singularities of a singular del Pezzo surface with
Du Val singularities is a weak del Pezzo surface of the same degree.

  * •

Since we do not assume del Pezzo surfaces are nonsingular, the degree of a del
Pezzo surface need not be an integer.

We now collect known properties of effective anti-canonical divisors on del
Pezzo surfaces of degree 1 and 2 with at worst Du Val singularities which will
use in Section 4.

######  Lemma 1.5.7.

Let $S$ be a del Pezzo surface of degree 1 with at most Du Val singularities.
Let $D$ be an anti-canonical $\mathbb{Q}$-divisor of $S$.

  1. (a)

Every element of $|-K_{S}|$ is reduced and irreducible, and $|-K_{S}|$ has a
unique base point $o$ which is a smooth point of $S$.

  2. (b)

If $C\in|-K_{S}|$ and the pair $(S,C)$ is not log-canonical at a smooth point
$p\in S$, then $C$ is a rational cuspidal curve with a cusp at $p$.

  3. (c)

The log pair $(S,D)$ is log-canonical at all but finitely many points,
including at the base point $o$.

  4. (d)

Let $q$ be a either a smooth point different from $o$ or a singular point of
type $A_{1},A_{2},A_{3}$ or $D_{4}$ and let $C$ be the unique curve in
$|-K_{S}|$ through $q$. If $(S,D)$ is not log-canonical at $q$ then

    1. (i)

the log pair $(S,C)$ is not log-canonical at $p$;

    2. (ii)

the support of $D$ contains the support of $C$.

###### Proof.

Part (a) is well-known. See Lemma 3.1 of [Par03] for the first claim and the
start of Section 3 in [CPW16b] for the second.

We prove (b). By (a), $C$ is irreducible and reduced. Since $(S,C)$ is not
log-canonical at $p$ and $p$ is a smooth point of $S$, $p$ must be a singular
point of $C$. Corollaire 2 in [Dem77] implies that $C$ cannot pass through a
singular point of $S$. Let $\pi:W\to S$ denote the minimal resolution of
singularities of $S$ so that $W$ is a weak del Pezzo surface of degree 1. The
adjunction formula implies that
$2p_{a}(\tilde{C})-2=\tilde{C}\cdot(\tilde{C}+K_{W})=0$ so
$1=p_{a}(\tilde{C})=p_{a}(C)$. Since $C$ is singular and $p_{a}(C)=1$, we have
$0=p_{g}(C)=p_{a}(C)-\underset{Q}{\sum}\frac{1}{2}m_{Q}(m_{Q}-1)$ where the
points $Q$ are the centers of blowups in the normalization of $C$ and $m_{Q}$
is the multiplicity of $C$ at $Q$. (In particular, $C$ is rational.) It
follows that there is a unique singular point of multiplicity 2 that is either
a node or a cusp, and that this point is necessarily $p$. Since $(S,C)$ is not
log-canonical at $p$, $p$ must be a cusp. This proves (b). Parts (c) and (d)
can be found as Theorem 3.1 in [CPW16b].

∎

We obtain the following corollary.

######  Corollary 1.5.8.

Let $S$ be a weak del Pezzo surface of degree 1 and let $D$ be an anti-
canonical $\mathbb{Q}$-divisor on $S$. Let $\pi:S\to T$ denote the contraction
of all $(-2)$-curves on $S$ and let $E\subset S$ denote the exceptional locus
of $\pi$. Let $o$ denote the unique base point of $|-K_{S}|$. Suppose $p\in S$
is such that $p\neq o$ and let $C$ denote the unique curve in $|-K_{S}|$
passing through $p$. If $(S,D)$ is not log-canonical at $p$ and $p\notin E$,
then

  * •

$(S,C)$ is not log-canonical at $p$;

  * •

$\operatorname{{\rm Supp}}(C)\subseteq\operatorname{{\rm Supp}}(D)$.

###### Proof.

Since $p\notin E$, $(T,\pi_{*}(D))$ is not log-canonical at $\pi(p)$. Since
$\pi(p)$ is a smooth point of $T$ which is not the base point of $|-K_{T}|$
(by Lemma 1.5.7 (c)) and $\pi_{*}(C)$ is the unique element of $|-K_{T}|$
which contains $\pi(p)$, Lemma 1.5.7 (d) implies that the pair
$(T,\pi_{*}(C))$ is not log-canonical at $\pi(p)$ and $\operatorname{{\rm
Supp}}(\pi_{*}(D))$ contains $\operatorname{{\rm Supp}}(\pi_{*}(C))$.
Consequently, $\operatorname{{\rm Supp}}(D)$ contains $\operatorname{{\rm
Supp}}(C)$. Let $\alpha:W\to T$ be a log resolution of $(T,\pi_{*}(C))$. Since
$\alpha:W\to T$ factors through $S$, it follows that $(S,C)$ is not log-
canonical at $p$.

∎

We state Lemma 3.4 of [CPW16c].

######  Lemma 1.5.9.

Let $S$ be a smooth del Pezzo surface of degree $2$ and let $D$ be an
effective $\mathbb{Q}$-divisor such that $D\sim_{\mathbb{Q}}-K_{S}$. Suppose
that the log pair $(S,D)$ is not log-canonical at $p$. Then

  1. (a)

there exists a unique divisor $C\in|-K_{S}|$ such that $(S,C)$ is not log-
canonical at $p$.

  2. (b)

The support of $D$ contains all the irreducible components of $C$ where $C$ is
either

     * •

an irreducible rational curve with a cusp at $p$

     * •

a union of two $(-1)$-curves meeting tangentially at $p$.

###  1.6 Cylinders and Polar Cylinders

######  Definition 1.6.1.

A scheme $U$ is a cylinder if $U\cong C\times{\mathbb{A}^{1}}$ for some affine
scheme $C$. A scheme $X$ contains a cylinder if there exists a non-empty open
set $U\subseteq X$ such that $U$ is a cylinder. Note that $U$ is affine.

######  Definition 1.6.2.

Let $H$ be a $\mathbb{Q}$-divisor on a projective normal variety $X$ over
${\rm\bf k}$. Let $U\subseteq X$ be a cylinder of $X$. Then, the cylinder
$U\subseteq X$ is called $H$-polar if $U=X\setminus\operatorname{{\rm
Supp}}(D)$ for some effective $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor
$D\in\operatorname{{\rm Div}}(X,\mathbb{Q})$ and some $s\in\mathbb{Q}^{+}$
such that $D\sim sH$. In the special case where $H\sim-K_{X}$, any $H$-polar
cylinder $U$ is called an anti-canonical polar cylinder.

######  Remark 1.6.3.

Let $H$ and $H^{\prime}$ be $\mathbb{Q}$-divisors on a projective normal
variety $X$. Assume that there exist $q,q^{\prime}\in\mathbb{Q}^{+}$ such that
$qH\sim q^{\prime}H^{\prime}$. Then a cylinder $U$ is $H$-polar if and only if
it is $H^{\prime}$-polar.

#### Demazure’s Construction and Polar Cylinders

######  Definition 1.6.4.

Let $B=\bigoplus_{i\in\mathbb{N}}B_{i}$ be an $\mathbb{N}$-graded integral
domain. An element $\xi\in\operatorname{{\rm Frac}}B$ is homogeneous if
$\xi=\frac{a}{b}$ for some homogeneous elements $a,b\in B$ with $b\neq 0$. If
$\xi$ is homogeneous, then its degree is $\deg(\xi)=\deg(a)-\deg(b)$.

The following is a special case of the Theorem below Section 3.5 of [Dem88].

######  Theorem 1.6.5.

Let $B=\bigoplus_{n\in\mathbb{N}}B_{n}$ be an $\mathbb{N}$-graded normal
domain that is finitely generated over ${\rm\bf k}$ and such that $e(B)=1$ and
$0pt(B_{+})>1$. Let $X=\operatorname{{\rm Proj}}B$. Then, there exists a
homogeneous element $T$ of $\operatorname{{\rm Frac}}B$ of degree 1, and for
each such $T$, there exists a unique $\mathbb{Q}$-divisor $H$ of $X$ such that
$B_{n}=\text{\rm H}^{0}(X,\operatorname{\mathcal{O}}_{X}(nH))T^{n}$ for all
$n\in\mathbb{N}$. Moreover, $H$ is ample and
$\operatorname{\mathcal{O}}_{X}(n)\cong\operatorname{\mathcal{O}}_{X}(nH)$ for
all $n\in\mathbb{Z}$.

######  Caution 1.6.6.

In Theorem 1.6.5, $nH$ is a $\mathbb{Q}$-divisor of $X$, so
$\operatorname{\mathcal{O}}_{X}(nH)$ is an abbreviation for
$\operatorname{\mathcal{O}}_{X}(\lfloor nH\rfloor)$ by Definition 1.3.2. Thus,
Theorem 1.6.5 asserts that
$\operatorname{\mathcal{O}}_{X}(n)\cong\operatorname{\mathcal{O}}_{X}(\lfloor
nH\rfloor)$ and $B_{n}={\rm{H}}^{0}(X,\operatorname{\mathcal{O}}_{X}(\lfloor
nH\rfloor))T^{n}$ for all $n\in\mathbb{N}$.

######  1.6.7.

We recall the following material from Section 5.4 of [CD22]. Let $B$ be an
$\mathbb{N}$-graded Noetherian normal domain such that the prime ideal
$B_{+}=\underset{i>0}{\bigoplus}B_{i}$ has height greater than $1$. Let
${\Omega}=\operatorname{{\rm Spec}}B$ and $X=\operatorname{{\rm Proj}}B$. We
shall now define a $\mathbb{Q}$-linear map $D\mapsto D^{*}$ from
$\operatorname{{\rm Div}}(X,\mathbb{Q})$ to $\operatorname{{\rm
Div}}({\Omega},\mathbb{Q})$.

Let $K({\Omega})$ and $K(X)$ be the function fields of ${\Omega}$ and $X$
respectively. Let $X^{(1)}$ be the set of homogeneous prime ideals of $B$ of
height $1$. Since $0pt{B_{+}}>1$, $X^{(1)}=\big{\\{}\,x\in
X\,:\,\dim\operatorname{\mathcal{O}}_{X,x}=1\,\big{\\}}$. For each
$\mathfrak{p}\in X^{(1)}$, $B_{\mathfrak{p}}\supset B_{(\mathfrak{p})}$ is an
extension of discrete valuations rings; let $e_{\mathfrak{p}}$ denote the
ramification index of this extension. Then
$e_{\mathfrak{p}}\in\mathbb{N}\setminus\\{0\\}$. If
$v^{X}_{\mathfrak{p}}:K(X)^{*}\to\mathbb{Z}$ and
$v^{\Omega}_{\mathfrak{p}}:K({\Omega})^{*}\to\mathbb{Z}$ denote the
normalized333The word “normalized” means that the maps $v^{X}_{\mathfrak{p}}$
and $v^{\Omega}_{\mathfrak{p}}$ are surjective. valuations of
$B_{(\mathfrak{p})}$ and $B_{\mathfrak{p}}$ respectively, then
$v^{\Omega}_{\mathfrak{p}}(\xi)=e_{\mathfrak{p}}v^{X}_{\mathfrak{p}}(\xi)$ for
all $\xi\in K(X)^{*}$. Let $C_{\mathfrak{p}}^{X}$ (resp.
$C_{\mathfrak{p}}^{\Omega}$) denote the closure of $\\{\mathfrak{p}\\}$ in $X$
(resp. in ${\Omega}$). Then $C_{\mathfrak{p}}^{X}$ (resp.
$C_{\mathfrak{p}}^{\Omega}$) is a prime divisor of $X$ (resp. of ${\Omega}$),
and every prime divisor of $X$ is a $C^{X}_{\mathfrak{p}}$ for some
$\mathfrak{p}\in X^{(1)}$. We define
$(C_{\mathfrak{p}}^{X})^{*}=e_{\mathfrak{p}}C_{\mathfrak{p}}^{\Omega}$ for
each $\mathfrak{p}\in X^{(1)}$, and extend linearly to a $\mathbb{Q}$-linear
map $\operatorname{{\rm Div}}(X,\mathbb{Q})\to\operatorname{{\rm
Div}}({\Omega},\mathbb{Q})$, $D\mapsto D^{*}$. It can be verified that the
linear map $D\mapsto D^{*}$ is injective and has the following two properties:

  1. (a)

$\big{(}\operatorname{{\rm div}}_{X}(\xi)\big{)}^{*}=\operatorname{{\rm
div}}_{\Omega}(\xi)$ for all $\xi\in K(X)^{*}$;

  2. (b)

if $f$ is a nonzero homogeneous element of $B$ and $D\in\operatorname{{\rm
Div}}(X,\mathbb{Q})$ satisfies $D^{*}=\operatorname{{\rm div}}_{\Omega}(f)$,
then $D\geq 0$ and $\operatorname{{\rm Supp}}(D)=V_{+}(f)$.

######  Lemma 1.6.8.

Let the assumptions and notation be as in Theorem 1.6.5.

  1. (a)

If $T$ and $H$ are as in Theorem 1.6.5 then $H^{*}=\operatorname{{\rm
div}}_{\Omega}(T)$.

  2. (b)

Let $T_{1},T_{2}$ be homogeneous elements of $\operatorname{{\rm Frac}}B$ of
degree $1$ and for $i=1,2$ let $H_{i}$ be the $\mathbb{Q}$-divisor of $X$ that
corresponds to $T_{i}$ as in Theorem 1.6.5. Then $T_{1}/T_{2}\in K(X)^{*}$ and
$\operatorname{{\rm div}}_{X}(T_{1}/T_{2})=H_{1}-H_{2}$. In particular,
$H_{1}\sim H_{2}$.

###### Proof.

Part (a) is stated in the first sentence on page 52 of [Dem88]. For (b), using
the notation of 1.6.7, clearly $T_{1}/T_{2}\in K(X)^{*}$ and

$(H_{1}-H_{2})^{*}=H_{1}^{*}-H_{2}^{*}=\operatorname{{\rm
div}}_{\Omega}(T_{1})-\operatorname{{\rm
div}}_{\Omega}(T_{2})=\operatorname{{\rm
div}}_{\Omega}(T_{1}/T_{2})=\big{(}\operatorname{{\rm
div}}_{X}(T_{1}/T_{2})\big{)}^{*},$

where the second equality follows from part (a) and the last one follows from
1.6.7 (a). Since $D\mapsto D^{*}$ is injective, $\operatorname{{\rm
div}}_{X}(T_{1}/T_{2})=H_{1}-H_{2}$, proving (b). ∎

The following is a special case of Lemma 5.20(a) in [CD22]. One direction was
originally shown in Remark 1.14 of [KPZ13]; we suspect both directions were
likely known at the time.

######  Lemma 1.6.9.

Let the assumptions and notation be as in Theorem 1.6.5. Fix some choice of
homogeneous $T\in\operatorname{{\rm Frac}}B$ of degree 1 as well as its
corresponding $\mathbb{Q}$-divisor $H$. Then, a cylinder $U$ of $X$ is
$H$-polar if and only if there exist $n\geq 1$ and $h\in
B_{n}\setminus\\{0\\}$ such that $U=D_{+}(h)$.

#### Cylinders and $\mathbb{P}^{1}$-fibrations on Normal Projective Surfaces

We now recall basic geometric consequence of the existence of cylinders on
normal projective surfaces. (See [CPPZ21, Section 2.1].)

######  Definition 1.6.10.

A surjective morphism $\phi:V\to B$ from a nonsingular projective surface to a
nonsingular projective curve is a called $\mathbb{P}^{1}$-fibration if a
general closed fiber of $\phi$ is isomorphic to $\mathbb{P}^{1}$. It is called
a $\mathbb{P}^{1}$-bundle if every fiber is isomorphic to $\mathbb{P}^{1}$.

######  Remark 1.6.11.

Suppose $\phi:V\to B$ be a $\mathbb{P}^{1}$-fibration and $L$ is a general
fiber of $\phi$. If $C\subset V$ is an irreducible curve such that $C\cdot
L=1$, then $C$ is a section of $\phi$.

