Birationally superrigid Fano 3-folds of codimension 4

TL;DR

This paper establishes the birational superrigidity of certain prime Fano 3-folds of codimension 4, expanding the classification of these algebraic varieties and confirming recent constructions.

Contribution

It proves birational superrigidity for a class of Fano 3-folds of codimension 4 with no projection centers, including recent examples by Coughlan and Ducat.

Findings

Proves birational superrigidity for specific Fano 3-folds

Confirms recent constructions of Fano 3-folds are superrigid

Poses questions and conjectures on classification

Abstract

We determine birational superrigidity for a quasi-smooth prime Fano 3-fold of codimension 4 with no projection centers. In particular we prove birational superrigidity for Fano 3-folds of codimension 4 with no projection centers which are recently constructed by Coughlan and Ducat. We also pose some questions and a conjecture regarding the classification of birationally superrigid Fano 3-folds.

Equations

$$R(X,-K_X)=\underset{m\in {\mathbb{Z}}_{\ge 0}}{⨁}{H}^0(X,-m{K}_X),$$

$$c(X,\mathcal{H})=\frac{a_E(K_X)}{m_E(\mathcal{H})}<\frac{1}{n},$$

$$(E^3)=\frac{r^2}{a(r-a)}.$$

$$e:=\min \{{\mathrm{ord}}_E(D_i)/n_i\mid 1\le i\le k\},$$

$$e^{\prime }:=\min \{{\mathrm{ord}}_E(D_i^{\prime })/n_i^{\prime }\mid 1\le i\le k\}\ge e,$$

$$(-K_Y\cdot \tilde{S}\cdot \tilde{L})\le 0,$$

$$(-K_Y\cdot \tilde{T}\cdot \tilde{L})=nl(-K_X{)}^3\ge 0.$$

$$0\ge (-K_Y\cdot \tilde{S}\cdot \tilde{L})=\sum m_i(-K_Y\cdot {\tilde{S}}_i\cdot \tilde{L})+(-K_Y\cdot \tilde{T}\cdot \tilde{L}),$$

$$\tilde{S}\cdot {\tilde{L}}_{\lambda }=\sum _i{c}_i{\tilde{C}}_{\lambda ,i}+\Xi ,$$

$$0\ge (-K_Y\cdot \tilde{S}\cdot {\tilde{L}}_{\lambda })\ge \sum _i{c}_i(-K_Y\cdot {\tilde{C}}_{\lambda ,i}).$$

$$\{{\tilde{C}}_{\lambda }^{\circ }\mid \text{λ∈P1 is general}\}$$

$$\gcd \{a_i\mid 0\le i\le n,i\ne j\}=1$$

$$F_1=F_2=\cdots =F_N=0,$$

$$(F_1=F_2=\cdots =F_N=0)\subset {\mathbb{A}}^{n+1}=\mathrm{Spec}\mathbb{C}[x_0,\dots ,x_n]$$

$$\begin{array}{rl}\Pi (\mathcal{C})& :=\bigcap _{z\in \mathcal{C}}(z=0)\subset \mathbb{P},\\ {\Pi }_X(\mathcal{C})& :=\Pi (\mathcal{C})\cap X\subset X.\end{array}$$

$$\Pi (\mathcal{C})=\Pi (x_{i_1},\dots ,x_{i_m}),{\Pi }_X(\mathcal{C})={\Pi }_X(x_{i_1},\dots ,x_{i_m}),$$

$$b:=\max \{a_i\mid 0\le i\le n,\text{ai is odd}\}.$$

$$(-b{\phi }^{\ast }{K}_X-\frac{1}{2}E)\cdot (-K_Y{)}^2=b(-K_X{)}^3-\frac{1}{2}\le 0.$$

$${\mathrm{ivr}}_{\mathsf{p}}(\mathcal{C}):=\underset{1\le j\le m}{\min }\left\{\frac{\overline{a_{i_j}}}{a_{i_j}{a}_k}\right\},$$

$$\mathrm{wp}(\mathsf{p}):=a(r-a),$$

$$e:=\min \{{\mathrm{ord}}_E(D_{i_j})/a_{i_j}\mid 1\le j\le m\}\ge {\mathrm{ivr}}_{\mathsf{p}}(\mathcal{C}).$$

$$(-{\phi }^{\ast }{K}_X-{\mathrm{ivr}}_{\mathsf{p}}(\mathcal{C})E)(-K_Y{)}^2=(-K_X{)}^3-\frac{{\mathrm{ivr}}_{\mathsf{p}}(\mathcal{C})}{\mathrm{wp}(\mathsf{p})}\le 0$$

$$F_l={\alpha }_l{x}_k^{m_l}{x}_{i_l}+(\text{other terms}),\text{for 1≤l≤n−3},$$

$$\{x_0,x_{i_1},\dots ,x_{i_{n-3}},x_{j_1},x_{j_2},x_{j_3}\}=\{x_0,\dots ,x_n\}.$$

$$\frac{1}{r}(a_{j_1},a_{j_2},a_{j_3}).$$

$$\mathbf{w}(x_1,\dots ,x_n)=\frac{1}{r}(b_1,\dots ,b_n)$$

$$(b_{j_1},b_{j_2},b_{j_3})=(\overline{a_{j_1}},\overline{a_{j_2}},\overline{a_{j_3}}).$$

$$E_{\mathbf{w}}:=(f_1=\cdots =f_{n-3}=0)\subset \mathbb{P}(b_1,\dots ,b_n),$$

$${\mathbf{w}}^{\prime }(x_1,\dots ,x_n)=\frac{1}{r}(b_1^{\prime },\dots ,b_n^{\prime }),$$

$$\Sigma :={\Pi }_X(\{x_0,\dots ,x_n\}\setminus \mathcal{C})$$

Full text

1. Introduction

A prime Fano $3$-fold is a normal projective $\mathbb{Q}$-factorial $3$-fold $X$ with only terminal singularities such that $-K_{X}$ is ample and the class group $\operatorname{Cl}(X)\cong\mathbb{Z}$ is generated by $-K_{X}$. For such $X$, there corresponds the anticanonical graded ring

[TABLE]

and by choosing generators we can embed $X$ into a weighted projective space. By the codimension of $X$ we mean the codimension of $X$ in the weighted projective space. Based on the analysis by Altınok, Brown, Iano-Fletcher, Kasprzyk, Prokhorov, Reid, etc. (see for example [4]), there is a database [5] of numerical data (such as Hilbert series) coming from graded rings that can be the anticanonical graded ring of a prime Fano $3$-fold. Currently it is not a classification, but it serves as an overlist, meaning that the anticanonical graded ring of a prime Fano $3$-fold appears in the database.