######  1.6.12.

Assume throughout this subsection that $U\cong Z\times\mathbb{A}^{1}$ is a
cylinder contained in a normal projective surface $S$ (so $Z$ is an affine
smooth curve). The projection $\mathrm{pr}_{Z}:U\to Z$ extends to a rational
map $\rho:S\dashrightarrow\bar{Z}$ where $\bar{Z}$ is the smooth projective
model of $Z$ and the general fibers of $\rho$ in the rational sense are the
closures in $S$ of the fibers of $\mathrm{pr}_{Z}$. Since the fibers of
$\mathrm{pr}_{Z}$ are isomorphic to $\mathbb{A}^{1}$, their closures have a
unique point at infinity. This implies that either
$\rho:S\dashrightarrow\bar{Z}$ is an everywhere defined
$\mathbb{P}^{1}$-fibration having one of the irreducible components of
$S\setminus U$ as a section or it is a strictly rational map with a unique
proper base point on $S$, equal to the intersection of these closures. By
resolving the singularities of $S$ as well as, if any, the indeterminacy of
the rational map $\rho$ (by blowing-up its unique proper base point and then
all the subsequent infinitely near points), we obtain the following
commutative diagram where $\phi:W\to\bar{Z}$ is a $\mathbb{P}^{1}$-fibration
and $W$ is smooth.

$\textstyle{U\cong\mathbb{A}^{1}\times
Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}_{Z}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\phi}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bar{Z}}$

Let $C_{1},\dots,C_{n}$ be the irreducible curves in $S$ such that $S\setminus
U=\cup_{i=1}^{n}C_{i}$. Let $E_{1},\dots,E_{r},$ denote the exceptional curves
of $\alpha$ and let $\tilde{C_{i}}$ denote the proper transform of $C_{i}$ in
$W$. Then exactly one curve among
$\tilde{C}_{1},\dots,\tilde{C}_{n},E_{1},\dots,E_{r}$ (say either
$\tilde{C}_{1}$ or $E_{r}$) is a section of $\phi$ and all other curves
$\tilde{C}_{i}$ and $E_{j}$ are contained in closed fibers of $\phi$. Moreover
$\rho$ is a morphism if and only if $\tilde{C}_{1}$ is a section of $\rho$.

######  Remark 1.6.13.

With the notation of 1.6.12, since $U$ is smooth and affine and $S$ is proper
over ${\rm\bf k}$, the $i=0$ case of Theorem 4.3 in [Har72] implies that
$S\setminus U$ is connected.

The following lemma is almost identical to Lemma 3.7 of [CPW16b], and the
proof is essentially the same.

######  Lemma 1.6.14.

Let $D=\sum_{i=1}^{n}a_{i}C_{i}$ be an effective anti-canonical divisor on
normal projective surface $S$ with quotient singularities such that
$U=X\setminus\mathrm{Supp}(D)$ is a cylinder. Then the log pair $(S,D)$ is not
log-canonical. Moreover, if the map $\rho:S\dashrightarrow\bar{Z}$ as in
1.6.12 has a proper base point $s\in S$, then the log pair $(S,D)$ is not log-
canonical at $s$.

###### Proof.

We use the notation of 1.6.12 throughout and assume further that $\alpha:W\to
S$ is a resolution such that the union of the proper transform $D_{W}$ of $D$
on $W$ and of the exceptional locus
$E=\overset{r}{\underset{j=1}{\bigcup}}E_{i}$ is an SNC divisor on $W$. Since
$D$ is an effective anti-canonical $\mathbb{Q}$-divisor on $S$, we have
$K_{S}+D\sim_{\mathbb{Q}}0$, so the ramification formula for $\alpha$ reads

$K_{W}+D_{W}\sim_{\mathbb{Q}}\alpha^{*}(K_{S}+D)+\sum_{j=1}^{r}b_{j}E_{j}\sim_{\mathbb{Q}}\sum_{j=1}^{r}b_{j}E_{j}.$

If $\rho:S\to\bar{Z}$ is a morphism, hence a $\mathbb{P}^{1}$-fibration, then
there exists a unique component of $D$, say $C_{1}$, whose proper transform
$\tilde{C}_{1}$ in $W$ is a section of $\phi$ and the proper transforms of all
other irreducible components of $D$ as well as the exceptional curves of
$\alpha$ are contained in closed fibers of $\phi$. For a general fiber $L$ of
$\phi$, we obtain the equality

$0=(\sum_{j=1}^{r}b_{j}E_{j})\cdot L=(K_{W}+D_{W})\cdot L=-2+a_{1}$

which implies that $a_{1}=2$. Hence, for every point $p$ on $C_{1}$, the log
pair $(S,D)$ is not log-canonical at $p$.

If $\rho$ is not a morphism, then the irreducible components of $D_{W}$ are
contained in closed fibers of $\phi$. Consequently, a general fiber
$L\cong\mathbb{P}^{1}$ of $\phi:W\to\mathbb{P}^{1}$ is a $(0)$-curve that does
not intersect $D_{W}$ and (by 1.6.12) intersects exactly one of the curves
$E_{j}$, say $E_{r}$, transversely at a unique point. We have

$-2=K_{W}\cdot L=(K_{W}+D_{W})\cdot L=(\sum_{j=1}^{r}b_{j}E_{j})\cdot
L=b_{r}.$

Thus, $b_{r}=-2<-1$ which shows that the log pair $(S,D)$ is not log-canonical
at the point $s$. ∎

The following is Theorem 1.5 in [CPW16b].

######  Theorem 1.6.15.

Let $S_{d}$ be a del Pezzo surface of degree $d$ with at most du Val
singularities. The surface $S_{d}$ does not admit a $-K_{S_{d}}$-polar
cylinder when:

  1. (1)

$d=1$ and $S_{d}$ allows only singular points of types
$A_{1},A_{2},A_{3},D_{4}$ if any;

  2. (2)

$d=2$ and $S_{d}$ allows only singular points of type $A_{1}$ if any;

  3. (3)

$d=3$ and $S_{d}$ allows no singular point.

Otherwise, $S_{d}$ admits a $-K_{S_{d}}$-polar cylinder.

The following corollary follows easily.

######  Corollary 1.6.16.

Let $T$ be a surface listed in Theorem 1.6.15. Let $\pi:W\to T$ be the minimal
resolution of singularities of $T$ and let $E\subset W$ denote the exceptional
locus of $\pi$. If $U=W\setminus\operatorname{{\rm Supp}}(D)$ is an anti-
canonical polar cylinder of $W$, then $E\not\subseteq\operatorname{{\rm
Supp}}(D)$.

###### Proof.

If $U=W\setminus\operatorname{{\rm Supp}}(D)$ is an anti-canonical polar
cylinder of $W$ and $E\subseteq\operatorname{{\rm Supp}}(D)$, then
$\pi(U)=T\setminus\operatorname{{\rm Supp}}(\pi_{*}(D))$ is an anti-canonical
polar cylinder of $T$. This contradicts Theorem 1.6.15. ∎

We collect below some additional classical facts about
$\mathbb{P}^{1}$-fibrations which we will use in Section 4.

######  Lemma 1.6.17.

$($[Miy00, Lemma I.2.11.2]$)$ Let $f:V\to B$ be a $\mathbb{P}^{1}$-fibration.
Then $f$ has a section and there are no multiple fibers for $f$. Let
$F=\sum_{i=1}^{n}m_{i}C_{i}$ be a singular fiber of $f$. Then, the following
hold:

  1. (1)

Among the irreducible components $C_{i}$ of $F$, there is a $(-1)$-component,
say $C_{1}$. If $m_{1}=1$ for a $(-1)$-component $C_{1}$, then there is
another $(-1)$-component in the fiber $F$.

  2. (2)

Let $\sigma_{1}:V\to V_{1}$ be the contraction of a $(-1)$-component $C_{1}$
and let $f_{1}=f\cdot\sigma_{1}^{-1}$. Then $f_{1}$ is a
$\mathbb{P}^{1}$-fibration and $F_{1}={\sigma_{1}}_{*}(F)$ is a fiber of
$f_{1}$. If $F_{1}$ is a singular fiber we obtain a $\mathbb{P}^{1}$-fibration
$f_{2}:V_{2}\to B$ by contracting a $(-1)$-component of $F_{1}$. By repeating
this process, we obtain a $\mathbb{P}^{1}$-fibration for which the direct
image of $F$ is isomorphic to a smooth fiber $\mathbb{P}^{1}$.

  3. (3)

Every irreducible component of $F$ is isomorphic to $\mathbb{P}^{1}$. No three
components of $F$ share a common point, and $\operatorname{{\rm Supp}}(F)$ has
no loops.

######  Lemma 1.6.18.

Suppose $F=\sum_{i=0}^{r}a_{i}C_{i}$ (where $a_{i}\in\mathbb{N}^{+}$) is a
singular fiber of a $\mathbb{P}^{1}$-fibration $\rho:W\to\mathbb{P}^{1}$ such
that the following hold:

  * •

$C_{0}$ is a $(-n)$-curve where $n\in\mathbb{N}^{+}$ and $a_{0}=1$;

  * •

for all $i>0$, $C_{i}$ intersects $C_{0}$.

Then

  1. (a)

if $i,i^{\prime}>0$ are distinct then $C_{i}$ and $C_{i^{\prime}}$ are
disjoint;

  2. (b)

$r=n$;

  3. (c)

$C_{i}^{2}=-1$ for all $i>0$;

  4. (d)

$a_{i}=1$ for all $i$.

###### Proof.

Part (a) follows immediately from the fact that the support of a singular
fiber of a $\mathbb{P}^{1}$-fibration has no loops (as stated in Lemma
1.6.17). Since $F$ must contain a $(-1)$-curve and the image of $F$ after
contracting a $(-1)$-curve is a fiber of a new $\mathbb{P}^{1}$-fibration
which also must contain a $(-1)$-curve, there must be at least $n-1$ distinct
$(-1)$-curves among the components of $F$ and so $r\geq n-1$. Let $\kappa:W\to
W^{\prime}$ denote the contraction of $n-1$ distinct $(-1)$-curves, say
$C_{1},\dots,C_{n-1}$. Then $\kappa_{*}(C_{0})$ is a $(-1)$-curve in the fiber
$\kappa_{*}(F)$. Since $\kappa_{*}(C_{0})$ can be contracted and no three
curves in a fiber of a $\mathbb{P}^{1}$-fibration can intersect (again by
Lemma 1.6.17), there can be at most 2 additional curves that intersect
$C_{0}$. Since $\kappa_{*}(C_{0})$ is not a $0$-curve, $\kappa_{*}(F)$ is not
irreducible. This shows that $r=n$ or $r=n+1$.

Assume $r=n+1$. Then
$\kappa_{*}(F)=\bar{C}_{0}+a_{n}\bar{C}_{n}+a_{n+1}\bar{C}_{n+1}$ where we
define $\bar{C}_{i}=\kappa_{*}C_{i}$ for both $i\in\\{n,n+1\\}$. Again using
Lemma 1.6.17, it can be checked that the only possibility is that
$\bar{C}_{n}$ and $\bar{C}_{n+1}$ are $(-2)$-curves. However, since
$\kappa_{*}(F)^{2}=0$, and $a_{0}=1$, we have

$0=-1-2(a_{n}^{2}+a_{n+1}^{2})+2(a_{n}+a_{n+1})$

which is impossible for $a_{n},a_{n+1}\in\mathbb{N}^{+}$. This shows that
$r=n$ proving (b). Since contracting $\kappa_{*}(C_{0})$ must yield a
$(0)$-curve, (c) follows from (b). Since

$0=F^{2}=(\sum_{i=0}^{n}a_{i}C_{i})^{2}=-n+2\sum_{i=1}^{n}a_{i}+\sum_{i=1}^{n}a_{i}^{2}(-1)=-n+\sum_{i=1}^{n}a_{i}(2-a_{i}),$

it follows that $\sum_{i=1}^{n}a_{i}(2-a_{i})=n$. Using that
$a_{i}\in\mathbb{N}^{+}$, it follows by induction on $n$ that $a_{i}=1$ for
all $i$, proving (d). ∎

##  2 Reductions of the Conjecture

In this section, we show that Main Conjecture holds for all graded rings
$B_{a_{0},\ldots,a_{n}}$ provided that it holds for those whose associated
quasismooth hypersurfaces

$\operatorname{{\rm
Proj}}(B_{a_{0},\dots,a_{n}})=\\{\sum_{i=0}^{n}X_{i}^{a_{i}}=0\\}\subset\mathbb{P}=\mathbb{P}(w_{0},\dots,w_{n})$

are well-formed. The proof builds in part on a characterization of these rings
as being precisely those for which the $(n+1)$-tuple $(a_{0},\ldots,a_{n})$
has cotype $0$. We then recall, following a general principle introduced in
[KPZ13],the relationship between the rigidity of $B_{a_{0},\ldots,a_{n}}$ and
the non-existence of certain polar cylinders in $\operatorname{{\rm
Proj}}(B_{a_{0},\dots,a_{n}})$.

###  2.1 Reduction to well-formed hypersurfaces

######  2.1.1.

Let $n\geq 2$ and let $S=(a_{0},\dots,a_{n})\in(\mathbb{N}^{+})^{n+1}$. Let
$f=X_{0}^{a_{0}}+\dots+X_{n}^{a_{n}}\in{\rm\bf k}[X_{0},\dots,X_{n}]$. Let
$L=\operatorname{{\rm lcm}}(a_{0},\dots,a_{n})$ and let
$\deg(X_{i})=w_{i}=L/a_{i}$ for each $i\in\\{0,1,\dots,n\\}$. Then $f$ is
homogeneous of degree $L$, $B_{S}={\rm\bf
k}_{w_{0},\dots,w_{n}}[X_{0},\dots,X_{n}]/\langle f\rangle$ is an
$\mathbb{N}$-graded ring, and $\deg(x_{i})=w_{i}$ for every $i=0,\dots,n$.
Since $B_{a_{0},\ldots,a_{n}}$ is regular in codimension one and Cohen-
Macaulay (since it is a hypersurface), it follows from Serre’s Normality
Criterion that $B_{a_{0},\ldots,a_{n}}$ is normal. Consequently the variety

$X_{f}=\operatorname{{\rm Proj}}({\rm\bf
k}_{w_{0},\dots,w_{n}}[X_{0},\dots,X_{n}]/\langle f\rangle)=\operatorname{{\rm
Proj}}(B_{a_{0},\dots,a_{n}})$

is a normal quasismooth weighted projective hypersurface of degree $L$ in the
weighted projective space $\mathbb{P}(w_{0},\ldots,w_{n})$.

The following definition is introduced in [CD22].

######  Definition 2.1.2.

Let $B=\bigoplus_{i\in\mathbb{Z}}B_{i}$ be a $\mathbb{Z}$-graded domain. The
saturation index of $B$ is defined as
$e(B)=\gcd\big{\\{}\,i\in\mathbb{Z}\,:\,B_{i}\neq 0\,\big{\\}}.$ The graded
ring $B$ is saturated in codimension 1 if $e(B/\mathfrak{p})=e(B)$ for every
homogeneous height one prime ideal $\mathfrak{p}$ of $B$.

The following remark is established in Example 3.16 in [CD22].

######  Remark 2.1.3.

Let $n\geq 2$ and let $B=B_{a_{0},\dots,a_{n}}$. Then the following are
equivalent:

  * •

$B$ is saturated in codimension 1;

  * •

$\operatorname{{\rm cotype}}(a_{0},\dots,a_{n})=0$.

The following was originally proven as Proposition 2.4.25 in [Chi23b]. We
include a simpler proof here.

######  Proposition 2.1.4.

Let $f=X_{0}^{a_{0}}+\dots+X_{n}^{a_{n}}$ where $n\geq 2$ and $a_{i}\geq 1$
for all $i$. Let $L=\operatorname{{\rm lcm}}(a_{0},\dots,a_{n})$ and let
$w_{i}=L/a_{i}$. Then, the weighted hypersurface
$X_{f}\subset\mathbb{P}(w_{0},\dots,w_{n})$ is well-formed and quasismooth if
and only if $\operatorname{{\rm cotype}}(a_{0},\dots,a_{n})=0$.

###### Proof.

Since $X_{f}$ is always a quasismooth weighted hypersurface, it suffices to
prove that $X_{f}$ is well-formed if and and only if $\operatorname{{\rm
cotype}}(a_{0},\dots,a_{n})=0$. By Proposition 4.6 of [Chi23a], $X_{f}$ is
well-formed if and only if $B_{a_{0},\dots,a_{n}}$ is saturated in codimension
1. The result follows from Remark 2.1.3. ∎

We will now show that if the Main Conjecture holds for rings
$B_{a_{0},\ldots,a_{n}}$ with $\operatorname{{\rm
cotype}}(a_{0},\ldots,a_{n})=0$ then it holds in general.

######  Notation 2.1.5.

Let $B$ be a $\mathbb{Z}$-graded ring and let $A$ be a graded subring of $B$.
Define $I_{A}=\big{\\{}\,i\in\mathbb{Z}\,:\,A_{i}\neq 0\,\big{\\}}$ and define
$\mathbb{Z}(A)$ to be the subgroup of $\mathbb{Z}$ generated by $I_{A}$.

The following is the $G=\mathbb{Z}$ case of Definition 5.1 of [DFMJ17].

######  Definition 2.1.6.

Let $B$ be a $\mathbb{Z}$-graded ring. A nonzero homogeneous element $x$ of
$B$ is $\mathbb{Z}$-critical if there exists a graded subring $A\subset B$
such that $\mathbb{Z}(A)\neq\mathbb{Z}(B)$ and $B=A[x]$.

The following is a special case of Theorem 6.2 of [DFMJ17].

######  Lemma 2.1.7.

Let $B$ be a $\mathbb{Z}$-graded integral domain, and let
$D\in\operatorname{{\rm LND}}(B)$ be homogeneous. For every
$\mathbb{Z}$-critical element $x\in B$, $D^{2}(x)=0$.

The following is Lemma 3.1 of [FMJ13]. See Definition 1.1.2 for the notation
$|f|_{R}$.

######  Lemma 2.1.8.

Assume $R$ is a ${\rm\bf k}$-domain, $f\in R$, $n\geq 2$, and $f+Z^{n}$ is a
prime element of $R[Z]$. If $|f|_{R}\leq 1$, then $B=R[Z]/\langle
Z^{n}+f\rangle$ is not rigid.

We leave the following straightforward algebraic lemma to the reader. Its
proof can be found as Lemma 1.3.6 in [Chi23b].

######  Lemma 2.1.9.

Let $n\geq 2$, let $S=(a_{0},\dots,a_{n})\in(\mathbb{N}^{+})^{n+1}$, and
consider

$B_{S}={\rm\bf k}[X_{0},\dots,X_{n}]/\langle
X_{0}^{a_{0}}+\cdots+X_{n}^{a_{n}}\rangle={\rm\bf k}[x_{0},\dots,x_{n}].$

Let $m\in\mathbb{N}^{+}$, let $S^{m}=(a_{0},\dots,a_{n-1},ma_{n})$ and write

$B_{S^{m}}={\rm\bf k}[Y_{0},\dots,Y_{n}]/\langle
Y^{a_{0}}+\dots+Y_{n-1}^{a_{n-1}}+Y_{n}^{ma_{n}}\rangle={\rm\bf
k}[y_{0},\dots,y_{n}].$

Then there is a ${\rm\bf k}$-isomorphism

$B_{S}[Z]/\langle Z^{m}-x_{n}\rangle\cong B_{S^{m}}$

where $Z$ is an indeterminate over $B_{S}$.

######  Proposition 2.1.10.

Let $n\geq 2$ and suppose $(a_{0},\dots,a_{n})\in(\mathbb{N}^{+})^{n+1}$
satisfies

  * •

$a_{n}\nmid\operatorname{{\rm lcm}}(a_{0},\dots,a_{n-1})$,

  * •

there exists $m\in\mathbb{N}^{+}$ such that $B_{a_{0},\dots,a_{n-1},ma_{n}}$
is rigid.

Then $B_{a_{0},\dots,a_{n-1},a_{n}}$ is rigid.

###### Proof.

Let $R=B_{a_{0},\dots,a_{n}}={\rm\bf k}[x_{0},\dots,x_{n}]$. Let $A={\rm\bf
k}[x_{0},\dots,x_{n-1}]$ and observe that $R=A[x_{n}]$. By 2.1.1,
$\mathbb{Z}(R)=\mathbb{Z}$. Let $L^{\prime}=\operatorname{{\rm
lcm}}(a_{0},\dots,a_{n-1})$, and let $L=\operatorname{{\rm
lcm}}(a_{n},L^{\prime})$. Since $a_{n}\nmid L^{\prime}$,
$\mathbb{Z}(A)=(L/L^{\prime})\mathbb{Z}\subset\mathbb{Z}$. It follows that
$x_{n}$ is a $\mathbb{Z}$-critical element of $R$. Let $m\in\mathbb{N}^{+}$ be
such that $B_{a_{0},\dots,a_{n-1},ma_{n}}$ is rigid, let
$R^{\prime}=R[Z]/\langle Z^{m}-x_{n}\rangle$ and note that Lemma 2.1.9 implies
that $R^{\prime}\cong B_{a_{0},\dots,a_{n-1},ma_{n}}$.

Assume that $R$ is not rigid. By Proposition 1.1.5, there exists
$D\in\operatorname{{\rm LND}}(R)$ which is nonzero and homogeneous; since
$x_{n}$ is a $\mathbb{Z}$-critical element of $R$, Lemma 2.1.7 implies that
$D^{2}(x_{n})=0$. In particular, $|x_{n}|_{R}\leq 1$. By Lemma 2.1.8,
$R^{\prime}$ is not rigid, hence $B_{a_{0},\dots,a_{n-1},ma_{n}}$ is not
rigid. This is a contradiction, so $R$ is rigid. ∎

######  Notation 2.1.11.

For each $n\geq 2$, define

$\Gamma_{n}=\big{\\{}\,(a_{0},\dots,a_{n})\in(\mathbb{N}^{+})^{n+1}\,:\,\text{$\min(a_{0},\dots,a_{n})>1$
and at most one $i$ satisfies $a_{i}=2$}\,\big{\\}}.$

and consider the statements:

$\displaystyle P(n):$ $\displaystyle\text{ $B_{a_{0},\dots,a_{n}}$ is rigid
for all $(a_{0},\dots,a_{n})\in\Gamma_{n}$ }.$ $\displaystyle P(n,i):$
$B_{a_{0},\dots,a_{n}}$ is rigid for all $(a_{0},\dots,a_{n})\in\Gamma_{n}$
satisfying $\operatorname{{\rm cotype}}(a_{0},\dots,a_{n})=i$.

The following appears as Proposition 4.9 of [CD20] with slightly different
notation.

######  Proposition 2.1.12.

Let $n\geq 2$, let $S,S^{*},S^{\prime}\in(\mathbb{N}^{+})^{n+1}$ and let
$i\in\\{0,\dots,n\\}$.

  1. (a)

If $S^{\prime}\leq^{i}S$ and $B_{S^{\prime}}$ is rigid then $B_{S}$ is rigid.

  2. (b)

If $S^{\prime}<^{i}S$ and $S^{\prime}<^{i}S^{*}$, then $B_{S}$ is rigid if and
only if $B_{S^{*}}$ is rigid.

######  Theorem 2.1.13.

Let $n\geq 3$. If $P(n-1)$ and $P(n,0)$ hold, then $P(n)$ holds.

###### Proof.

Suppose $S=(a_{0},\dots,a_{n})\in\Gamma_{n}$ and assume that $P(n-1)$ and
$P(n,0)$ hold. We must show that $B_{S}$ is rigid. If $\operatorname{{\rm
cotype}}(S)=0$ we are done, since $P(n,0)$ holds by assumption. Assume
henceforth that $\operatorname{{\rm cotype}}(S)\geq 1$.

Suppose $\operatorname{{\rm cotype}}(S)\geq 2$. By contradiction, assume that
there exists $D\in\operatorname{{\rm LND}}(B_{S})\setminus\\{0\\}$. Without
loss of generality, by Proposition 1.1.5 together with Theorem 1.1.3 (d) we
may assume $D$ is homogeneous and irreducible. For each $i\in\\{0,\dots,n\\}$,
let $w_{i}=\deg(x_{i})$ be as in 2.1.1. Let $H_{i}=\langle
w_{0},\dots,w_{i-1},\hat{w_{i}},w_{i+1},\dots,w_{n}\rangle\subseteq\mathbb{Z}$
and $S_{i}=(a_{0},\dots,a_{i-1},\hat{a_{i}},a_{i+1},\dots a_{n})$. Since
$\operatorname{{\rm cotype}}(S)\geq 2$, there exist distinct
$j,k\in\\{0,\dots,n\\}$ such that $H_{j}\subset\mathbb{Z}$ and
$H_{k}\subset\mathbb{Z}$. By Corollary 6.3 (b) of [DFMJ17], either
$x_{j}\in\ker(D)$ or $x_{k}\in\ker(D)$. Without loss of generality, we may
assume $j=n$ or $k=n$, so that $x_{n}\in\ker(D)$. Then, since $D$ is
irreducible, $D$ induces a nonzero locally nilpotent derivation on
$B_{S}/\langle x_{n}\rangle\cong B_{S_{n}}$. But since $S\in\Gamma_{n}$, it
follows that $S_{n}\in\Gamma_{n-1}$. This is a contradiction since $P(n-1)$
holds. So $B_{S}$ is rigid when $\operatorname{{\rm cotype}}(S)\geq 2$.

Finally, assume $\operatorname{{\rm cotype}}(S)=1$. Then, up to permuting the
$a_{i}$, we may arrange that $a_{n}\nmid\operatorname{{\rm
lcm}}(a_{0},\dots,a_{n-1})$. Let $L=\operatorname{{\rm
lcm}}(a_{0},\dots,a_{n-1})$. By Proposition 2.1.10, it suffices to prove that
$B_{a_{0},\dots,a_{n-1},a_{n}L}$ is rigid. We have
$(a_{0},\dots,a_{n-1},L)<^{n}(a_{0},\dots,a_{n-1},a_{n}L)$ and
$\operatorname{{\rm cotype}}(a_{0},\dots,a_{n-1},L)=0$. Since $L>2$,
$(a_{0},\dots,a_{n-1},L)\in\Gamma_{n}$. By assumption, $P(n,0)$ holds so
$B_{a_{0},\dots,a_{n-1},L}$ is rigid and hence $B_{a_{0},\dots,a_{n}L}$ is
rigid by Proposition 2.1.12 (a). ∎

Combining Theorem 2.1.13 with the fact that the base case $P(2)$ holds by
[KZ00b, Lemma 4], we derive by induction the following corollary:

######  Corollary 2.1.14.

The Main Conjecture holds if and only if for every $n\geq 3$ and every
$(a_{0},\ldots a_{n})\in\Gamma_{n}$ satisfying $\operatorname{{\rm
cotype}}(a_{0},\ldots,a_{n})=0$ the ring $B_{a_{0},\ldots,a_{n}}$ is rigid.

Corollary 2.1.14 reduces the proof of the Main Conjecture to the class of
graded rings $B_{a_{0},\ldots,a_{n}}$ where
$(a_{0},\ldots,a_{n})\in\Gamma_{n}$ and the associated quasismooth
hypersurface $\operatorname{{\rm Proj}}(B_{a_{0},\dots,a_{n}})$ is well-
formed.

###  2.2 Reduction to the non-existence of polar cylinders

The following is a special case of Theorem 1.2 of [CD22], which in turn is a
generalization of Theorem 0.6 in [KPZ13].

######  Theorem 2.2.1.

Let $B=\bigoplus_{i\in\mathbb{N}}B_{i}$ be an $\mathbb{N}$-graded affine
$\mathbb{C}$-domain such that the transcendence degree of $B$ over $B_{0}$ is
at least $2$. The following are equivalent.

  1. (a)

There exists $d\geq 1$ such that $B^{(d)}$ is not rigid.

  2. (b)

There exists a homogeneous element $h\in B\setminus\\{0\\}$ of positive degree
such that the open subset $D_{+}(h)$ of $\operatorname{{\rm Proj}}B$ is a
cylinder.

Moreover, if $B$ is normal and is saturated in codimension 1 then the above
conditions are equivalent to

  1. (c)

$B$ is not rigid.

######  Corollary 2.2.2.

Let $n\geq 2$ and suppose $\operatorname{{\rm cotype}}(a_{0},\dots,a_{n})=0$.
The following are equivalent:

  * •

$B_{a_{0},\dots,a_{n}}$ is rigid;

  * •

$(B_{a_{0},\dots,a_{n}})^{(d)}$ is rigid for all $d\in\mathbb{N}^{+}$.

###### Proof.

It suffices to check that $B_{a_{0},\dots,a_{n}}$ satisfies the assumptions of
Theorem 2.2.1. The only non-trivial things to check are normality and
saturation in codimension 1. Normality is shown in 2.1.1 and Remark 2.1.3
implies that $B_{a_{0},\dots,a_{n}}$ is saturated in codimension 1. ∎

######  Remark 2.2.3.

The assumption that $\operatorname{{\rm cotype}}(a_{0},\dots,a_{n})=0$ is
necessary for Corollary 2.2.2 to hold. We will see later that $B_{2,3,3,4}$ is
rigid whereas $(B_{2,3,3,4})^{(2)}\cong B_{2,3,3,2}$ is not rigid (as
discussed in the Introduction).

The following appears as Lemma 4.1.8 in [Chi23b]. The proof given here is
simpler.

######  Lemma 2.2.4.

With the notation of 1.6.7, let $B=B_{a_{0},\dots,a_{n}}$ where
$\operatorname{{\rm cotype}}(a_{0},\dots,a_{n})=0$ and let
$\mathfrak{p}_{i}=\langle x_{i}\rangle\lhd B$. For each $i=0,\dots,n$,
$\mathfrak{p}_{i}\in X^{(1)}$ and $e_{\mathfrak{p}_{i}}=1$.

###### Proof.

By Remark 2.1.3, $B$ is saturated in codimension 1. By Corollary 9.4 in
[Dai23], $e_{\mathfrak{q}}=1$ for all $\mathfrak{q}\in X^{(1)}$. In
particular, the result is true when $\mathfrak{q}=\mathfrak{p}_{i}$. ∎

######  Theorem 2.2.5.

Let $n\geq 2$ and consider $(a_{0},\dots,a_{n})\in(\mathbb{N}^{+})^{n+1}$ of
cotype $0$. Let $B=B_{a_{0},\dots,a_{n}}$, let $X=\operatorname{{\rm
Proj}}(B)$, and let $\alpha$ be the amplitude of $X$. Then,

  1. (a)

$\omega_{X}\cong\operatorname{\mathcal{O}}_{X}(\alpha)$.

  2. (b)

Let $T$ be a homogeneous element of $\operatorname{{\rm Frac}}B$ of degree 1
and let $H$ be the unique $\mathbb{Q}$-divisor of $X$ determined by $T$ as in
Theorem 1.6.5. Then $H\in\operatorname{{\rm Div}}(X)$, $K_{X}\sim\alpha H$ and
$K_{X}$ is $\mathbb{Q}$-Cartier.

  3. (c)

If $B_{\alpha}\neq 0$, then $X$ does not contain a cylinder and $B^{(d)}$ is
rigid for all $d\geq 1$. In particular, this holds when $\alpha=0$.

  4. (d)

Assume that $\alpha\neq 0$ and define
$s=\frac{\alpha}{|\alpha|}\in\\{1,-1\\}$. Then $sK_{X}$ is ample and the
following are equivalent:

    1. (i)

$B$ is not rigid;

    2. (ii)

for some $d\geq 1$, $B^{(d)}$ is not rigid;

    3. (iii)

there exists a homogeneous element $h\in B\setminus\\{0\\}$ of positive degree
such that the open subset $D_{+}(h)$ of $\operatorname{{\rm Proj}}B$ is a
cylinder;

    4. (iv)

there exists a $(sK_{X})$-polar cylinder of $X$.

###### Proof.

First, since $\operatorname{{\rm cotype}}(a_{0},\dots,a_{n})=0$, $X$ is a
well-formed quasismooth weighted complete intersection (by Proposition 2.1.4)
so (a) follows from Theorem 1.4.9.