The database contains a huge number of candidates, which suggests difficulty of biregular classification of Fano $3$-folds. The aim of this paper is to shed light on the classification of birationally superrigid Fano 3-folds. Here, a Fano $3$-fold of Picard number $1$ is said to be birationally superrigid if any birational map to a Mori fiber space is biregular. We remark that, in [1], a possible approach to achieving birational classification of Fano $3$-folds is suggested by introducing notion of solid Fano $3$-folds, which are Fano $3$-folds not birational to neither a conic bundles nor a del Pezzo fibration.

Up to codimension $3$, we have satisfactory results on the classification of quasi-smooth prime Fano $3$-folds: the classification is completed in codimensions $1$ and $2$ ([12], [8], [3]) and in codimension $3$ the existence is known for all $70$ numerical data in the database. Moreover birational superrigidity of quasi-smooth prime Fano $3$-folds of codimension at most $3$ has been well studied as well (see [14], [9], [7], [19], [2], [1], and see also [20], [21] for solid cases in codimension $2$).

For quasi-smooth prime Fano $3$-folds of codimension $4$, there are 145 candidates of numerical data in [5]. In [6], existence for 116 data is proved, where the construction is given by birationally modifying a known variety. This process is called unprojection and, as a consequence, a constructed Fano $3$-fold corresponding to each of the 116 data admits a Sarkisov link to a Mori fiber space, hence it is not birationally superrigid. The 116 families of Fano $3$-folds are characterized as those that possesses a singular point which is so called a type $\mathrm{I}$ projection center (see [6] for datails). There are other types of projection centers (such as types $\mathrm{II}_{1},\dots,\mathrm{II}_{7},\mathrm{IV}$ according to the database [5]). Through the known results in codimensions $1$, $2$ and $3$, we can expect that the existence of a projection center violates birational superrigidity. Therefore it is natural to consider prime Fano $3$-folds without projection centers for the classification of birational superrigid Fano $3$-folds (see also the discussion in Section 5).

According to the database [5], there are $5$ candidates of quasi-smooth prime Fano $3$-folds of codimension $4$ with no projection centers. Those are identified by database numbers $\#25$, $\#166$, $\#282$, $\#308$ and $\#29374$. Among them, $\#29374$ corresponds to smooth prime Fano $3$-folds of degree $10$ embedded in $\mathbb{P}^{7}$, and it is proved in [11] that they are not birationally superrigid (not even birationally rigid, a weaker notion than superrigidity). Recently Coughlan and Ducat [10] constructed many prime Fano $3$-folds including those corresponding to $\#25$ and $\#282$ and we sometimes refer to these varieties as cluster Fano $3$-folds. There are two constructions, $\mathsf{G}^{(4)}_{2}$ and $\mathsf{C}_{2}$ formats (see [10, Section 5.6] for details and see Section 4.1 for concrete descriptions) for $\#282$ and they are likely to sit in different components of the Hilbert scheme.

Theorem 1.1**.**

Let $X$ be a quasi-smooth prime Fano $3$-fold of codimension $4$ and of numerical type $\#282$ which is constructed in either $\mathsf{G}^{(4)}_{2}$ format or $\mathsf{C}_{2}$ format. If $X$ is constructed in $\mathsf{C}_{2}$ format, then we assume that $X$ is general. Then $X$ is birationally superrigid.

For the remaining three candidates $\#25$, $\#166$ and $\#282$, we can prove birational superrigidity in a stronger manner; we are able to prove birational superrigidity for these $3$ candidates by utilizing only numerical data. Here, by numerical data for a candidate Fano $3$-fold $X$, we mean the weights of the weighted projective space, degrees of the defining equations, the anticanonical degree $(-K_{X})^{3}$ and the basket of singularities of $X$ (see Section 3). Note that we do not know the existence of Fano $3$-folds for $\#166$ and $\#308$.

Theorem 1.2**.**

Let $X$ be a quasi-smooth prime Fano $3$-fold of codimension $4$ and of numerical type $\#25$, $\#166$ or $\#308$. Then $X$ is birationally superrigid.

Acknowledgments**.**

The author would like to thank Stephen Coughlan for giving me fruitful information on cluster Fano 3-folds. He is partially supported by JSPS KAKENHI Grant Number JP18K03216.

2. Birational superrigidity

2.1. Basic properties

Throughout this subsection, we assume that $X$ is a Fano $3$-fold of Picard number $1$, that is, $X$ is a normal projective $\mathbb{Q}$-factorial $3$-fold such that $X$ has only terminal singularities, $-K_{X}$ is ample and $\operatorname{rank}\operatorname{Pic}(X)=1$.

Definition 2.1**.**

We say that $X$ is birationally superrigid if any birational map $\sigma\colon X\dashrightarrow Y$ to a Mori fiber space $Y\to T$ is biregular.

By an extremal divisorial extraction $\varphi\colon(E\subset Y)\to(\Gamma\subset X)$, we mean an extremal divisorial contraction $\varphi\colon Y\to X$ from a normal projective $\mathbb{Q}$-factorial variety $Y$ with only terminal singularities such that $E$ is the $\varphi$-exceptional divisor and $\Gamma=\varphi(E)$.

Definition 2.2**.**

Let $\mathcal{H}\sim_{\mathbb{Q}}-nK_{X}$ be a movable linear system, where $n$ is a positive integer. A maximal singularity of $\mathcal{H}$ is an extremal extraction $\varphi\colon(E\subset Y)\to(\Gamma\subset X$ such that

[TABLE]

where

• •

c(X,\mathcal{H}):=\max\{\,\lambda\mid\text{K_{X}+\lambda\mathcal{H} is canonical}\,\} is the canonical threshold of $(X,\mathcal{H})$,

• •

$a_{E}(K_{X})$ is the discrepancy of $K_{X}$ along $E$, and

• •

$m_{E}(\mathcal{H})$ is the multiplicity along $E$ of the proper transform $\varphi_{*}^{-1}\mathcal{H}$ on $Y$.

We say that an extremal divisorial extraction is a maximal singularity if if there exists a movable linear system $\mathcal{H}$ such that the extraction is a maximal singularity of $\mathcal{H}$. A subvariety $\Gamma\subset X$ is called a maximal center if there is an maximal singularity $Y\to X$ whose center is $\Gamma$.

Theorem 2.3**.**

If $X$ admits no maximal center, then $X$ is birationally superrigid.