We prove (b). By Lemma 1.6.8 (b), if assertion (b) is true for one particular
choice of a homogeneous element $T$ in $\operatorname{{\rm Frac}}B$ of degree
1, then it is true for every choice of such a $T$. As such, we assume
henceforth that $T=\prod_{i=0}^{n}x_{i}^{b_{i}}$ where
$b_{0},\dots,b_{n}\in\mathbb{Z}$ are such that
$\sum_{i=0}^{n}b_{i}\deg(x_{i})=1$. Then, by Lemma 1.6.8 (a),
$H^{*}=\operatorname{{\rm
div}}_{\Omega}(T)=\sum_{i=0}^{n}b_{i}C^{\Omega}_{\mathfrak{p}_{i}}$. Letting
$E=\sum_{i=0}^{n}b_{i}C^{X}_{\mathfrak{p}_{i}}\in\operatorname{{\rm Div}}(X)$
we obtain that
$E^{*}=\sum_{i=0}^{n}b_{i}C^{\Omega}_{\mathfrak{p}_{i}}=\operatorname{{\rm
div}}_{\Omega}(T)=H^{*}$ (since by Lemma 2.2.4, $e_{\mathfrak{p}_{i}}=1$ for
all $i=0,\dots,n$). Since $D\mapsto D^{*}$ is injective, we obtain that $E=H$,
so $H\in\operatorname{{\rm Div}}(X)$. By (a) together with Theorem 1.6.5,
$\operatorname{\mathcal{O}}_{X}(K_{X})\cong\omega_{X}\cong\operatorname{\mathcal{O}}_{X}(\alpha)\cong\operatorname{\mathcal{O}}_{X}(\alpha
H)$ and so $K_{X}\sim\alpha H$. Finally, $K_{X}$ is $\mathbb{Q}$-Cartier by
Proposition 1.4.8 (e), proving (b).

For (c), assume that $B_{\alpha}\neq\\{0\\}$. Let $\tilde{X}$ be a resolution
of singularities of $X$. Since $X$ has rational singularities,
$p_{g}(\tilde{X})=h^{2}(\tilde{X},\operatorname{\mathcal{O}}_{\tilde{X}})=h^{2}(X,\operatorname{\mathcal{O}}_{X})$
and Lemma 1.4.10 gives that

$p_{g}(\tilde{X})=h^{2}(\tilde{X},\operatorname{\mathcal{O}}_{\tilde{X}})=h^{2}(X,\operatorname{\mathcal{O}}_{X})=\dim_{\mathbb{C}}B_{\alpha}\neq
0.$

In particular, $\tilde{X}$ and $X$ are not ruled, so $X$ certainly does not
contain a cylinder and thus $B$ is rigid by Theorem 2.2.1. By Corollary 2.2.2,
$B^{(d)}$ is rigid for all $d\geq 1$. This proves (c).

We prove (d). Let $H$ be as in part (b). Since $H$ is ample, $rH$ is ample for
every integer $r>0$. Since $K_{X}\sim\alpha H$, $sK_{X}\sim|\alpha|H$ is
ample. By Remark 2.1.3, $B$ is saturated in codimension 1. By Theorem 2.2.1,
(i), (ii) and (iii) are equivalent. Assume (iii) holds, and let $h\in
B\setminus\\{0\\}$ be a homogeneous element of positive degree such that
$D_{+}(h)\subset X$ is a cylinder. By Lemma 1.6.9, $D_{+}(h)$ is $H$-polar and
since $sK_{X}\sim|\alpha|H$ it is also $(sK_{X})$-polar by Remark 1.6.3, so
(iii) implies (iv). Conversely, assume (iv) holds. Let $U$ be an
$(sK_{X})$-polar cylinder of $X$. Since $sK_{X}\sim|\alpha|H$, $U$ is
$H$-polar by Remark 1.6.3. By Lemma 1.6.9, (iii) holds and so (iv) implies
(iii). This proves (d). ∎

##  3 Proof of the Main Conjecture in dimension $3$: simpler cotype 0 cases

######  Notation 3.0.1.

Let $L=\operatorname{{\rm lcm}}(a_{0},a_{1},a_{2},a_{3})$ and let
$\alpha=L-\sum_{i=0}^{n}\frac{L}{a_{i}}$. (Note that $\alpha\in\mathbb{Z}$.)
Define the following sets:

$\displaystyle\Gamma_{3}^{+}$
$\displaystyle=\big{\\{}\,(a_{0},a_{1},a_{2},a_{3})\in\Gamma_{3}\,:\,\operatorname{{\rm
cotype}}(a_{0},a_{1},a_{2},a_{3})=0\text{ and }\alpha\geq 0\,\big{\\}}$
$\displaystyle\Gamma_{3}^{-}$
$\displaystyle=\big{\\{}\,(a_{0},a_{1},a_{2},a_{3})\in\Gamma_{3}\,:\,\operatorname{{\rm
cotype}}(a_{0},a_{1},a_{2},a_{3})=0\text{ and }\alpha<0\,\big{\\}}.$

Theorem 2.1.13 and Corollary 2.1.14 have the immediate consequence that to
prove the $n=3$ case of the Main Conjecture , it suffices to show that
$B_{a_{0},a_{1},a_{2},a_{3}}$ is rigid for all
$(a_{0},a_{1},a_{2},a_{3})\in\Gamma_{3}^{+}\cup\Gamma_{3}^{-}$.

###  3.1 The case of non-negative amplitude

In this section, we show that every ring $B_{a_{0},a_{1},a_{2},a_{3}}$ with
$(a_{0},a_{1},a_{2},a_{3})\in\Gamma_{3}^{+}$ is rigid. We will use the
following classical argument whose proof is similar to that of Lemma 1.6.14.

######  Lemma 3.1.1.

Let $X$ be a normal projective surface with quotient singularities such that
$K_{X}$ is pseudo-effective. Then $X$ does not contain any cylinder.

###### Proof.

Assume $X$ contains a cylinder $U\cong Z\times\mathbb{A}^{1}$. Let
$\sigma:\tilde{X}\to X$ be a resolution of indeterminacy of the induced
rational map $\rho:X\dasharrow\bar{Z}$, where $\bar{Z}$ is the smooth
projective model of Z such that $\sigma:\tilde{X}\to X$ is also a log-
resolution of singularities of $X$. Let $\tilde{\rho}:\tilde{X}\to\bar{Z}$ be
the induced $\mathbb{P}^{1}$-fibration and let $C\subset\tilde{X}$ be the
unique curve contained in $\sigma^{-1}(X\setminus U)$ which is a section of
$\tilde{\rho}$. Consider the ramification formula
$K_{\tilde{X}}=\sigma^{*}K_{X}+\sum_{i=1}^{r}b_{i}E_{i}$ for $\sigma$, where
the $E_{i}$ are the exceptional curves of $\sigma$ and where $b_{i}>-1$ since
by assumption $X$ has quotient, hence klt, singularities. For a general closed
fiber $F\cong\mathbb{P}^{1}$ of $\tilde{\rho}$, we have

$-2=K_{\tilde{X}}\cdot F=(\sigma^{*}K_{X}+\sum_{i=1}^{r}b_{i}E_{i})\cdot
F=K_{X}\cdot\sigma_{*}F+\sum_{i=1}^{r}b_{i}(E_{i}\cdot F).$

Since $K_{X}$ is pseudo-effective, $K_{X}\cdot\sigma_{*}F\geq 0$. Since
$E_{i}\cdot F\geq 0$ for all $E_{i}$, the only possibility is that $C=E_{i}$
for some $i$. Assuming that $C=E_{1}$, we have, since $E_{1}\cdot F=1$, that
$b_{1}\leq-2$, which is impossible. ∎

######  Proposition 3.1.2.

Let $(a_{0},a_{1},a_{2},a_{3})\in\Gamma_{3}^{+}$ and let
$B=B_{a_{0},a_{1},a_{2},a_{3}}$. Then $B^{(d)}$ is rigid for every $d\geq 1$.

###### Proof.

Since $\operatorname{{\rm cotype}}(a_{0},a_{1},a_{2},a_{3})=0$, Proposition
2.1.4 implies that $X=\operatorname{{\rm Proj}}(B)$ is a well-formed
quasismooth weighted complete intersection. By Proposition 1.4.8, $X$ is
normal and has cyclic quotient singularities. Moreover, Theorem 2.2.5 (using
the notation of said theorem) implies that $H\in\operatorname{{\rm Div}}(X)$,
$H$ is ample, and $K_{X}=\alpha H$, where, by assumption, $\alpha\geq 0$. So,
$K_{X}$ is either trivial or ample and in particular is pseudo-effective. Now
suppose that there exists $d\geq 1$ such that $B^{(d)}$ is not rigid. Then, by
Theorem 2.2.1, there exists a homogeneous element $h\in B\setminus\\{0\\}$ of
positive degree such that the open subset $D_{+}(h)$ of $\operatorname{{\rm
Proj}}B$ is a cylinder, which is impossible by Lemma 3.1.1. (Note that the
$\alpha=0$ case also follows directly from Theorem 2.2.5 (c).) ∎

###  3.2 The case of negative amplitude: known results

######  Proposition 3.2.1.

Let $(a_{0},a_{1},a_{2},a_{3})\in\Gamma_{3}^{-}$, let
$B=B_{a_{0},a_{1},a_{2},a_{3}}$ and let $X=\operatorname{{\rm Proj}}B$. Then
the following hold:

  1. (a)

$X$ is a del Pezzo surface.

  2. (b)

There exists $d\in\mathbb{N}^{+}$ such that $B^{(d)}$ is not rigid if and only
if $X$ contains a $-K_{X}$-polar cylinder.

###### Proof.

By Theorem 2.2.5 (d) and (b), $-K_{X}$ is an ample $\mathbb{Q}$-Cartier
divisor. Since $X$ has quotient singularities (by Proposition 1.4.8), $X$ is a
del Pezzo surface. This proves (a). Part (b) follows from the equivalence of
(ii) and (iv) in Theorem 2.2.5 (d). ∎

The following lemma taken from [Chi23b, Lemma 4.2.4] further reduces the proof
of the $n=3$ case of the Main Conjecture to the study of finitely many
specific cases:

######  Lemma 3.2.2.

Up to a permutation of $a_{0},a_{1},a_{2},a_{3}$, the set $\Gamma_{3}^{-}$
consists of the following $4$-tuples:

$\\{(2,3,3,6),\ \ (2,3,6,6),\ \ (2,4,4,4),\ \ (3,3,3,3),\ \ (3,3,4,4),\ \
(3,3,5,5),\ \ (2,3,4,12),\ \ (2,3,5,30)\\}.$

#### Rigidity of $B_{3,3,4,4}$ and $B_{3,3,5,5}$.

######  Proposition 3.2.3.

Let $B=B_{3,3,c,c}$ where $c=4$ or $5$, and let $X=\operatorname{{\rm
Proj}}B$. Then $X$ does not contain a $-K_{X}$-polar cylinder. Moreover, for
every $d\geq 1$, $B^{(d)}$ is rigid.

###### Proof.

Lemmas 4.1 and 5.1 in [CPS10] imply in particular that for every effective
anti-canonical $\mathbb{Q}$-divisor $D$ on $X$, the log pair $(X,D)$ is log-
canonical. Thus, by Lemma 1.6.14, $X$ does not contain any $-K_{X}$-polar
cylinder and so, by Proposition 3.2.1 (b), $B^{(d)}$ is rigid for every $d\geq
1$. ∎

#### Rigidity of $B_{3,3,3,3}$, $B_{2,4,4,4}$, $B_{2,3,6,6}$ and $B_{2,3,3,6}$
.

######  Lemma 3.2.4.

Let $S\in\\{(3,3,3,3),(2,4,4,4),(2,3,6,6)\\}$. Then $\operatorname{{\rm
Proj}}B_{S}$ is a nonsingular del Pezzo surface.

###### Proof.

In all three cases the proof is identical. We have that $\operatorname{{\rm
cotype}}(S)=0$, so Proposition 2.1.4 implies that $\operatorname{{\rm
Proj}}B_{S}$ is a well-formed quasismooth hypersurface. Applying Propositions
1.4.5 and 1.4.6, we obtain that $\operatorname{{\rm Sing}}(\operatorname{{\rm
Proj}}B_{S})=\emptyset$. Finally, $\operatorname{{\rm Proj}}B_{S}$ is a del
Pezzo surface by Proposition 3.2.1. ∎

######  Lemma 3.2.5.

$($[Chi23b, Corollary 3.2.23]$)$ Let $X=\operatorname{{\rm Proj}}B_{2,3,3,6}$.
Then $X$ has exactly 3 singular points of type $A_{1}$. In particular $X$ is
Gorenstein.

###### Proof.

We may view $X$ as a closed subvariety of $\mathbb{P}(3,2,2,1)$. Let
$\phi_{1}:X\to Y=\mathbb{P}(3,2,1)=\operatorname{{\rm
Proj}}(\mathbb{C}_{3,2,1}[X_{0},X_{2},X_{3}])$ be the projection morphism onto
the first, third and fourth coordinates. Let $U=D_{+}(gx_{2})\subset X$ where
$g=x_{0}^{2}+x_{2}^{3}+x_{3}^{6}$. Let
$G=X_{0}^{2}+X_{2}^{3}+X_{3}^{6}\in\mathbb{C}_{3,2,1}[X_{0},X_{2},X_{3}]$. It
can be verified that $\phi_{1}|_{U}:U\to D_{+}(GX_{2})\subset Y$ is étale. Let
$y=[0:1:0]\in Y$. For each of the distinct points $x\in\operatorname{{\rm
Sing}}(X)=\\{[0:-1:1:0],[0:\zeta_{6}:1:0],[0:\zeta_{6}^{-1}:1:0]\\}$ we have
$\phi_{1}(x)=y$, so (by Proposition 17.6.3 in [Gro67])
$\widehat{\operatorname{\mathcal{O}}}_{X,x}\cong\widehat{\operatorname{\mathcal{O}}}_{Y,y}$.
Since $y\in\mathbb{P}(3,2,1)$ is a Du Val singularity of type $A_{1}$, it
follows that each of the three singular points of $X$ is of type $A_{1}$ as
well. ∎

######  Lemma 3.2.6.

Let $B$ be any of $B_{3,3,3,3}$, $B_{2,4,4,4}$, $B_{2,3,6,6}$, $B_{2,3,3,6}$.
Let $X=\operatorname{{\rm Proj}}B$ and let $\alpha$ be the amplitude of $X$.
Then,

  1. (a)

$K_{X}$ is Cartier;

  2. (b)

$K_{X}^{2}=\dim_{\mathbb{C}}B_{-\alpha}-1$

###### Proof.

We prove (a). By Lemma 3.2.4, $\operatorname{{\rm Proj}}B_{3,3,3,3}$,
$\operatorname{{\rm Proj}}B_{2,4,4,4}$ and $\operatorname{{\rm
Proj}}B_{2,3,6,6}$ are nonsingular and by Lemma 3.2.5, $\operatorname{{\rm
Proj}}B_{2,3,3,6}$ is Gorenstein, proving (a).

For (b), consider a homogeneous $T\in K(X)^{*}$ of degree $1$ and let $H$ be
the unique $\mathbb{Q}$-divisor of $X$ determined by $T$ as in Theorem 1.6.5.
By Theorem 2.2.5 (b), $H\in\operatorname{{\rm Div}}(X)$ and $K_{X}=\alpha H$
where $H$ is ample. By (a), $K_{X}$ is Cartier. By Lemma 1.5.4, we have

$(K_{X})^{2}=\chi(\operatorname{\mathcal{O}}_{X})-2\chi(\operatorname{\mathcal{O}}_{X}(K_{X}))+\chi(\operatorname{\mathcal{O}}_{X}(K_{X})\otimes_{\operatorname{\mathcal{O}}_{X}}\operatorname{\mathcal{O}}_{X}(K_{X}))=\chi(\operatorname{\mathcal{O}}_{X})-2\chi(\operatorname{\mathcal{O}}_{X}(\alpha))+\chi(\operatorname{\mathcal{O}}_{X}(2\alpha)),$
(1)

where we use that $K_{X}$ is Cartier together with Theorem 1.6.5 in the second
equality. Define $a_{i}=\dim_{\mathbb{C}}B_{i}$ for $i\in\mathbb{Z}$ and note
that $a_{0}=1$, and $a_{\alpha}=a_{2\alpha}=0$ since $\alpha<0$. Applying
Lemma 1.4.10, we find

$\displaystyle\chi(\operatorname{\mathcal{O}}_{X})$
$\displaystyle=a_{0}+a_{\alpha}=1,$
$\displaystyle\chi(\operatorname{\mathcal{O}}_{X}(\alpha))$
$\displaystyle=a_{\alpha}+a_{0}=1,$
$\displaystyle\chi(\operatorname{\mathcal{O}}_{X}(2\alpha))$
$\displaystyle=a_{2\alpha}+a_{-\alpha}=a_{-\alpha}.$

Plugging these values into (1), we obtain $K_{X}^{2}=a_{-\alpha}-1$ proving
(b). ∎

######  Proposition 3.2.7.

Let $B=B_{a_{0},a_{1},a_{2},a_{3}}$ where
$(a_{0},a_{1},a_{2},a_{3})\in\\{(2,3,3,6),(2,3,6,6),(2,4,4,4),(3,3,3,3)\\}$.
Then $B^{(d)}$ is rigid for all $d\in\mathbb{N}^{+}$.