For a proof of birational superrigidity of a given Fano $3$-fold $X$ of Picard number $1$, we need to exclude each subvariety of $X$ as a maximal center. In the next subsection we will explain several methods of exclusion under a relatively concrete setting. Here we discuss methods of excluding terminal quotient singular points in a general setting.

For a terminal quotient singular point $\mathsf{p}\in X$ of type $\frac{1}{r}(1,a,r-a)$, where $r$ is coprime to $a$ and $0<a<r$, there is a unique extremal divisorial extraction $\varphi\colon(E\subset Y)\to(\mathsf{p}\in X)$, which is the weighted blowup with weight $\frac{1}{r}(1,a,r-a)$, and we call it the Kawamata blowup (see [17] for details). The integer $r>1$ is called the index of $\mathsf{p}\in X$. For the Kawamata blowup $\varphi\colon(E\subset Y)\to(\mathsf{p}\in X)$, we have $K_{Y}=\varphi^{*}K_{X}+\frac{1}{r}E$ and

[TABLE]

For a divisor $D$ on $X$, the order of $D$ along $E$, denote by $\operatorname{ord}_{E}(D)$, is defined to be the coefficient of $E$ in $\varphi^{*}D$.

We first explain the most basic method.

Lemma 2.4** ([9, Lemma 5.2.1]).**

Let $\mathsf{p}\in X$ a terminal quotient singular point and $\varphi\colon(E\subset Y)\to(\mathsf{p}\in X)$ the Kawamata blowup. If $(-K_{Y})^{2}\notin\operatorname{Int}\operatorname{\overline{NE}}(Y)$, then $\mathsf{p}$ is not a maximal center.

For the application of the above lemma, we need to find a nef divisor on $Y$. The following result, which is a slight generalization of [20, Lemma 6.6], is useful.

Lemma 2.5**.**

Let $\mathsf{p}\in X$ be a terminal quotient singular point and $\varphi\colon(E\subset Y)\to(\mathsf{p}\in X)$ the Kawamata blowup. Assume that there are effective Weil divisors $D_{1},\dots,D_{k}$ such that the intersection $D_{1}\cap\cdots\cap D_{k}$ does not contain a curve through $\mathsf{p}$. We set

[TABLE]

where $n_{i}$ is the positive rational number such that $D_{i}\sim_{\mathbb{Q}}-n_{i}K_{X}$. Then $-\varphi^{*}K_{X}-\lambda E$ is a nef divisor for $0\leq\lambda\leq e$.

Proof.

We may assume $e>0$, that is, $D_{i}$ passes through $\mathsf{p}$ for any $i$. For an effective divisor $D\sim_{\mathbb{Q}}-nK_{X}$, we call $\operatorname{ord}_{E}(D)/n$ the vanishing ratio of $D$ along $E$. For $1\leq i\leq k$, we choose a component of $D_{i}$, denoted $D^{\prime}_{i}$, which has maximal vanishing ratio along $E$ among the components of $D_{i}$. Clearly we have $D^{\prime}_{1}\cap\cdots\cap D^{\prime}_{k}$ does not contain a curve through $\mathsf{p}$ and we have

[TABLE]

where $n^{\prime}_{i}\in\mathbb{Q}$ is such that $D^{\prime}_{i}\sim_{\mathbb{Q}}-n^{\prime}_{i}K_{X}$. Since $D^{\prime}_{1},\dots,D^{\prime}_{k}$ are prime divisors, we can apply [20, Lemma 6.6] and conclude that $-\varphi^{*}K_{X}-e^{\prime}E$ is nef. Then so is $-\varphi^{*}K_{X}-\lambda E$ for any $0\leq\lambda\leq e^{\prime}$ since $-\varphi^{*}K_{X}$ is nef, and the proof is completed. ∎

We have another method of exclusion which can be sometimes effective when Lemma 2.4 is not applicable.

Lemma 2.6**.**

Let $\mathsf{p}\in X$ be a terminal quotient singular point and $\varphi\colon(E\subset Y)\to(\mathsf{p}\in X)$ the Kawamata blowup. Suppose that there exists an effective divisor $S$ on $X$ passing through $\mathsf{p}$ and a linear system $\mathcal{L}$ of divisors on $X$ passing through $\mathsf{p}$ with the following properties.

1. (1)

$S\cap\operatorname{Bs}\mathcal{L}$* does not contain a curve passing through $\mathsf{p}$, and* 2. (2)

For a general member $L\in\mathcal{L}$, we have

[TABLE]

where $\tilde{S},\tilde{L}$ are the proper transforms of $S,L$ on $Y$, respectively.

Then $\mathsf{p}$ is not a maximal center.

Proof.

We write $D\sim-nK_{X}$. Write $S=\sum m_{i}S_{i}+T$, where $m_{i}>0$, $S_{i}$ is a prime divisor and $T$ is an effective divisor which does not pass through $\mathsf{p}$. We have $T\sim-lK_{X}$ for some $l\geq 0$ and

[TABLE]

Since

[TABLE]

there is a component $S_{i}$ for which $(-K_{Y}\cdot\tilde{S}_{i}\cdot\tilde{L})\leq 0$. Since $\mathsf{p}\in S_{i}\cap\operatorname{Bs}\mathcal{L}\subset S\cap\operatorname{Bs}\mathcal{L}$, we may assume that $S$ is a prime divisor by replacing $S$ by $S_{i}$.

Write $\mathcal{L}=\{L_{\lambda}\mid\lambda\in\mathbb{P}^{1}\}$. For $\lambda\in\mathbb{P}^{1}$, we write $S\cdot L_{\lambda}=\sum_{i}c_{i}C_{\lambda,i}$, where $c_{i}\geq 0$ and $C_{\lambda,i}$ is an irreducible and reduced curve on $X$. For a curve or a divisor $\Delta$ on $X$, we denote by $\tilde{\Delta}$ its proper transform on $Y$. Then,

[TABLE]

where $\Xi$ is an effective $1$-cycle supported on $E$. Since any component of $\Xi$ is contracted by $\varphi$ and $-K_{Y}$ is $\varphi$-ample, we have $(-K_{Y}\cdot\Xi)\geq 0$. Thus, for a general $\lambda\in\mathbb{P}^{1}$, we have

[TABLE]

It follows that $(-K_{Y}\cdot\tilde{C}_{\lambda,i})\leq 0$ for some $i$. We choose such a $\tilde{C}_{\lambda,i}$ and denote it as $\tilde{C}^{\circ}_{\lambda}$. By the assumption (1), the set

[TABLE]

consists of infinitely many distinct curves. We have $(-K_{Y}\cdot\tilde{C}^{\circ}_{\lambda})\leq 0$ by the construction. We see that $(E\cdot\tilde{C}_{\lambda}^{\circ})>0$ since $\tilde{C}_{\lambda}$ is the proper transform of a curve passing through $\mathsf{p}$. Therefore $\mathsf{p}$ is not a maximal center by [20, Lemma 2.20]. ∎

2.2. Fano varieties in a weighted projective space

Let $\mathbb{P}=\mathbb{P}(a_{0},\dots,a_{n})$ be a weighted projective space with homogeneous coordinates $x_{0},\dots,x_{n}$ of $\deg x_{i}=a_{i}$. We assume that $\mathbb{P}$ is well formed, that is,

[TABLE]

for $j=0,1,\dots,n$. Throughout the present subsection, let $X\subset\mathbb{P}$ be a normal projective $3$-fold defined by the equations

[TABLE]

where $F_{i}\in\mathbb{C}[x_{0},\dots,x_{n+1}]$ is a homogeneous polynomial of degree $d_{i}$ with respect to the grading $\deg x_{i}=a_{i}$.

Definition 2.7**.**

We say that $X$ is quasi-smooth if the affine cone

[TABLE]

is smooth outside the origin.

In the following we assume that $X$ is a quasi-smooth prime Fano $3$-fold. For $0\leq i\leq n$, we define $\mathsf{p}_{x_{i}}=(0\!:\!\cdots\!:\!1\!:\!\cdots\!:\!0)\in\mathbb{P}$, where the unique $1$ is in the $(i+1)$st position, and we define $D_{i}=(x_{i}=0)\cap X$ which is a Weil divisor such that $D_{i}\sim-a_{i}K_{X}$.

Lemma 2.8**.**

If $(-K_{X})^{3}\leq 1$, then no curve on $X$ is a maximal center.

Proof.

The same proof of [1, Lemma 2.1] applies in this setting without any change. ∎

Lemma 2.9**.**

Assume that $a_{0}\leq a_{1}\leq\dots\leq a_{n}$. If $a_{n-1}a_{n}(-K_{X})^{3}\leq 4$, then no nonsingular point of $X$ is a maxinal center.

Proof.

The proof is almost identical to that of [1, Lemma 2.6]. ∎

Definition 2.10**.**

Let $\mathcal{C}\subset\{x_{0},\dots,x_{n}\}$ be a set of homogeneous coordinates. We define

[TABLE]

We also denote

[TABLE]

when $\mathcal{C}=\{x_{i_{1}},\dots,x_{i_{m}}\}$.

Lemma 2.11**.**

Let $\mathsf{p}\in X$ be a singular point of type $\frac{1}{2}(1,1,1)$ and let

[TABLE]

If $2b(-K_{X})^{3}\leq 1$, then $\mathsf{p}$ is not a maximal center.

Proof.

Let $\mathcal{C}=\{x_{i_{1}},\dots,x_{i_{m}}\}$ be the set of homogeneous coordinates of odd degree. The set $\Pi_{X}(\mathcal{C})=D_{i_{1}}\cap\cdots\cap D_{i_{m}}$ consists of singular points since $X$ is quasi-smooth and has only terminal quotient singularities (which are isolated), In particular $\Pi_{X}(\mathcal{C})$ is a finite set of points. Let $\varphi\colon(E\subset Y)\to(\mathsf{p}\in X)$ be the Kawamata blowup. Then $\operatorname{ord}_{E}(D_{i_{j}})\geq 1/2$ since $2D_{i_{j}}$ is a Cartier divisor passing through $\mathsf{p}$ and thus $-b\varphi^{*}K_{X}-\frac{1}{2}E$ is nef by Lemma 2.5. We have

[TABLE]

This shows that $(-K_{Y})^{2}\notin\operatorname{\overline{NE}}(Y)$ and $\mathsf{p}$ is not a maximal center by Lemma 2.4. ∎

Definition 2.12**.**

Let $\mathsf{p}=\mathsf{p}_{x_{k}}\in X$ be a terminal quotient singular point of type $\frac{1}{a_{k}}(1,c,a_{k}-c)$ for some $c$ with $1\leq c\leq a_{k}/2$. We define

[TABLE]

where $\mathcal{C}=\{x_{i_{1}},\dots,x_{i_{m}}\}$ and $\overline{a_{j_{i}}}$ is the integer such that $1\leq\overline{a_{j_{i}}}\leq a_{k}$ and $\overline{a_{j_{i}}}$ is congruent to $a_{j_{i}}$ modulo $a_{k}$, and call it the initial vanishing ratio of $\mathcal{C}$ at $\mathsf{p}$.

Definition 2.13**.**

For a terminal quotient singularity $\mathsf{p}$ of type $\frac{1}{r}(1,a,r-a)$, we define

[TABLE]

and call it the weight product of $\mathsf{p}$.

Lemma 2.14**.**

Let $\mathsf{p}=\mathsf{p}_{x_{k}}\in X$ be a terminal quotient singular point. Suppose that there exists a subset $\mathcal{C}\subset\{x_{0},\dots,x_{n}\}$ satisfying the following properties.

1. (1)

$\mathsf{p}\in\Pi_{X}(\mathcal{C})$, or equivalently $x_{k}\notin\mathcal{C}$. 2. (2)

$\Pi_{X}(\mathcal{C}\cup\{x_{k}\})=\emptyset$. 3. (3)

$\operatorname{ivr}_{\mathsf{p}}(\mathcal{C})\geq\operatorname{wp}(\mathsf{p})(-K_{X})^{3}$.

Then $\mathsf{p}$ is not a maximal center.

Proof.

We write $\mathcal{C}=\{x_{i_{1}},\dots,x_{i_{m}}\}$. We claim that $\Pi_{X}(\mathcal{C})=D_{i_{1}}\cap\cdots\cap D_{i_{m}}$ is a finite set of points. Indeed, if it contains a curve, then $\Pi_{X}(\mathcal{C}\cup\{x_{k}\})=\Pi_{X}(\mathcal{C})\cap D_{k}\neq\emptyset$ since $D_{k}$ is an ample divisor on $X$. This is impossible by the assumption (2). Note that we have $\operatorname{ord}_{E}(D_{i_{j}})\geq\overline{a_{i_{j}}}/a_{k}$ (cf. [1, Section 3]) so that

[TABLE]

By Lemma 2.5, $-\varphi^{*}K_{X}-\operatorname{ivr}_{\mathsf{p}}(\mathcal{C})E$ is nef and we have

[TABLE]

by the assumption (3). Therefore $(-K_{Y})^{2}\notin\operatorname{\overline{NE}}(Y)$ and $\mathsf{p}$ is not a maximal center. ∎