###### Proof.

Let $X=\operatorname{{\rm Proj}}B$. If $(a_{0},a_{1},a_{2},a_{3})=(2,3,3,6)$
then by Lemmas 3.2.5 and 3.2.6, $X$ is a degree 2 del Pezzo surface with at
most $A_{1}$ singularities. In the other three cases, $X=\operatorname{{\rm
Proj}}B$ is nonsingular by Lemma 3.2.4 and by Lemma 3.2.6 can be seen to
satisfy $K_{X}^{2}\leq 3$. Consequently, $X$ satisfies one of conditions
(i)-(iii) in Theorem 1.6.15 and hence $X$ does not contain a $-K_{X}$ polar
cylinder. By Theorem 3.2.1, $B^{(d)}$ is rigid for all $d\in\mathbb{N}^{+}$. ∎

##  4 Proof of the Main Conjecture in dimension $3$: the remaining cases of
cotype $0$ and negative amplitude

To finish the proof of the Main Conjecture in dimension 3, it remains to prove
that $B_{2,3,5,30}$ and $B_{2,3,4,12}$ are rigid.

###  4.1 Rigidity of $B_{2,3,5,30}$

######  4.1.1.

Let $B=B_{2,3,5,30}$ and let $S=\operatorname{{\rm
Proj}}B\subset\mathbb{P}(15,10,6,1)$. Consider the homogeneous element
$T=x_{3}\in\operatorname{{\rm Frac}}(B)$. Then
$\Delta=V_{+}(x_{3})\in\operatorname{{\rm Div}}(S)$ is the unique
$\mathbb{Q}$-divisor satisfying $B=\bigoplus_{n\in\mathbb{N}}B_{n}$ where
$B_{n}=\text{\rm H}^{0}(S,\operatorname{\mathcal{O}}_{S}(n\Delta))T^{n}$ for
all $n\in\mathbb{N}$ (as defined in Theorem 1.6.5). Considering $\Delta$ as a
closed subvariety of $S$, we find $\Delta\cong\mathbb{P}^{1}.$

Since $\operatorname{{\rm cotype}}(2,3,5,30)=0$, Proposition 2.1.4 implies
that $S$ is a well-formed hypersurface. Since (by Proposition 1.4.8) $S$ is
normal and Cohen-Macaulay, we have
$\omega_{S}\overset{\ref{NormalCMCoincide}}{\cong}\omega_{S}^{o}\overset{\ref{dualizing}}{\cong}\operatorname{\mathcal{O}}_{S}(-2)\overset{\ref{Demazure}}{\cong}\operatorname{\mathcal{O}}_{S}(-2\Delta)$.

######  Remarks 4.1.2.

The following hold:

  1. (a)

$2\Delta$ is an ample anti-canonical divisor of $S$;

  2. (b)

$(K_{S})^{2}=\frac{2}{15}$.

In particular, $S$ is a singular del Pezzo surface.

###### Proof.

Part (a) is clear in view of 4.1.1, so we need only prove (b). Note that
$15\Delta$ is a Cartier divisor on $S$ and so
$(2\Delta)^{2}=(-K_{S})^{2}=\frac{1}{225}(15\Delta)^{2}=\frac{1}{225}(-15\Delta)^{2}$.
It suffices to show that $(-15\Delta)^{2}=30$. By Lemma 1.5.4, we have

$\begin{split}(-15\Delta)^{2}&=\chi(\operatorname{\mathcal{O}}_{S})-2\chi(\operatorname{\mathcal{O}}_{S}(-30))+\chi(\operatorname{\mathcal{O}}_{S}(-60))\\\
&=h^{0}(S,\operatorname{\mathcal{O}}_{S})-h^{1}(S,\operatorname{\mathcal{O}}_{S})+h^{2}(S,\operatorname{\mathcal{O}}_{S})\\\
&-2[h^{0}(S,\operatorname{\mathcal{O}}_{S}(-30))-h^{1}(S,\operatorname{\mathcal{O}}_{S}(-30))+h^{2}(S,\operatorname{\mathcal{O}}_{S}(-30))]\\\
&+h^{0}(S,\operatorname{\mathcal{O}}_{S}(-60))-h^{1}(S,\operatorname{\mathcal{O}}_{S}(-60))+h^{2}(S,\operatorname{\mathcal{O}}_{S}(-60))\\\
&=1-2\dim_{\mathbb{C}}B_{28}+\dim_{\mathbb{C}}B_{58}\\\
&=1-2(15)+59=30,\end{split}$ (2)

where we use Lemma 1.4.10 throughout. (Note that in the calculations of
$\dim_{\mathbb{C}}B_{28}$ and $\dim_{\mathbb{C}}B_{58}$, we use that $B_{28}$
has
$\big{\\{}\,xy^{a}z^{b}t^{c}\,:\,10a+6b+c=13\,\big{\\}}\cup\big{\\{}\,y^{a}z^{b}t^{c}\,:\,10a+6b+c=28\,\big{\\}}$
as a ($\mathbb{C}$-module) basis and $B_{58}$ has
$\big{\\{}\,xy^{a}z^{b}t^{c}\,:\,10a+6b+c=43\,\big{\\}}\cup\big{\\{}\,y^{a}z^{b}t^{c}\,:\,10a+6b+c=58\,\big{\\}}$
as a basis, where in all cases $a,b,c\in\mathbb{N}$.) ∎

######  Proposition 4.1.3.

The singular points of $S\subset\mathbb{P}(15,10,6,1)$ are
$\\{[0:1:-1:0],[1:0:-1:0],[1:-1:0:0]\\}$. Moroever,

  1. (a)

$[1:-1:0:0]$ is a $\frac{1}{5}(1,1)$ singularity.

  2. (b)

$[1:0:-1:0]$ is a $\frac{1}{3}(1,1)$ singularity;

  3. (c)

$[0:1:-1:0]$ is a $\frac{1}{2}(1,1)$ singularity;

  4. (d)

For each $P\in\operatorname{{\rm Sing}}(S)$, $\operatorname{{\rm
mult}}_{P}(\Delta)=1$.

###### Proof.

The singular points can be identified by combining Propositions 1.4.5 and
1.4.6.

We will prove (a) and (b). Let $\mathbb{P}=\mathbb{P}(15,10,6,1)$. First,
observe that both points $[1:0:-1:0],[1:-1:0:0]\in S\subseteq\mathbb{P}$ are
contained in the affine open subset $U=D_{+}(X_{0})\subset\mathbb{P}$. The
open subset $U$ is the algebraic quotient of
$Y=\mathbb{C}^{*}\times\mathbb{C}^{3}$ by the $\mathbb{C}^{*}$-action
$\lambda\cdot(y_{0},y_{1},y_{2},y_{3})=(\lambda^{15}y_{0},\lambda^{10}y_{1},\lambda^{6}y_{2},\lambda
y_{3})$. Consider the étale Galois cover
$\phi:\mathbb{C}^{*}\to\mathbb{C}^{*}$ where $u\mapsto u^{15}=y_{0}$ so that
$\bar{\phi}=\phi\times\operatorname{{\rm
Id}}:X=\mathbb{C}^{*}\times\mathbb{C}^{3}\to\mathbb{C}^{*}\times\mathbb{C}^{3}=Y$
is a degree 15 cover with Galois group $\Gamma=\mathbb{Z}_{15}$. Consider the
$\mathbb{C}^{*}$ action on $X$ given by $\mu\cdot(u,x_{1},x_{2},x_{3})=(\mu
u,\mu^{10}x_{1},\mu^{6}x_{2},\mu x_{3})$ and let $V$ denote the algebraic
quotient of $X$ by $\mu$. Then $\bar{\phi}\circ\lambda=\mu\circ\bar{\phi}$,
giving the following commutative diagram.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\phi}}$$\scriptstyle{\mu}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\phi}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\textstyle{Y}$

The quotient $V/\bar{\phi}=(X/\mu)/\bar{\phi}$ is defined and is equal to $U$.
Observe that the equivalence class $[u:x_{1}:x_{2}:x_{3}]\in(X/\mu)$
corresponds to the point $(a,b,c)\in\mathbb{C}^{3}$ where
$a=u^{-10}x_{1},b=u^{-6}x_{2},c=u^{-1}x_{3}$. Now, let $\epsilon$ be a
generator of $\Gamma$. Then the action of $\Gamma$ on
$(a,b,c)\in\mathbb{C}^{3}$ is determined by
$\epsilon\cdot(a,b,c)=(\epsilon^{10}a,\epsilon^{6}b,\epsilon c)$. Let
$\tilde{S}=V(1+a^{3}+b^{5}+c^{30})\subset\mathbb{C}^{3}$. Then $S\cap U$ is
the quotient of $\tilde{S}$ by the induced action of $\Gamma$. Moreover, the
singular points of $D_{+}(x_{0})=S\cap U$ correspond to orbits of points in
$\tilde{S}$ with non-trivial stabilizer subgroups of $\Gamma$.

The points of $\tilde{S}$ with non-trivial stabilizers are those of form
$(a,0,0)$ where $a^{3}=-1$ and $(0,b,0)$ where $b^{5}=-1$, i.e.,

  * •

$\\{(\zeta_{6},0,0),(-1,0,0),(\zeta_{6}^{5},0,0)\\}$

  * •

$\\{(0,\zeta_{10},0),(0,\zeta_{10}^{3},0),(0,-1,0),(0,\zeta_{10}^{7},0),(0,\zeta_{10}^{9},0)\\}$.

The three points $\\{(\zeta_{6},0,0),(-1,0,0),(\zeta_{6}^{5},0,0)\\}$ form a
unique $\Gamma$-orbit. Letting $\eta=\epsilon^{3}$, the stabilizer of each
point in this orbit is the subgroup
$\langle\eta\rangle\subseteq\mathbb{Z}_{15}$, which is isomorphic to
$\mathbb{Z}_{5}$. Let $s=(-1,0,0)\in\tilde{S}$ and let $\mathfrak{m}_{s}$
denote the unique maximal ideal of the local ring
$\operatorname{\mathcal{O}}_{\tilde{S},s}$. Then, the induced action of
$\mathbb{Z}_{5}$ on the cotangent space
$\mathfrak{m}_{s}/\mathfrak{m}_{s}^{2}=\mathbb{C}\langle b,c\rangle$ is given
by $(\eta^{6}b,\eta c)=(\eta b,\eta c)$. So the singular point $[1:-1:0:0]$ is
a $\frac{1}{5}(1,1)$ singularity. This proves (a).

Similarly, the five points
$\\{(0,\zeta_{10},0),(0,\zeta_{10}^{3},0),(0,-1,0),(0,\zeta_{10}^{7},0),(0,\zeta_{10}^{9},0)\\}$
form a unique $\Gamma$-orbit. Letting $\xi=\epsilon^{5}$, the stabilizer of
each point in this orbit is the subgroup
$\langle\xi\rangle\subseteq\mathbb{Z}_{15}$, which is isomorphic to
$\mathbb{Z}_{3}$. Let $t=(0,-1,0)$ and let $\mathfrak{m}_{t}$ denote the
unique maximal ideal of the local ring
$\operatorname{\mathcal{O}}_{\tilde{S},t}$. Then, the induced action of
$\mathbb{Z}_{3}$ on the cotangent space
$\mathfrak{m}_{t}/\mathfrak{m}_{t}^{2}=\mathbb{C}\langle a,c\rangle$ is given
by $(\xi^{10}a,\xi c)=(\xi a,\xi c)$. So the corresponding singular point
$[1:0:-1:0]$ is a $\frac{1}{3}(1,1)$ singularity. This proves (b).

We leave (c) to the reader. Part (d) is clear since each point
$P\in\operatorname{{\rm Sing}}(X)$ is supported on $\Delta$ and
$\Delta\cong\mathbb{P}^{1}$. ∎

######  4.1.4.

For each $k\in\\{2,3,5\\}$, let $P_{k}$ denote the $\frac{1}{k}(1,1)$
singularity of $S$. Let $\sigma:\tilde{S}\to S$ be the minimal resolution of
singularities of $S$ and let $\tilde{E}_{k}$ denote the exceptional curve
lying over $P_{k}$. Then for each $k\in\\{2,3,5\\}$, $\tilde{E}_{k}$ is an
irreducible curve isomorphic to $\mathbb{P}^{1}$ with self intersection $-k$.
Applying Proposition 1.5.2 we find

$K_{\tilde{S}}=\sigma^{*}K_{S}-\frac{1}{3}\tilde{E}_{3}-\frac{3}{5}\tilde{E}_{5}.$
(3)

Let $\tilde{\Delta}$ denote the proper transform of $\Delta$ on $\tilde{S}$.
Since $1=\operatorname{{\rm mult}}_{P_{2}}(\Delta)=\operatorname{{\rm
mult}}_{P_{3}}(\Delta)=\operatorname{{\rm mult}}_{P_{5}}(\Delta)$ we have
$\sigma^{*}\Delta=\tilde{\Delta}+\frac{1}{2}\tilde{E}_{2}+\frac{1}{3}\tilde{E}_{3}+\frac{1}{5}\tilde{E}_{5}$.
Since $2\Delta$ is an anti-canonical divisor of $S$, we obtain using (3) that

$2\tilde{\Delta}+\tilde{E}_{2}+\tilde{E}_{3}+\tilde{E}_{5}\text{ is an anti-
canonical divisor of $\tilde{S}$ and $K_{\tilde{S}}^{2}=-2$. }$ (4)

The support of $\sigma^{*}(\Delta)$ is given by the following weighted graph

$\begin{array}[]{ccccc}&&(\tilde{E}_{2},-2)\\\ &&|\\\
(\tilde{E}_{3},-3)&-&(\tilde{\Delta},-1)&-&(\tilde{E}_{5},-5).\end{array}$

#### Auxiliary surfaces and birational morphisms

######  4.1.5.

We now describe some birational morphisms from $\tilde{S}$. The following
diagram of morphisms and surfaces will be used repeatedly in the arguments
that follow. Note that $S,\tilde{S}$ and $\sigma$ are already defined, whereas
the other surfaces and morphisms will be defined below.

$\textstyle{\tilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\tau_{1}}$$\textstyle{S^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{2}}$$\textstyle{\breve{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{3}}$$\textstyle{\hat{S}}$$\textstyle{S}$

Define $\tau_{1}:\tilde{S}\to S^{\prime}$ to be the contraction of
$\tilde{\Delta}$ onto the smooth point of $S^{\prime}$ which we denote by
$x_{0}^{\prime}$. For each $i\in\\{2,3,5\\}$, let
$E_{i}^{\prime}={\tau_{1}}_{*}(\tilde{E}_{i})$. The support of
${\tau_{1}}_{*}(\sigma^{*}\Delta)=\frac{1}{2}E_{2}^{\prime}+\frac{1}{3}E_{3}^{\prime}+\frac{1}{5}E_{5}^{\prime}$
has the weighted graph

$\textstyle{({E_{2}}^{\prime},-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({E_{3}}^{\prime},-2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({E_{5}}^{\prime},-4).}$

Define $\tau_{2}:S^{\prime}\to\breve{S}$ to be the contraction of
${E_{2}}^{\prime}$ onto a smooth point of $\breve{S}$ which we denote by
$\breve{x_{0}}$. Let $\breve{E}_{i}={\tau_{2}}_{*}(E_{i}^{\prime})$ for each
$i\in\\{3,5\\}$. Then $\breve{E_{3}}^{2}=-1$, $\breve{E_{5}}^{2}=-3$ and
$\breve{E_{3}},\breve{E_{5}}$ are projective lines that intersect
tangentially; in particular $\breve{E_{3}}\cdot\breve{E_{5}}=2$. Finally,
define $\tau_{3}:\breve{S}\longrightarrow\hat{S}$ to be the contraction of
$\breve{E_{3}}$ onto a smooth point of $\hat{S}$ which we denote by
$\hat{x_{0}}$. Let $\hat{E}_{5}={\tau_{3}}_{*}(\breve{E}_{5})$. Since
$\breve{E}_{3}\cdot\breve{E}_{5}=2$, $\operatorname{{\rm
mult}}_{\hat{x_{0}}}(\hat{E}_{5})=2$ and consequently
$\hat{E}_{5}^{2}=\breve{E}_{5}^{2}+(2)(2)=1$. It follows that $\hat{E}_{5}$ is
a singular projective curve containing an affine line, hence is a cuspidal
curve with a cusp at $\hat{x_{0}}$.