Let $\mathsf{p}\in X$ be a singular point such that it can be transformed to $\mathsf{p}_{x_{k}}$ by a change of coordinates. For simplicity of the description we assume $\mathsf{p}=\mathsf{p}_{x_{0}}$ and we set $r=a_{0}>1$. Let $\varphi\colon(E\subset Y)\to(\mathsf{p}\in X)$ be the Kawamata blowup. We explain a systematic way to estimate $\operatorname{ord}_{E}(x_{i})$ for coordinates $x_{i}$ and also an explicit description of $\varphi$. It is a consequence of the quasi-smoothness of $X$ that after re-numbering the defining equation we can write

[TABLE]

where $\alpha_{l}\in\mathbb{C}\setminus\{0\}$, $m_{l}$ is a positive integer and $x_{0},x_{i_{1}},\dots,x_{i_{n-3}}$ are mutually distinct so that by denoting the other $3$ coordinates as $x_{j_{1}},x_{j_{2}},x_{j_{3}}$ we have

[TABLE]

In this case we can choose $x_{j_{1}},x_{j_{2}},x_{j_{3}}$ as local orbi-coordinates of $X$ at $\mathsf{p}$ and the singular point $\mathsf{p}$ is of type

[TABLE]

Definition 2.15** ([1, Definitions 3.6, 3.7]).**

For an integer $a$, we denote by $\bar{a}$ the positive inter such that $\bar{a}\equiv a\pmod{r}$ and $0<\bar{a}\leq r$. We say that

[TABLE]

is an admissible weight at $\mathsf{p}$ if $b_{i}\equiv a_{i}\pmod{r}$ for any $i$.

For an admissible weight $\mathbf{w}$ at $\mathsf{p}$ and a polynomial $f=f(x_{0},\dots,x_{n})$, we denote by $f^{\mathbf{w}}$ the lowest weight part of $f$, where we assume that $\mathbf{w}(x_{0})=0$.

We say that an admissible weight $\mathbf{w}$ at $\mathsf{p}$ satisfies the KBL condition if $x_{0}^{e_{l}}x_{i_{l}}\in F_{l}^{\mathbf{w}}$ for $1\leq l\leq n-3$ and

[TABLE]

Let $\mathbf{w}(x_{1},\dots,x_{n})=\frac{1}{r}(b_{1},\dots,b_{n})$ be an admissible weight at $\mathsf{p}$ satisfying the KBL condition. We denote by $\Phi_{\mathbf{w}}\colon Q_{\mathbf{w}}\to\mathbb{P}$ at $\mathsf{p}$ with weight $\mathbf{w}$, and by $Y_{\mathbf{w}}$ the proper transform of $X$ via $\Phi_{\mathbf{w}}$. Then the induced morphism $\varphi_{\mathbf{w}}=\Phi_{\mathbf{w}}|_{Y_{\mathbf{w}}}\colon Y_{\mathbf{w}}\to X$ coincides with the Kawamata blowup at $\mathsf{p}$. From this we see that the exceptional divisor $E$ is isomorphic to

[TABLE]

where $f_{l}=F_{l}^{\mathbf{w}}(1,x_{1},\dots,x_{n})$. We refer readers to [1, Section 3] for details.

Lemma 2.16** ([1, Lemma 3.9]).**

Let $\mathbf{w}(x_{1},\dots,x_{n})=\frac{1}{r}(b_{1},\dots,b_{n})$ be an admissible weight at $\mathsf{p}\in X$ satisfying the KBL condition. Then the following assertions hold.

1. (1)

We have $\operatorname{ord}_{E}(D_{i})\geq b_{i}/r$ for any $i$. 2. (2)

If $F_{l}^{\mathbf{w}}=\alpha_{l}x_{0}^{m_{l}}x_{i_{l}}$, where $\alpha_{i}\in\mathbb{C}\setminus\{0\}$, for some $1\leq l\leq n-3$, then the weight

[TABLE]

where $b^{\prime}_{j}=b_{j}$ for $j\neq l$ and $b^{\prime}_{l}=b_{l}+r$, satisfies the KBL condition. In particular, $\operatorname{ord}_{E}(D_{l})\geq(b_{l}+r)/r$.

We will use the following notation for a polynomial $f=f(x_{0},\dots,x_{n})$.

• •

For a monomial $p=x_{0}^{e_{0}}\cdots x_{n}^{e_{n}}$, we write $p\in f$ if $p$ appears in $f$ with non-zero (constant) coefficient.

• •

For a subset $\mathcal{C}\subset\{x_{0},\dots,x_{n}\}$ and $\Pi=\Pi(\mathcal{C})$, we denote by $f|_{\Pi}$ the polynomial in variables $\{x_{0},\dots,x_{n}\}\setminus\mathcal{C}$ obtained by putting $x_{i}=0$ for $x_{i}\in\mathcal{C}$ in $f$.

Remark 2.17**.**

We explain some consequences of quasi-smoothness, which will be frequently used in Section 3. We keep the above notation and assumption. In particular $X\subset\mathbb{P}(a_{0},\dots,a_{n})$ is assumed to be quasi-smooth.

1. (1)

Let $\mathcal{C}=\{x_{i_{1}},\dots,x_{i_{m}}\}$ be the coordinates such that $r:=\gcd\{a_{i_{1}},\dots,a_{i_{m}}\}>1$ and $a_{j}$ is coprime to $r$ for any $j\neq i_{1},\dots,i_{m}$. Then

[TABLE]

is contained in the singular locus of $X$ and $X$ has a quotient singular point of index $r$ at each point of $\Sigma$. In particular, if $X$ has only isolated singularities (e.g. $\dim X=3$ and $X$ has only terminal singularities), then either $\Sigma=\emptyset$ of $\Sigma$ consists of finite set of singular points of index $r$. 2. (2)

Let $x_{k}$ be the coordinate such that $a_{j}\neq a_{k}$ for any $j\neq k$. If $X$ does not contain a singular point of index $r$, then $\mathsf{p}_{k}\notin X$, that is, a power of $x_{k}$ appears in one of the defining polynomials with non-zero coefficient.

3. Proof of birational superrigidity by numerical data

We prove birational superrigidity of codimension $4$ quasi-smooth prime Fano $3$-folds with no projections by utilizing only numerical data. The numerical data for each Fano $3$-fold will be described in the beginning of the corresponding subsection. The Fano $3$-folds are embedded in a weighted projective $7$-space, denoted by $\mathbb{P}$, and we use the symbol $p,q,r,s,t,u,v,w$ for the homogeneous coordinates of $\mathbb{P}$. We use the following terminologies: Let $X\subset\mathbb{P}$ be a codimension $4$ quasi-smooth prime Fano $3$-fold. For a homogeneous coordinate $z\in\{p,q,\dots,w\}$,

• •

$D_{z}:=(z=0)\cap X$ is the Weil divisor on $X$ cut out by $z$, and

• •

$\mathsf{p}_{z}\in\mathbb{P}$ is the point at which only the coordinate $z$ does not vanish.

Note that Theorem 1.2 will follow from Theorems 3.1, 3.2, 3.4.

3.1. Fano $3$-folds of numerical type $\#25$

Let $X$ be a quasi-smooth prime Fano $3$-fold of numerical type $\#25$, whose data consist of the following.

• •

$X\subset\mathbb{P}(2_{p},5_{q},6_{r},7_{s},8_{t},9_{u},10_{v},11_{w})$.

• •

$(-K_{X})^{3}=1/70$.

• •

$\deg(F_{1},F_{2},F_{3},F_{4},F_{5},F_{6},F_{7},F_{8},F_{9})=(16,17,18,18,19,20,20,21,22)$.

• •

$\mathcal{B}_{X}=\left\{7\times\frac{1}{2}(1,1,1),\frac{1}{5}(1,1,4),\frac{1}{7}(1,2,5)\right\}$.

Here the subscripts $p,q,\dots,w$ of the weights means that they are the homogeneous coordinates of the indicated degrees, and $\mathcal{B}_{X}$ indicates the numbers and the types of singular points of $X$.

Theorem 3.1**.**

Let $X$ be a quasi-smooth prime codimension $4$ Fano $3$-fold of numerical type $\#25$. Then $X$ is birationally superrigid.

Proof.

By Lemmas 2.8 and 2.9, no curve and no nonsingular point on $X$ is a maximal center. By Lemma 2.11, singular points of type $\frac{1}{2}(1,1,1)$ are not maximal centers.

Let $\mathsf{p}$ be the singular point of type $\frac{1}{5}(1,1,4)$. Replacing the coordinate $v$ if necessary, we may assume $\mathsf{p}=\mathsf{p}_{q}$. We set $\mathcal{C}=\{p,s,u,v\}$. We have

[TABLE]

By Lemma 2.14, it remains to show that $\Pi_{X}:=\Pi_{X}(\mathcal{C}\cup\{q\})=\emptyset$. We set $\Pi:=\Pi(\mathcal{C}\cup\{q\})\subset\mathbb{P}$ so that $\Pi_{X}=\Pi\cap X$. Since $\mathsf{p}_{t}\notin X$, one of the defining polynomials contain a power of $t$. By looking at the degrees of $F_{1},\dots,F_{9}$, we have $t^{2}\in F_{1}$. Similarly, we have $r^{3}\in F_{3}$ and $w^{2}\in F_{9}$ after possibly interchanging $F_{3}$ and $F_{4}$. The monomial $t^{2}$ (resp. $r^{3}$) is the only monomial of degree $16$ (resp. $18$) consisting of the variables $r,t,w$. The monomials $w^{2}$ and $t^{2}r$ are the only monomials of degree $22$ consisting the variables $r,t,w$. Hence, re-scaling $r,t,w$, we can write

[TABLE]

for some $\alpha\in\mathbb{C}$. The set $\Pi_{X}$ is contained in the common zero loci of the above 3 polynomials inside $\Pi$. The equations have only trivial solution and this shows that $\Pi_{X}=\emptyset$. Thus $\mathsf{p}$ is not a maximal center.

Let $\mathsf{p}=\mathsf{p}_{s}$ be the singular point of type $\frac{1}{7}(1,2,5)$ and set $\mathcal{C}=\{p,q,r\}$. We have

[TABLE]

By Lemma 2.14, it remains to show that $\Pi_{X}:=\Pi_{X}(\mathcal{C}\cup\{s\})=\emptyset$. We set $\Pi:=\Pi(\mathcal{C}\cup\{s\})\subset\mathbb{P}$ so that $\Pi_{X}=\Pi\cap X$. Since $\mathsf{p}_{t},\mathsf{p}_{u},\mathsf{p}_{v},\mathsf{p}_{w}\notin X$, we may assume $t^{2}\in F_{1}$, $u^{2}\in F_{3}$, $v^{2}\in F_{6}$ and $w^{2}\in F_{9}$ after possibly interchanging defining polynomials of the same degree. Then we can write

[TABLE]

for some $\alpha,\beta,\gamma\in\mathbb{C}$. This shows that $\Pi_{X}=\emptyset$ and thus $\mathsf{p}$ is not a maximal center. This completes the proof. ∎

3.2. Fano $3$-folds of numerical type $\#166$

Let $X$ be a quasi-smooth prime Fano $3$-fold of numerical type $\#166$, whose data consist of the following.

• •

$X\subset\mathbb{P}(2_{p},2_{q},3_{r},3_{s},4_{t},4_{u},5_{v},5_{w})$.

• •

$(-K_{X})^{3}=1/6$.

• •

$\deg(F_{1},F_{2},F_{3},F_{4},F_{5},F_{6},F_{7},F_{8},F_{9})=(8,8,8,9,9,9,10,10,10)$.

• •

$\mathcal{B}_{X}=\left\{11\times\frac{1}{2}(1,1,1),\frac{1}{3}(1,1,2)\right\}$.

Theorem 3.2**.**

Let $X$ be a quasi-smooth prime codimension $4$ Fano $3$-fold of numerical type $\#166$. Then $X$ is birationally superrigid.

Proof.

By Lemmas 2.8 and 2.9, no curve and no nonsingular point is a maximal center.

Let $\mathsf{p}$ be a singular point of type $\frac{1}{2}(1,1,1)$. After replacing coordinates, we may assume $\mathsf{p}=\mathsf{p}_{p}$. We set $\mathcal{C}=\{q,r,s,t,u\}$. We have

[TABLE]

Moreover we have $\Pi_{X}(\mathcal{C}\cup\{p\})=\emptyset$ because $X$ is quasi-smooth and it does not have a singular point of index $5$. Thus, by Lemma 2.14, $\mathsf{p}$ is not a maximal center.