######  Proposition 4.1.6.

With the notation of 4.1.5,

  1. (a)

$\hat{E}_{5}$ is an ample anti-canonical divisor of $\hat{S}$;

  2. (b)

$\hat{S}$ is a smooth del Pezzo surface of degree 1;

###### Proof.

Observe first that

$\hat{E}_{5}^{2}=(-\hat{E_{5}})^{2}=\breve{E_{5}}^{2}+(2)(2)=1.$ (5)

We prove (a). It follows from (4) and from the definitions of
$\tau_{1},\tau_{2},\tau_{3}$ that $\hat{E}_{5}$ is an anti-canonical divisor
of $\hat{S}$. By Theorem 1.3.6, it suffices to show that for any curve
$\hat{C}\subset\hat{S}$, $\hat{C}\cdot\hat{E}_{5}>0$. Observe that the
birational map $S\dashrightarrow\hat{S}$ restricts to an isomorphism between
$S\setminus\Delta$ and $\hat{S}\setminus\hat{E}_{5}$. If
$\hat{C}=\hat{E}_{5}$, we’re done by (5). If $\hat{C}\neq\hat{E}_{5}$, the
proper transform $\tilde{C}\in\tilde{S}$ of $\hat{C}\in\hat{S}$ is equal to
the proper transform of some irreducible curve $C$ in $S$ other than $\Delta$.
Since $\Delta$ is ample, $C\cdot\Delta>0$ and so $\tilde{C}$ intersects
$\tilde{\Delta}\cup\tilde{E}_{2}\cup\tilde{E}_{3}\cup\tilde{E}_{5}$. This
implies that $\hat{C}$ intersects $\hat{E}_{5}$ and so
$\hat{C}\cdot\hat{E}_{5}>0$. This proves (a).

For (b), $\hat{S}$ is smooth because $\tau_{1},\tau_{2},\tau_{3}$ contract
$(-1)$-curves on smooth surfaces. The other claims in (b) follow from part (a)
and (5). ∎

######  4.1.7.

Since by Proposition 4.1.6, $\hat{S}$ is a smooth del Pezzo surface of degree
1, there exist 8 disjoint $(-1)$-curves $\hat{F}_{1},\dots,\hat{F}_{8}$ on
$\hat{S}$. For any $(-1)$-curve $\hat{F}_{i}$ on $\hat{S}$
($i\in\\{1,\dots,8\\}$), the adjunction formula implies

$-2=\hat{F}_{i}^{2}+\hat{F}_{i}\cdot K_{\hat{S}},$

so $\hat{F}_{i}\cdot(-K_{\hat{S}})=\hat{F}_{i}\cdot\hat{E}_{5}=1$. This shows
that

for each $i\in\\{1,\dots,8\\}$, $\hat{F}_{i}$ intersects $\hat{E}_{5}$
transversely at a point other than $\hat{x}_{0}$ (the singular point of
$\hat{E}_{5}$). (6)

After blowing down these 8 curves, the image of $\hat{E}_{5}$ is an
irreducible cuspidal curve $H$ in $\mathbb{P}^{2}$ with self-intersection
equal to $\hat{E}_{5}^{2}+8=9$. It follows that $H=V_{+}(h)$ for some
homogeneous $h\in\mathbb{C}^{[3]}$ of degree $3$, so $H$ is an irreducible
cuspidal cubic curve in $\mathbb{P}^{2}$. Let
$\breve{F}_{i}\in\operatorname{{\rm Div}}(\breve{S})$ (resp.
$F_{i}^{\prime}\in\operatorname{{\rm Div}}(S^{\prime})$) denote the proper
transform of $\hat{F}_{i}$ in $\breve{S}$ (resp. $S^{\prime}$). Since the
curves $\hat{F}_{i}$ do not intersect $\hat{x}_{0}$, the curves
$\breve{F}_{i}$ (resp. $F_{i}^{\prime}$) are $(-1)$-curves intersecting
$\breve{E}_{5}$ (resp. $E_{5}^{\prime}$) transversely. For any choice of two
$(-1)$-curves in $\hat{S}$, say $\hat{F}_{1}$ and $\hat{F}_{2}$, we obtain the
following commutative diagram

$\textstyle{\tilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\tau_{1}}$$\textstyle{S^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{2}}$$\scriptstyle{\kappa^{\prime}}$$\textstyle{\breve{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{3}}$$\scriptstyle{\breve{\kappa}}$$\textstyle{\hat{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{S}$$\textstyle{\ddot{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\bar{\tau_{2}}}$$\textstyle{\dot{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\tau_{3}}}$$\textstyle{S^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T}$$\textstyle{\mathbb{P}^{2}}$

where

  * •

$\kappa:\hat{S}\to S^{3}$ is the contraction of $\hat{F}_{1}$ and
$\hat{F}_{2}$

  * •

$\breve{\kappa}:\breve{S}\to\dot{S}$ is the contraction of $\breve{F}_{1}$ and
$\breve{F}_{2}$

  * •

$\kappa^{\prime}:S^{\prime}\to\ddot{S}$ is the contraction of $F_{1}^{\prime}$
and $F_{2}^{\prime}$

  * •

$\bar{\tau_{2}}:\ddot{S}\to\dot{S}$ is the contraction of the $(-1)$-curve
$\ddot{E}_{2}=\kappa^{\prime}_{*}(E_{2}^{\prime})$

  * •

$\bar{\tau_{3}}:\dot{S}\to S^{3}$ is the contraction of the $(-1)$-curve
$\dot{E}_{3}=\breve{\kappa}_{*}(\breve{E}_{3})$

  * •

$\pi:\ddot{S}\to T$ is the contraction of the $(-2)$-curves
$\ddot{E}_{3}=\kappa^{\prime}_{*}(E_{3}^{\prime})$ and
$\ddot{E}_{5}=\kappa^{\prime}_{*}(E_{5}^{\prime})$.

######  Proposition 4.1.8.

With the notation of 4.1.7, the following hold:

  1. (a)

$S^{3}$ is a smooth del Pezzo surface of degree $3$;

  2. (b)

$\dot{S}$ is a smooth del Pezzo surface of degree $2$;

  3. (c)

$\ddot{S}$ is a smooth weak del Pezzo surface of degree $1$;

  4. (d)

the only $(-2)$-curves on $\ddot{S}$ are $\ddot{E}_{3}$ and $\ddot{E}_{5}$;

  5. (e)

$T$ is a singular del Pezzo surface with a unique $A_{2}$ singularity.

###### Proof.

Part (a) is clear. We prove part (b). Using the calculations in 4.1.5, we have
$(-K_{\breve{S}})^{2}=(\breve{E_{3}}+\breve{E_{5}})^{2}=-1+(2)(2)-3=0$, and
hence $(-K_{\dot{S}})^{2}=(-K_{\breve{S}})^{2}+2=2$. Observe that
$\dot{E}_{3}$ and $\dot{E}_{5}$ are both $(-1)$-curves in $\dot{S}$. We now
show that the anti-canonical divisor
$\dot{E}_{3}+\dot{E}_{5}\in\operatorname{{\rm Div}}(\dot{S})$ is ample. Let
$\dot{C}$ be an irreducible curve in $\dot{S}$. Suppose that
$\dot{C}=(\kappa\circ\tau_{2}\circ\tau_{1})_{*}(\tilde{C})$ where $\tilde{C}$
is the proper transform of some curve $C$ in $S$. Since $\Delta$ is ample,
$C\cdot\Delta>0$ and so $\dot{C}\cdot(\dot{E}_{3}+\dot{E}_{5})>0$. The only
other possibilities are $\dot{C}=\dot{E}_{3}$ or $\dot{C}=\dot{E}_{5}$. In
both cases, we have $\dot{C}\cdot(\dot{E}_{3}+\dot{E}_{5})=1$. This shows that
$-K_{\dot{S}}$ is ample, and proves (b).

We prove (c). Note first that by part (b) we have
$(-K_{\ddot{S}})^{2}=(-K_{\dot{S}})^{2}-1=1$ so it suffices to show that
$-K_{\ddot{S}}$ is nef and big. We first show that $-K_{\ddot{S}}$ is nef. Let
$\ddot{C}$ be any curve in $\ddot{S}$. Suppose
$\ddot{C}=(\kappa^{\prime}\circ\tau_{1})_{*}(\tilde{C})$ where $\tilde{C}$ is
the proper transform of a curve $C$ in $S$. Since $C\cdot\Delta>0$ (because
$\Delta$ is ample), we have $\ddot{C}\cdot\ddot{E}_{i}>0$ for some $i=2,3,5$.
Thus
$-K_{\ddot{S}}\cdot\ddot{C}=(\ddot{E}_{2}+\ddot{E}_{3}+\ddot{E}_{5})\cdot\ddot{C}>0$.
The only other possibility is that $\ddot{C}=\ddot{E}_{i}$ for some $i=2,3,5$.
Since by adjunction $-K_{\ddot{S}}\cdot\ddot{E}_{2}=1$ and
$-K_{\ddot{S}}\cdot\ddot{E}_{3}=0=-K_{\ddot{S}}\cdot\ddot{E}_{5}$, it follows
that $-K_{\ddot{S}}$ is nef. Since $\ddot{S}$ is smooth and $-K_{\ddot{S}}$ is
nef, Proposition 1.3.7 implies that $-K_{\ddot{S}}$ is big, showing (c).

We prove (d). First, note that $\ddot{E}_{3}$ and $\ddot{E}_{5}$ are indeed
$(-2)$-curves. Suppose there exists some $(-2)$-curve $\ddot{E}$ on $\ddot{S}$
other than $\ddot{E}_{3}$ and $\ddot{E}_{5}$. By adjunction, we have

$\ddot{E}\cdot(\ddot{E}_{2}+\ddot{E}_{3}+\ddot{E}_{5})=0$

and consequently, since $\ddot{E}$ is none of $\ddot{E}_{2}$, $\ddot{E}_{3}$,
$\ddot{E}_{5}$, $\ddot{E}$ is disjoint from $\operatorname{{\rm
Supp}}(\ddot{E}_{2}+\ddot{E}_{3}+\ddot{E}_{5})$. It follows that
$(\bar{\tau_{3}}\circ\bar{\tau_{2}})_{*}(\ddot{E})$ is a $(-2)$-curve on the
smooth del Pezzo surface $S^{3}$. This contradicts Remark 1.5.6, and proves
(d).

Part (e) follows immediately from (c) and (d) together with the fact that
$\ddot{E}_{3}$ and $\ddot{E}_{5}$ intersect transversely at a single point. ∎

####  4.1.9 Exclusion of anti-canonial polar cylinders.

This section is devoted to establishing that the surface $S=\operatorname{{\rm
Proj}}B_{2,3,5,30}$ does not contain an anti-canonical polar cylinder. The
proof of this claim is by contradiction, and so from this point until the end
of Section 4.1 we will regularly consider the consequences of the following
assumption:

######  Assumption 4.1.10.

Assume the surface $S=\operatorname{{\rm Proj}}B_{2,3,5,30}$ contains an anti-
canonical polar cylinder $U\cong\mathbb{A}^{1}\times
Z=S\setminus\operatorname{{\rm Supp}}(D)$. Given $D$, define

$\tilde{D}=\sigma^{*}(D)+\frac{1}{3}\tilde{E_{3}}+\frac{3}{5}\tilde{E}_{5}\in\operatorname{{\rm
Div}}(\tilde{S},\mathbb{Q}).$

######  Proposition 4.1.11.

Assume 4.1.10 holds. Then,

  1. (a)

$\operatorname{{\rm Supp}}(D)$ contains $\operatorname{{\rm Sing}}(S)$;

  2. (b)

$\tilde{E}_{2},\tilde{E}_{3},\tilde{E}_{5}$ are contained in
$\operatorname{{\rm Supp}}(\tilde{D})$;

  3. (c)

$\tilde{D}$ is an effective anti-canonical $\mathbb{Q}$-divisor on
$\tilde{S}$;

  4. (d)

$\sigma^{-1}(U)=\tilde{S}\setminus\operatorname{{\rm
Supp}}(\sigma^{*}(D))=\tilde{S}\setminus\operatorname{{\rm Supp}}(\tilde{D})$
is an anti-canonical polar cylinder in $\tilde{S}$.

###### Proof.

Since $S$ is a normal surface and $U$ is a cylinder, $U$ must be smooth and so
$\operatorname{{\rm Sing}}(S)\subset\operatorname{{\rm Supp}}(D)$ proving (a).
Part (b) follows from the definition of $\tilde{D}$ together with the fact
that $P_{2},P_{3},P_{5}\in\operatorname{{\rm Supp}}(D)$. Part (c) follows from
the definition of $\tilde{D}$ together with (3). We prove (d). Since
$P_{2},P_{3},P_{5}\in\operatorname{{\rm Supp}}(D)$, $\sigma|_{\sigma^{-1}(U)}$
is an isomorphism onto its image. In particular,
$\sigma^{-1}(U)=\tilde{S}\setminus\operatorname{{\rm
Supp}}(\sigma^{*}(D))=\tilde{S}\setminus\operatorname{{\rm Supp}}(\tilde{D})$
is a cylinder which by (c) is an anti-canonical polar cylinder.

∎

######  Notation 4.1.12.

Assuming 4.1.10, the discussion in 1.6.12 implies that we have the following
commutative diagram:

---
$\textstyle{U\cong\mathbb{A}^{1}\times
Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}_{Z}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\tilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\tilde{\rho}}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\phi}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\bar{Z}\cong\mathbb{P}^{1}}$

where $\tilde{\rho}$ denotes the rational map associated to the cylinder
$\sigma^{-1}(U)\subset\tilde{S}$, $\tilde{\mathcal{L}}$ is the linear system
associated to $\tilde{\rho}$, and $\pi$ is the resolution of indeterminacy of
$\tilde{\rho}$ so that $\phi$ is a $\mathbb{P}^{1}$-fibration. In what
follows, we define $\tilde{U}=\sigma^{-1}(U)$.

######  Lemma 4.1.13.

Assume 4.1.10 holds. Then $\tilde{\Delta}$ is not an irreducible component of
$\tilde{D}$.

###### Proof.

Assume that $\tilde{\Delta}$ is an irreducible component of $\tilde{D}$. Since
$\tilde{\Delta}\subset\operatorname{{\rm Supp}}{\tilde{D}}$ it follows from
Proposition 4.1.11 (b) that $\operatorname{{\rm Supp}}{\tilde{D}}$ contains
$\tilde{\Delta}\cup\tilde{E_{2}}\cup\tilde{E_{3}}\cup\tilde{E_{5}}$. Let
$\tau=\tau_{3}\circ\tau_{2}\circ\tau_{1}$. The contraction
$\tau:\tilde{S}\to\hat{S}$ restricts to an isomorphism between
$\tilde{U}=\tilde{S}\setminus\operatorname{{\rm Supp}}(\tilde{D})$ and its
image $\tau(\tilde{U})=\hat{S}\setminus\operatorname{{\rm Supp}}{\hat{D}}$
where $\hat{D}=\tau_{*}(\tilde{D})$. Since $\hat{D}$ is an effective anti-
canonical $\mathbb{Q}$-divisor on $\hat{S}$, it follows that $\hat{S}$
contains an anti-canonical polar cylinder. Since $\hat{S}$ is a smooth del
Pezzo surface of degree 1, this contradicts Theorem 1.6.15. ∎

######  Lemma 4.1.14.

Assume 4.1.10 holds. Then the linear system $\tilde{\mathcal{L}}$ has a unique
proper base point.

###### Proof.

Since by 1.6.12 $\mathcal{L}$ has at most one base point,
$\tilde{\mathcal{L}}$ has at most one base point. So we need only show that
$\tilde{\mathcal{L}}$ is not base point free.

Assume for the sake of a contradiction that $\tilde{\mathcal{L}}$ is base-
point free. Then by 1.6.12 $\tilde{D}$ contains a section $\Gamma$ of the
$\mathbb{P}^{1}$-fibration $\tilde{\rho}$. In particular, for a general fiber
$L$ of $\tilde{\rho}$, the adjunction formula implies

$2=-K_{\tilde{S}}\cdot L=2\tilde{\Delta}\cdot L+\tilde{E}_{2}\cdot
L+\tilde{E}_{3}\cdot L+\tilde{E}_{5}\cdot L.$ (7)

Let $I=\\{2,3,5\\}$. If $\Gamma=\tilde{E}_{i}$ for some $i\in I$, then (7)
implies that $\tilde{\Delta}\cdot L=0$ and (by Remark 1.6.11) that
$\tilde{E}_{j}$ is a section for some $j\in I\setminus\\{i\\}$. This is
impossible since by 1.6.12 $\operatorname{{\rm Supp}}(\tilde{D})$ contains
exactly one section of $\tilde{\rho}$ in its support. For each $i\in I$, since
$\tilde{E}_{i}\subset\operatorname{{\rm Supp}}(\tilde{D})$ and is not a
section, we have by 1.6.12 that $\tilde{E}_{i}$ is contained in a closed fiber
of $\tilde{\rho}$. By (7), $\tilde{\Delta}\cdot L=1$ and so $\tilde{\Delta}$
is a section by Remark 1.6.11.

Since the three curves $\tilde{E}_{i}$ intersect $\tilde{\Delta}$ at different
points and at most one irreducible curve in each fiber can intersect a section
of a $\mathbb{P}^{1}$-fibration, we obtain that

each curve $\tilde{E}_{i}$ is contained in a different fiber of
$\tilde{\rho}$. (8)

We will show

every singular fiber of $\tilde{\rho}$ contains some curve $\tilde{E}_{i}$.
(9)

Recall that by Lemma 1.6.17 every singular fiber of $\tilde{\rho}$ is a tree
of irreducible curves isomorphic to $\mathbb{P}^{1}$. Since all fibers are
linearly equivalent curves in $\tilde{S}$ and $\tilde{\Delta}\cdot L=1$, every
fiber contains exactly one component that intersects $\tilde{\Delta}$, and
that intersection is transverse. Assume $\tilde{F}$ is a singular fiber
containing multiple curves $\tilde{H}_{1},\dots,\tilde{H}_{r}$. Since $\Delta$
is ample, $\tilde{H}_{j}=\tilde{E}_{i}$ for some $j\in\\{1,\dots,r\\}$ and all
other curves $\tilde{H}_{k}$ in $\tilde{F}$ intersect $\tilde{H}_{j}$. This
proves (9).

By combining (8) and (9), there are exactly $3$ singular fibers. Let us denote
these $3$ fibers by $\tilde{F}_{2},\tilde{F}_{3},\tilde{F}_{5}$ where
$\tilde{F}_{i}$ is the fiber containing $\tilde{E}_{i}$. We claim:

the singular fibers $\tilde{F}_{i}$ have form
$\tilde{F}_{i}=\tilde{E}_{i}+\tilde{R}_{i}$ where $\tilde{R}_{i}$ consists of
$i$ disjoint $(-1)$-curves intersecting $\tilde{E}_{i}$ at distinct points of
$\tilde{E}_{i}\setminus\tilde{\Delta}$. (10)

Consider the fiber $\tilde{F}_{i}$, and write
$\tilde{F}_{i}=\tilde{E}_{i}+\tilde{R}_{i}$ where
$\tilde{R}_{i}=\sum_{\ell=1}^{q}a_{i,\ell}\tilde{G}_{i,\ell}$, where $q$
depends on $i$. Note that $\tilde{E}_{i}$ is the only component of
$\tilde{F}_{i}$ that intersects $\tilde{\Delta}$. Since $\Delta$ is ample,
every component of $\tilde{F}_{i}$ must intersect $\tilde{E}_{i}$ and can
intersect no other exceptional curve $\tilde{E}_{j}$ where $j\in
I\setminus\\{i\\}$. Since (by Lemma 1.6.17) $\tilde{F}_{i}$ is a tree and all
components $\tilde{G}_{i,\ell}$ intersect $\tilde{E}_{i}$, the components
$\tilde{G}_{i,\ell}$ must be disjoint and must intersect $\tilde{E}_{i}$ at
distinct points of $\tilde{E}_{i}\setminus\tilde{\Delta}$. Since
$\tilde{E}_{i}^{2}=-i$, Lemma 1.6.18 implies that $q=i$, $\tilde{G}_{i,\ell}$
is a $(-1)$-curve and $a_{i,\ell}=1$ for all $\ell=1,\dots,i$. This proves
(10).

Denote by
$\psi:\tilde{S}\to\mathbb{F}_{1}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-1))$
the contraction of all 10 $(-1)$-curves $\tilde{G}_{i,\ell}$ in the $3$
singular fibers of $\tilde{\rho}$. (The image of $\psi$ is a
$\mathbb{P}^{1}$-bundle over $\mathbb{P}^{1}$ and hence is a Hirzebruch
surface. Since the image of $\psi$ contains the $(-1)$-curve
$\psi_{*}(\tilde{\Delta})$, the image of $\psi$ is not a minimal surface, and
so must indeed be $\mathbb{F}_{1}$.) Letting
$\rho_{1}:\mathbb{F}_{1}\to\mathbb{P}^{1}$ be the $\mathbb{P}^{1}$-bundle
structure, we have $\tilde{\rho}=\rho_{1}\circ\psi$ by Lemma 1.6.17 (2).

Since $\operatorname{{\rm Supp}}(\tilde{D})$ is connected and contains
$\tilde{E}_{2},\tilde{E}_{3}$ and $\tilde{E}_{5}$ and since $\tilde{\rho}$
restricts to a trivial $\mathbb{A}^{1}$-bundle on
$\tilde{S}\setminus\operatorname{{\rm Supp}}(\tilde{D})$, it follows that
$\operatorname{{\rm Supp}}(\tilde{D})$ contains at least two irreducible
components of $\tilde{R}_{3}$ and at least four irreducible components of
$\tilde{R}_{5}$. Choose one $(-1)$-curve $\tilde{G}_{3,1}\subset\tilde{R}_{3}$
which intersects $\tilde{E}_{3}$ and three curves
$\tilde{G}_{5,1},\tilde{G}_{5,2},\tilde{G}_{5,3}\subset\tilde{R}_{5}$ which
intersect $\tilde{E}_{5}$. Let $\bar{\psi}:\tilde{S}\to\bar{S}$ denote the
contraction of the four curves
$\tilde{G}_{3,1},\tilde{G}_{5,1},\tilde{G}_{5,2},\tilde{G}_{5,3}$.

We claim

$\text{ $\bar{S}$ is a weak del Pezzo surface of degree 2}.$ (11)

First, let $\bar{\Delta}=\bar{\psi}_{*}(\tilde{\Delta})$, and let
$\bar{E}_{i}=\bar{\psi}_{*}(\tilde{E}_{i})$ for each $i\in I$. Note that
$\bar{E}_{i}$ is a $(-2)$-curve for each $i\in I$. Then
$(-K_{\bar{S}})^{2}=(2\bar{\Delta}+\bar{E}_{2}+\bar{E}_{3}+\bar{E}_{5})^{2}=2$,
so to prove 11 it suffices to show that $-K_{\bar{S}}$ is nef and big. Let
$\bar{C}$ be a curve in $\bar{S}$. If $\bar{C}$ is one of
$\bar{\Delta},\bar{E}_{2},\bar{E}_{3},\bar{E}_{5}$ then
$\bar{C}\cdot(2\bar{\Delta}+\bar{E}_{2}+\bar{E}_{3}+\bar{E}_{5})\geq 0$.
Otherwise, since $\Delta$ is an ample divisor on $S$, the proper transform of
$\bar{C}$, denoted $\tilde{C}$ must intersect one of
$\tilde{\Delta},\tilde{E}_{2},\tilde{E}_{3},\tilde{E}_{5}$, and consequently
$\bar{C}\cdot(2\bar{\Delta}+\bar{E}_{2}+\bar{E}_{3}+\bar{E}_{5})>0$. This
shows that $-K_{\bar{S}}$ is nef. Since $-K_{\bar{S}}$ is nef and
$(-K_{\bar{S}})^{2}=2$, Proposition 1.3.7 implies that $-K_{\bar{S}}$ is big.
This proves (11).

Next we show that

$\text{ the three curves $\bar{E}_{2},\bar{E}_{3},\bar{E}_{5}$ are the only
$(-2)$-curves on $\bar{S}$}.$ (12)

Let $\bar{E}$ be a $(-2)$-curve on $\bar{S}$ other than
$\bar{E}_{2},\bar{E}_{3},\bar{E}_{5}$. By adjunction,

$\bar{E}\cdot(2\bar{\Delta}+\bar{E}_{2}+\bar{E}_{3}+\bar{E}_{5})=0$

and since $\bar{E}$ is none of $\bar{E}_{2}$, $\bar{E}_{3}$, $\bar{E}_{5}$,
$\bar{\Delta}$, it follows that $\bar{E}$ is disjoint from $\operatorname{{\rm
Supp}}(2\bar{\Delta}+\bar{E}_{2}+\bar{E}_{3}+\bar{E}_{5})$. Consequently, the
proper transform $\tilde{E}=\bar{\phi}^{*}(\bar{E})$ is a $(-2)$-curve on
$\tilde{S}$ that is disjoint from
$\tilde{\Delta}\cup\tilde{E}_{2}\cup\tilde{E}_{3}$ and so
$\hat{E}=\tau_{*}(\tilde{E})$ is a $(-2)$-curve on $\hat{S}$. This contradicts
Remark 1.5.6, since $\hat{S}$ is a smooth del Pezzo surface by Proposition
4.1.6 (b). We conclude no such $\bar{E}$ exists, proving (12). It follows
immediately from (12) that the contraction of
$\bar{E}_{2},\bar{E}_{3},\bar{E}_{5}$ yields a morphism $\pi:\bar{S}\to T$
where $T$ is a degree 2 del Pezzo surface with exactly 3 $A_{1}$
singularities.

Since the four curves
$\tilde{G}_{3,1},\tilde{G}_{5,1},\tilde{G}_{5,2},\tilde{G}_{5,3}$ are
contained in $\operatorname{{\rm Supp}}(\tilde{D})$, the divisor
$\bar{D}=\bar{\psi}_{*}(\tilde{D})$ is an anti-canonical divisor on $\bar{S}$.
So $\bar{S}\setminus\bar{D}$ is an anti-canonical polar cylinder on $\bar{S}$
and $\bar{D}$ contains $\bar{E}_{2},\bar{E}_{3},\bar{E}_{5}$ in its support.
This contradicts Corollary 1.6.16 and completes the proof. ∎

######  Lemma 4.1.15.

Assume 4.1.10 holds and let $\tilde{s}$ be the unique base point of
$\tilde{\mathcal{L}}$. Then
$\tilde{s}\notin\tilde{\Delta}\cup\tilde{E}_{2}\cup\tilde{E}_{3}\cup\tilde{E}_{5}$.

###### Proof.

The proof consists of four parts. We will show

  1. (i)

$\tilde{s}\notin\tilde{\Delta}$,

  2. (ii)

$\tilde{s}\notin\tilde{E}_{5}$,

  3. (iii)

$\tilde{s}\notin\tilde{E}_{3}$,

  4. (iv)

$\tilde{s}\notin\tilde{E}_{2}$.

Assume $\tilde{s}\in\tilde{\Delta}$. By Lemma 4.1.13,
$\tilde{\Delta}\not\subseteq\operatorname{{\rm Supp}}(\tilde{D})$. Since
$\tilde{D}$ is an anti-canonical $\mathbb{Q}$-divisor on $\tilde{S}$ and
$\tilde{\Delta}$ is a $(-1)$-curve, we have

$\tilde{\Delta}\cdot\tilde{D}=\tilde{\Delta}\cdot(2\tilde{\Delta}+\tilde{E}_{2}+\tilde{E}_{3}+\tilde{E}_{5})=2\tilde{\Delta}^{2}+\tilde{\Delta}\cdot(\tilde{E}_{2}+\tilde{E}_{3}+\tilde{E}_{5})=-2+3=1$

and so

$1=\tilde{\Delta}\cdot\tilde{D}\geq\operatorname{{\rm
mult}}_{\tilde{s}}(\tilde{\Delta})\operatorname{{\rm
mult}}_{\tilde{s}}(\tilde{D})\geq\operatorname{{\rm
mult}}_{\tilde{s}}(\tilde{D})$ (13)

the first inequality because $\tilde{\Delta}\not\subseteq\operatorname{{\rm
Supp}}(\tilde{D})$, the second because $\tilde{s}\in\tilde{\Delta}$. On the
one hand, Lemma 1.5.3 together with (13) implies that $(\tilde{S},\tilde{D})$
is log-canonical at $\tilde{s}$. On the other hand, Lemma 1.6.14 implies that
$(\tilde{S},\tilde{D})$ is not log-canonical at $\tilde{s}$. This is a
contradiction, proving (i).

Suppose $\tilde{s}\in\tilde{E}_{5}$. By part (i),
$\tilde{s}\in\tilde{E}_{5}\setminus\tilde{\Delta}$. Let
$\tau=\tau_{3}\circ\tau_{2}\circ\tau_{1}$, let $\hat{D}=\tau_{*}(\tilde{D})$
and let $\hat{s}=\tau(\tilde{s})\in\hat{S}$. Since by Lemma 1.6.14,
$(\tilde{S},\tilde{D})$ is not log-canonical at $\tilde{s}$,
$(\hat{S},\hat{D})$ is not log-canonical at $\hat{s}$. We claim:

$\text{the image $\hat{E}_{5}$ of $\tilde{E}_{5}$ is the unique member of the
pencil $|-K_{\hat{S}}|$ passing through $\hat{s}$}.$ (14)

Indeed, by Lemma 1.5.7 $|-K_{\hat{S}}|$ has a unique proper base point $q$, so
it suffices to prove $\hat{s}\neq q$. If $\hat{s}=q$, then for a general
member $Z\in|-K_{\hat{S}}|$ passing through $\hat{s}=q$, we find
$1=\hat{D}\cdot Z\geq\operatorname{{\rm mult}}_{\hat{s}}\hat{D}$. But since
$(\hat{S},\hat{D})$ is not log-canonical at $\hat{s}$, we obtain a
contradiction to Lemma 1.5.3. This proves (14).

Since $\tilde{s}\in\tilde{E}_{5}\setminus\tilde{\Delta}$, $\hat{s}$ is not
equal to the singular point of $\hat{E}_{5}$ so $\operatorname{{\rm
mult}}_{\hat{s}}(\hat{E}_{5})=1$. Lemma 1.5.3 then implies that
$(\hat{S},\hat{E}_{5})$ is log-canonical at $\hat{s}$. On the one hand, since
$(\hat{S},\hat{D})$ is not log-canonical at $\hat{s}\in\operatorname{{\rm
Supp}}(\hat{D})$, Lemma 1.5.7 (d) implies that there exists some curve
$\hat{C}$ in the linear system $|-K_{\hat{S}}|$ passing through $\hat{s}$ and
satisfying $(\hat{S},\hat{C})$ is not log-canonical at $\hat{s}$. On the other
hand, (14) implies that $\hat{C}=\hat{E}_{5}$, so
$(\hat{S},\hat{C})=(\hat{S},\hat{E}_{5})$ is log-canonical at $\hat{s}$. This
is a contradiction, and proves (ii).

Assume $\tilde{s}\in\tilde{E}_{3}$ and recall the notation of 4.1.7. By part
(i), $\tilde{s}\in\tilde{E}_{3}\setminus\tilde{\Delta}$. Furthermore, the
image $\dot{D}=(\breve{\kappa}\circ\tau_{2}\circ\tau_{1})_{*}(\tilde{D})$ is
an effective anti-canonical $\mathbb{Q}$-divisor on $\dot{S}$ which contains
the integral anti-canonical divisor $\dot{E_{3}}+\dot{E_{5}}$ in its support.
Let $\dot{x_{0}}$ denote the point of intersection of $\dot{E_{3}}$ and
$\dot{E_{5}}$. Since $\tilde{s}\in\tilde{E}_{3}\setminus\tilde{\Delta}$ and
$\tilde{E}_{2}\cap\tilde{E}_{3}=\emptyset$,
$(\breve{\kappa}\circ\tau_{2}\circ\tau_{1})(\tilde{s})=\dot{s}\neq\dot{x_{0}}$.
Since by Lemma 1.6.14, $(\tilde{S},\tilde{D})$ is not log-canonical at
$\tilde{s}$, $(\dot{S},\dot{D})$ is not log-canonical at $\dot{s}$. By
Proposition 4.1.8, $\dot{S}$ is a smooth del Pezzo surface of degree 2. By
Lemma 1.5.9, the support of $\dot{D}$ must contain the support of an effective
anti-canonical integral divisor $\dot{C}$ on $\dot{S}$ such that
$(\dot{S},\dot{C})$ is not log-canonical at $\dot{s}$ and $\dot{C}$ is either

  * •

a pair of $(-1)$-curves intersecting tangentially at $\dot{s}$,

  * •

an irreducible rational curve with a cusp at $\dot{s}$.