Let $\mathsf{p}$ be the singular point of type $\frac{1}{3}(1,1,2)$. After replacing $r$ and $s$, we may assume $\mathsf{p}=\mathsf{p}_{s}$. We set $\mathcal{C}=\{p,q,r\}$. Then we have

[TABLE]

By Lemma 2.14, it remains to show that $\Pi_{X}:=\Pi_{X}(\mathcal{C}\cup\{s\})=\emptyset$. We set $\Pi:=\Pi(\mathcal{C}\cup\{s\})\subset\mathbb{P}$ so that $\Pi_{X}=\Pi\cap X$. We have

[TABLE]

We see that $F_{1}|_{\Pi},F_{2}|_{\Pi},F_{3}|_{\Pi}$ consist only of monomials in variables $t,u$, and $X$ does not have a singular point of index $4$. Hence the equation

[TABLE]

implies $t=u=0$. Similarly, $F_{7}|_{\Pi},F_{8}|_{\Pi},F_{9}|_{\Pi}$ consist only of the monomials in variables $v,w$, and $X$ does not contain a singular point of index $5$. Hence the equation

[TABLE]

implies $v=w=0$. It follows that $\Pi_{X}=\emptyset$ and $\mathsf{p}$ is not a maximal center. Therefore $X$ is birationally superrigid. ∎

3.3. Fano $3$-folds of numerical type $\#282$

Let $X$ be a quasi-smooth prime Fano $3$-fold of numerical type $\#282$, whose data consist of the following.

• •

$X\subset\mathbb{P}(1_{p},6_{q},6_{r},7_{s},8_{t},9_{u},10_{v},11_{w})$.

• •

$(-K_{X})^{3}=1/42$.

• •

$\deg(F_{1},F_{2},F_{3},F_{4},F_{5},F_{6},F_{7},F_{8},F_{9})=(16,17,18,18,19,20,20,21,22)$.

• •

$\mathcal{B}=\left\{2\times\frac{1}{2}(1,1,1),2\times\frac{1}{3}(1,1,2),\frac{1}{6}(1,1,5),\frac{1}{7}(1,1,6)\right\}$.

Proposition 3.3**.**

Let $X$ be a quasi-smooth prime codimension $4$ Fano $3$-fold of numerical type $\#282$. Then no curve and no point is a maximal center except possibly for the singular point of type $\frac{1}{6}(1,1,5)$.

Proof.

By Lemmas 2.8, 2.9 and 2.11, it remains to exclude singular points of type $\frac{1}{3}(1,1,2)$ and $\frac{1}{7}(1,1,6)$ as maximal centers.

Let $\mathsf{p}$ be a singular point of type $\frac{1}{3}(1,1,2)$ and let $\varphi\colon(E\subset Y)\to(\mathsf{p}\in X)$ be the Kawamata blowup. We claim that $\Pi_{X}(p,s,t,w)=D_{p}\cap D_{s}\cap D_{t}\cap D_{w}$ is a finite set of points (containing $\mathsf{p}$). Since $X$ does not contain a singular point of index $10$, we may assume that $v^{2}\in F_{6}$. Then, by re-scaling $v$, we have

[TABLE]

and this shows that $\Pi_{X}(p,s,t,w)=\Pi_{X}(p,s,t,v,w)$. The latter set consists of singular points $\{2\times\frac{1}{2}(1,1,2),\frac{1}{6}(1,1,5)\}$ and thus $\Pi_{X}(p,s,t,w)$ is a finite set of points. We have

[TABLE]

By Lemma 2.5, $N:=-\varphi^{*}K_{X}-\frac{1}{21}E$ is a nef divisor on $Y$ and we have $(N\cdot(-K_{Y})^{2})=0$. Thus $\mathsf{p}$ is not a maximal center.

Let $\mathsf{p}=\mathsf{p}_{s}$ be the singular point of type $\frac{1}{7}(1,1,6)$ and set $\mathcal{C}=\{p,q,r\}$. We have

[TABLE]

We set $\Pi:=\Pi(\mathcal{C}\cup\{s\})$. Since $\mathsf{p}_{t},\mathsf{p}_{u},\mathsf{p}_{v},\mathsf{p}_{w}\notin X$, we have $t^{2}\in F_{1}$, $w^{2}\in F_{9}$ and we may assume $u^{2}\in F_{3}$, $v^{2}\in F_{6}$. Then, by re-scaling $t,u,v,w$, we can write

[TABLE]

where $\alpha,\beta\in\mathbb{C}$. This shows that $\Pi_{X}(\mathcal{C}\cup\{s\})=\Pi\cap X=\emptyset$. Thus $\mathsf{p}$ is not a maximal center by Lemma 2.14 and the proof is completed. ∎

3.4. Fano $3$-folds of numerical type $\#308$

Let $X$ be a quasi-smooth prime Fano $3$-fold of numerical type $\#308$, whose data consist of the following.

• •

$X\subset\mathbb{P}(1_{p},5_{q},6_{r},6_{s},7_{t},8_{u},9_{v},10_{w})$.

• •

$(-K_{X})^{3}=1/30$.

• •

$\deg(F_{1},F_{2},F_{3},F_{4},F_{5},F_{6},F_{7},F_{8},F_{9})=(14,15,16,16,17,18,18,19,20)$.

• •

$\mathcal{B}_{X}=\left\{\frac{1}{2}(1,1,1),\frac{1}{3}(1,1,2),\frac{1}{5}(1,2,3),2\times\frac{1}{6}(1,1,5)\right\}$.

Theorem 3.4**.**

Let $X$ be a quasi-smooth prime Fano $3$-fold of numerical type $\#308$. Then $X$ is birationally superrigid.

Proof.

By Lemmas 2.8, 2.9 and 2.11, no curve and no nonsingular point is a maximal center and the singular point of type $\frac{1}{2}(1,1,1)$ is not a maximal center.

Let $\mathsf{p}$ be the singular point of type $\frac{1}{3}(1,1,2)$, which is necessary contained in $(p=q=t=u=w=0)$, and let $\varphi\colon(E\subset Y)\to(\mathsf{p}\in X)$ be the Kawamata blowup. We set $\mathcal{C}=\{p,q,u\}$ and $\Pi=\Pi(\mathcal{C})\subset\mathbb{P}$. Since $\mathsf{p}_{t},\mathsf{p}_{w}\notin X$, we have $t^{2}\in F_{1},w^{2}\in F_{9}$ and we can write

[TABLE]

where $\alpha,\beta\in\mathbb{C}$. Thus,

[TABLE]

and this consists of two $\frac{1}{6}(1,1,5)$ points and $\mathsf{p}$. In particular $D_{p}\cap D_{q}\cap D_{u}=\Pi_{X}(\mathcal{C})$ is a finite set of points. We have

[TABLE]

hence $N:=-8\varphi^{*}K_{X}-\frac{2}{3}E$ is a nef divisor on $Y$ by Lemma 2.5. We have

[TABLE]

By Lemma 2.4, $\mathsf{p}$ is not a maximal center.