If $\dot{E}_{3}\not\subseteq\operatorname{{\rm Supp}}(\dot{C})$, then since
$\dot{E}_{3}$ is a $(-1)$-curve and $\dot{C}$ is an anti-canonical divisor,
the adjunction formula implies

$1=\dot{C}\cdot\dot{E_{3}}\geq\operatorname{{\rm
mult}}_{\dot{s}}(\dot{C})\operatorname{{\rm mult}}_{\dot{s}}(\dot{E_{3}})\geq
2$

which is impossible, so $\dot{E}_{3}\subseteq\operatorname{{\rm
Supp}}(\dot{C})$. It follows that $\dot{C}=\dot{E}_{3}+\dot{C^{\prime}}$ for
some $(-1)$-curve $\dot{C^{\prime}}$ meeting $\dot{E}_{3}$ tangentially at
$\dot{s}$. If $\dot{C^{\prime}}=\dot{E}_{5}$, then $\dot{E}_{3}$ and
$\dot{E}_{5}$ meet tangentially at $\dot{s}$. By assumption $\dot{E}_{3}$ and
$\dot{E}_{5}$ meet only at $\dot{x_{0}}$, and since $\dot{s}\neq\dot{x_{0}}$
this case is impossible. It follows that $\dot{C^{\prime}}\neq\dot{E_{5}}$.
Since $\dot{E_{5}}$ is a $(-1)$-curve and $\dot{C^{\prime}}\neq\dot{E}_{5}$ we
find

$1=\dot{C}\cdot\dot{E_{5}}\geq\dot{E_{3}}\cdot\dot{E_{5}}=2$

which too is impossible. We conclude that $\tilde{s}\notin\tilde{E}_{3}$,
proving (iii).

Finally, assume $\tilde{s}\in\tilde{E}_{2}$. By part (i),
$\tilde{s}\in\tilde{E}_{2}\setminus\tilde{\Delta}$. By Lemma 1.6.14,
$(\tilde{S},\tilde{D})$ is not log-canonical at $\tilde{s}$. There exist two
$(-1)$-curves $\tilde{F}_{1}$ and $\tilde{F}_{2}$ that exist $\tilde{E}_{5}$
such that $\tilde{s}\notin\tilde{F}_{1}\cup\tilde{F}_{2}$. Recalling the
notation of 4.1.7, let
$\ddot{D}=(\kappa^{\prime}\circ\tau_{1})_{*}(\tilde{D})$ where
$\kappa^{\prime}$ is the contraction of $\tilde{F}_{1}$ and $\tilde{F}_{2}$.
Then $(S,\ddot{D})$ is not log-canonical at
$\ddot{s}=(\kappa^{\prime}\circ\tau_{1})(\tilde{s})$. Let
$\ddot{x_{0}}=\ddot{E}_{2}\cap\ddot{E}_{3}\cap\ddot{E}_{5}$. Since
$\tilde{s}\in\tilde{E}_{2}\setminus\tilde{\Delta}$ we have
$\ddot{s}\neq\ddot{x_{0}}$. By Proposition 4.1.8 (d),

the only two $-2$-curves on $\ddot{S}$ are $\ddot{E}_{5}$ and $\ddot{E}_{3}$.
(15)

and by Proposition 4.1.8 (e),

$T$ is a singular del Pezzo surface with a unique $A_{2}$ singularity. (16)

Since $\ddot{s}\neq\ddot{x_{0}}$, it follows from (ii), (iii) and (15) that
there is no $(-2)$-curve passing through $\ddot{s}$. So $s_{T}=\pi(\ddot{s})$
is a smooth point of $T$ and $D_{T}=\pi_{*}(\ddot{D})$ is an effective anti-
canonical $\mathbb{Q}$-divisor of $T$ which is not log-canonical at $s_{T}$.
Lemma 1.5.7 (c) implies that $s_{T}$ is not the base point of $|-K_{T}|$ and
so there exists a unique divisor in $|-K_{T}|$ passing through $s_{T}$.
Clearly this divisor is $E_{2,T}=\pi_{*}(\ddot{E_{2}})$. Lemma 1.5.7 (d) then
implies that $E_{2,T}\subseteq\operatorname{{\rm Supp}}(D_{T})$ and
$(T,E_{2,T})$ is not log-canonical at $s_{T}$. But $s_{T}$ is a regular point
of $E_{2,T}$, a contradiction to Lemma 1.5.3. We conclude that
$\tilde{s}\notin\tilde{E}_{2}$, proving (iv). ∎

######  Proposition 4.1.16.

Let $\tilde{D}$ be an effective anti-canonical $\mathbb{Q}$-divisor on
$\tilde{S}$. Then the log pair $(\tilde{S},\tilde{D})$ is log-canonical at all
points
$\tilde{s}\in\tilde{S}\setminus(\tilde{\Delta}\cup\tilde{E}_{2}\cup\tilde{E}_{3}\cup\tilde{E}_{5})$.

###### Proof.

Recall that by the discussion in 4.1.7, there exist $8$ disjoint $(-1)$-curves
$\tilde{F}_{i}$ (where $i\in\\{1,\dots,8\\}$) meeting $\tilde{E}_{5}$
transversely at points of $\tilde{E}_{5}\setminus\tilde{\Delta}$. (These are
the proper transforms of the $\hat{F}_{i}$ in $\tilde{S}$.) Given a point
$\tilde{s}\in\tilde{S}\setminus(\tilde{\Delta}\cup\tilde{E}_{2}\cup\tilde{E}_{3}\cup\tilde{E}_{5}$),
since the curves $\tilde{F}_{i}$ are disjoint, there exists a pair of curves,
say $\tilde{F}_{1}$ and $\tilde{F}_{2}$, such that
$\tilde{s}\notin\tilde{F}_{1}\cup\tilde{F}_{2}$. Let
$\tau=\tau_{3}\circ\tau_{2}\circ\tau_{1}$, and consider the commutative
diagram

$\textstyle{\tilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\kappa^{\prime}\circ\tau_{1}}$$\textstyle{\hat{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{\ddot{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\tau_{3}}\circ\bar{\tau_{2}}}$$\scriptstyle{\pi}$$\textstyle{S_{3}}$$\textstyle{T}$

where the surfaces and morphisms are precisely those described in 4.1.7.

Now assume that the log pair $(\tilde{S},\tilde{D})$ is not log-canonical at
$\tilde{s}$. Let $\hat{D}=\tau_{*}(\tilde{D})$, $\hat{s}=\tau(\tilde{s})$,
$\ddot{D}=(\kappa^{\prime}\circ\tau_{1})_{*}(\tilde{D})$,
$\ddot{s}=(\kappa^{\prime}\circ\tau_{1})(s)$ and observe that $\hat{D}$ and
$\ddot{D}$ are anti-canonical $\mathbb{Q}$-divisors on $\hat{S}$ and
$\ddot{S}$ respectively. Since $\tilde{s}$ does not belong to the exceptional
locus of $\tau$, the log pair $(\hat{S},\hat{D})$ is not log-canonical at
$\hat{s}$. Similarly, $(\ddot{S},\ddot{D})$ is not log-canonical at
$\ddot{s}$.

Since (by Proposition 4.1.6 (b)) $\hat{S}$ is a smooth del Pezzo surface of
degree 1, Lemma 1.5.7 (c) implies that $\hat{s}$ is not the base point of
$|-K_{\hat{S}}|$. Let $\hat{C}$ denote the unique curve in $|-K_{\hat{S}}|$
passing through $\hat{s}$. Then $\operatorname{{\rm Supp}}(\hat{D})$ contains
$\operatorname{{\rm Supp}}(\hat{C})$ and $(\hat{S},\hat{C})$ is not log-
canonical at $\hat{s}$. By Lemma 1.5.7, $\hat{C}$ is a reduced rational curve
with a cusp at $\hat{s}$. Since $\hat{E}_{5}$ and $\hat{C}$ are both anti-
canonical divisors on $\hat{S}$, $\hat{E}_{5}\cdot\hat{C}=1$. This implies
that $\hat{C}$ and $\hat{E}_{5}$ intersect transversely at a unique point,
necessarily different from the singular point $\hat{x_{0}}$ of $\hat{E}_{5}$
(recalling from 4.1.5 that $\hat{x_{0}}$ is the cusp of $\hat{E}_{5}$). As a
consequence, the proper transform of $\hat{C}$ on $\tilde{S}$, which we denote
by $\tilde{C}$, intersects $\tilde{E}_{5}$ transversely and is disjoint from
$\tilde{E}_{2}\cup\tilde{E}_{3}\cup\tilde{\Delta}$.

Recall that the contraction $\pi:\ddot{S}\to T$ of
$\ddot{E}_{3}\cup\ddot{E}_{5}$ is the minimal resolution of $T$, a del Pezzo
surface of degree $1$ with a unique $A_{2}$-singularity. Since $\tilde{s}$ is
not contracted by $\pi\circ\kappa^{\prime}\circ\tau_{1}$,
$D_{T}=(\pi\circ\kappa^{\prime}\circ\tau_{1})_{*}(\tilde{D})$ is an effective
anti-canonical $\mathbb{Q}$-divisor on $T$ for which the log pair $(T,D_{T})$
is not log-canonical at the point
$t=(\pi\circ\kappa^{\prime}\circ\tau_{1})(\tilde{s})$. By Lemma 1.5.7 (c), $t$
is not the base point of $|-K_{T}|$. Also note that since
$\tilde{s}\notin\tilde{E}_{3}\cup\tilde{E}_{5}\cup\tilde{F}_{1}\cup\tilde{F}_{2}$,
$t$ is a smooth point of $T$. Let $C^{\prime}_{T}$ be the unique divisor in
$|-K_{T}|$ passing through $t$. Then, by Lemma 1.5.7 (d), the log pair
$(T,C^{\prime}_{T})$ is not log-canonical at $t$ and $\operatorname{{\rm
Supp}}(C^{\prime}_{T})\subseteq\operatorname{{\rm Supp}}(D_{T})$. Again by
Lemma 1.5.7 (b), it follows that $C^{\prime}_{T}$ is a reduced and irreducible
rational curve with a cusp at $t$.

Let $E_{2,T}=\pi_{*}(\ddot{E}_{2})$. Observe that $E_{2,T}$ is an anti-
canonical divisor on $T$ containing the singular point of $T$. Also, $t\notin
E_{2,T}$. It follows that the anti-canonical divisor
$C^{\prime}_{T}\in|-K_{T}|$ does not pass through the singular point of $T$
(otherwise the singular point of $T$ would be a base point of $|-K_{T}|$ which
is impossible by Lemma 1.5.7 (a)) and $C^{\prime}_{T}$ intersects ${E}_{2,T}$
transversely at a smooth point of $T$. Let $\ddot{C}^{\prime}$ denote the
proper transform of $C^{\prime}_{T}$ in $\ddot{S}$. Then $\ddot{C}^{\prime}$
is an anti-canonical divisor on $\ddot{S}$ that is disjoint from
$\ddot{E}_{3}\cup\ddot{E}_{5}$ and intersects $\ddot{E}_{2}$ transversely at a
single point. This implies that $\tilde{C}^{\prime}$ (the proper transform of
$\ddot{C}$ in $\tilde{S}$) meets $\tilde{E}_{2}$ and is disjoint from
$\tilde{E}_{3}\cup\tilde{E}_{5}\cup\tilde{\Delta}$.

Observe that $\kappa_{*}(\hat{C})$ and
$\bar{\tau_{3}}\circ\bar{\tau_{2}}_{*}(\ddot{C})$ are both anti-canonical
divisors on $S_{3}$ and both have a cusp at the point
$s_{3}=\kappa\circ\tau(\tilde{s})=\bar{\tau_{3}}\circ\bar{\tau_{2}}\circ\kappa^{\prime}\circ\tau_{1}(\tilde{s})$.
Since $\tilde{C}$ intersects $\tilde{E}_{5}$ and $\tilde{C}^{\prime}$ does
not, it follows that the images of $\tilde{C}$ and $\tilde{C}^{\prime}$ in
$S_{3}$, which we denote by $C_{3}$ and $C_{3}^{\prime}$ are distinct. We
obtain

$3=C_{3}\cdot C_{3}^{\prime}\geq(C_{3}\cdot
C_{3}^{\prime})_{s_{3}}\geq\mathrm{mult}_{s_{3}}C_{3}\times\mathrm{mult}_{s_{3}}C_{3}^{\prime}=4$

a contradiction.

∎

######  Proposition 4.1.17.

The surface $S=\operatorname{{\rm Proj}}B_{2,3,5,30}$ does not contain an
anti-canonical polar cylinder.

###### Proof.

By contradiction, assume 4.1.10 holds. With notation as in 4.1.12, Lemma
4.1.15 implies that the linear system $\tilde{\mathcal{L}}$ has a unique
proper base point at a point
$\tilde{s}\in\tilde{S}\setminus(\tilde{\Delta}\cup\tilde{E}_{2}\cup\tilde{E}_{3}\cup\tilde{E}_{5})$.
Lemma 1.6.14 implies the log pair $(\tilde{S},\tilde{D})$ is not log-canonical
at $\tilde{s}$, contradicting Proposition 4.1.16. ∎

###  4.2 Rigidity of $B_{2,3,4,12}$.

######  4.2.1.

Let $B=B_{2,3,4,12}$ and let $S=\operatorname{{\rm
Proj}}B\subset\mathbb{P}(6,4,3,1)$. Consider the homogeneous element
$T=x_{3}\in\operatorname{{\rm Frac}}(B)$. Then
$\Delta=V_{+}(x_{3})\in\operatorname{{\rm Div}}(S)$ is the unique
$\mathbb{Q}$-divisor satisfying $B=\bigoplus_{n\in\mathbb{N}}B_{n}$ where
$B_{n}=\text{\rm H}^{0}(S,\operatorname{\mathcal{O}}_{S}(n\Delta))T^{n}$ for
all $n\in\mathbb{N}$ (as defined in Theorem 1.6.5). Considering $\Delta$ as a
closed subvariety of $S$, we find $\Delta\cong\mathbb{P}^{1}.$

Since $\operatorname{{\rm cotype}}(2,3,4,12)=0$, Proposition 2.1.4 implies
that $S$ is a well-formed hypersurface. Since (by Proposition 1.4.8) $S$ is
normal and Cohen-Macaulay, we have
$\omega_{S}\overset{\ref{NormalCMCoincide}}{\cong}\omega_{S}^{o}\overset{\ref{dualizing}}{\cong}\operatorname{\mathcal{O}}_{S}(-2)\overset{\ref{Demazure}}{\cong}\operatorname{\mathcal{O}}_{S}(-2\Delta)$.

######  Remarks 4.2.2.

The following hold:

  1. (a)

$2\Delta$ is an ample anti-canonical divisor of $S$;

  2. (b)

$(K_{S})^{2}=\frac{2}{3}$.

In particular, $S$ is a singular del Pezzo surface.

###### Proof.

Part (a) is clear and we leave it to the reader to verify (b) following the
argument given in Remarks 4.1.2 (b). ∎

######  Proposition 4.2.3.

The singular points of $S\subset\mathbb{P}(6,4,3,1)$ are
$\\{[1:-1:0:0],[1:0:\zeta_{8}:0],[1:0:\zeta_{8}^{-1}:0]\\}$ where $\zeta_{8}$
is a primitive $8^{th}$ root of unity. Moroever,

  1. (a)

$[1:-1:0:0]$ is a $\frac{1}{2}(1,1)$ singularity;

  2. (b)

$[1:0:\zeta_{8}:0]$ and $[1:0:\zeta_{8}^{-1}:0]$ are $\frac{1}{3}(1,1)$
singularities;

  3. (c)

For each $P\in\operatorname{{\rm Sing}}(S)$, $\operatorname{{\rm
mult}}_{P}(\Delta)=1$.

###### Proof.

We leave it to the reader to prove this Proposition following the argument of
Proposition 4.1.3, mutatis mutandis. ∎

######  4.2.4.

Let $P_{2}=[1:-1:0:0]$ denote the $\frac{1}{2}(1,1)$ singularity of $S$ and
let $P_{3+}$ and $P_{3-}$ denote the $\frac{1}{3}(1,1)$ singularities. Let
$\sigma:\tilde{S}\to S$ be the minimal resolution of singularities of $S$ and