Let $\mathsf{p}$ be a singular point of type $\frac{1}{6}(1,1,5)$. After replacing $r$ and $s$, we may assume $\mathsf{p}=\mathsf{p}_{s}$. We set $\mathcal{C}=\{p,q,r\}$. We have

[TABLE]

Since $\mathsf{p}_{t},\mathsf{p}_{u},\mathsf{p}_{v},\mathsf{p}_{w}\notin X$, we may assume $t^{2}\in F_{1},u^{2}\in F_{3},v^{2}\in F_{6},w^{2}\in F_{9}$ after possibly interchanging $F_{3}$ with $F_{4}$ and $F_{6}$ with $F_{7}$. Then, by setting $\Pi=\Pi(\mathcal{C}\cup\{s\})$ and by re-scaling $t,u,v,w$, we have

[TABLE]

where $\alpha,\beta\in\mathbb{C}$. This shows that $\Pi_{X}(\mathcal{C}\cup\{s\})=\emptyset$ and $\mathsf{p}$ is not a maximal center by Lemma 2.14.

Finally, let $\mathsf{p}$ be a singular point of type $\frac{1}{5}(1,2,3)$ and let $\varphi\colon(E\subset Y)\to(\mathsf{p}\in X)$ be the Kawamata blowup. Replacing the coordinate $w$, we may assume $\mathsf{p}=\mathsf{p}_{q}$. We write

[TABLE]

where $\lambda,\mu,\nu,\lambda^{\prime},\mu^{\prime},\nu^{\prime}\in\mathbb{C}$ and $f_{11},f_{16},g_{11},g_{16}\in\mathbb{C}[p,r,s,t,u,v,w]$ are homogeneous polynomials of the indicated degrees.

We first consider the case where $\mu\nu^{\prime}-\nu\mu^{\prime}\neq 0$. By replacing $r$ and $s$, we may assume that $\mu=\nu^{\prime}=1$ and $\lambda=\nu=\lambda^{\prime}=\mu^{\prime}=0$. We consider the initial weight at $\mathsf{p}$

[TABLE]

Then $F_{3}^{\mathbf{w}_{\operatorname{in}}}=q^{2}r$ and $F_{4}^{\mathbf{w}_{\operatorname{in}}}=q^{2}s$, and this implies $\operatorname{ord}_{E}(D_{r}),\operatorname{ord}_{E}(D_{s})\geq 6/5$. Note that $\operatorname{ord}_{E}(D_{p})\geq\mathbf{w}_{\operatorname{in}}(x)=1/5$. We set $\mathcal{C}=\{p,r,s\}$ and $\Pi=\Pi(\mathcal{C}\cup\{q\})$. By re-scaling $t,u,v,w$, we can write

[TABLE]

where $\alpha,\beta\in\mathbb{C}$. Hence $\Pi_{X}(\mathcal{C}\cup\{q\})=\emptyset$. Since $D_{q}$ is an apmle divisor, this implies that $D_{p}\cap D_{r}\cap D_{s}$ is a finite set of points (including $\mathsf{p}$). By Lemma 2.5, $N:=-\varphi^{*}K_{X}-\frac{1}{5}E$ is a nef divisor on $Y$. We have

[TABLE]

and this shows that $\mathsf{p}$ is not a maximal center.

Next we consider the case where $\mu\nu^{\prime}-\nu\mu^{\prime}=0$. By replacing $r$ and $s$ suitably and by possibly interchanging $F_{3}$ and $F_{4}$, we may assume that

[TABLE]

It is straightforward to see that $q^{3}p$ is the unique monomial in $F_{3}$ with initial weight $1/5$, so that $\operatorname{ord}_{E}(D_{p})\geq 6/5$. Let $\mathcal{L}\subset|-6K_{X}|$ be the pencil generated by the sections $r$ and $s$. Since $\operatorname{ord}_{E}(D_{r})=1/5$ and $\operatorname{ord}_{E}(D_{s})\geq 1/5$, a general member $L\in\mathcal{L}$ vanishes along $E$ to order $1/5$ so that $\tilde{L}\sim-6\varphi^{*}K_{X}-\frac{1}{5}E$. We have

[TABLE]

since $\operatorname{ord}_{E}(p)\geq 6/5$. By Lemma 2.6, $\mathsf{p}$ is not a maximal center and the proof is completed. ∎

4. Birational superrigidity of cluster Fano $3$-folds

In this section, we prove Theorem 1.1, which follow from Theorems 4.1 and 4.2 below.

4.1. $\#282$ by $\mathsf{G}_{2}^{(4)}$ format

Let $X$ be a codimension $4$ prime Fano $3$-fold of numerical type $\#282$ constructed in $\mathsf{G}_{2}^{(4)}$ format. Then, by [10, Example 5.5], $X$ is defined by the following polynomials in $\mathbb{P}(1_{p},6_{q},6_{r},7_{s},8_{t},9_{u},10_{v},11_{w})$.

[TABLE]

Here $P_{12},Q_{9}\in\mathbb{C}[p,q,r,s,t,u,v,w]$ are homogeneous polynomials of the indicated degree. Recall that $(-K_{X})^{3}=1/42$.

Theorem 4.1**.**

Let $X$ be a codimension $4$ Fano $3$-fold of numerical type $\#282$ constructed in $\mathsf{G}^{(4)}_{2}$ format. Then $X$ is birationally superrigid.

Proof.

By Proposition 3.3, it remains to exclude the singular point $\mathsf{p}\in X$ of type $\frac{1}{6}(1,1,5)$ as a maximal center. The point $\mathsf{p}$ corresponds to the unique solution solution of the equations

[TABLE]

and we have $\mathsf{p}=\mathsf{p}_{r}$. We set $\mathcal{C}=\{p,q\}$, $\Pi=\Pi(\mathcal{C})$ and $\Gamma:=\Pi_{X}(\mathcal{C})=\Pi\cap X$.

We will show that $\Gamma$ is an irreducible and reduced curve. We can write

[TABLE]

where $\lambda,\mu\in\mathbb{C}$. By the quasi-smoothness of $X$ at $\mathsf{p}$, we see that $\lambda,\mu\neq 0$. Then we have

[TABLE]

We work on the open subset $U$ on which $w\neq 0$. Then $\Gamma\cap U$ is isomorphic to the $\mathbb{Z}/11\mathbb{Z}$-quotient of the affine curve

[TABLE]

It is straightforward to check that the polynomial $\lambda r^{2}v+\mu^{3}rv^{6}-1$ is irreducible. Thus $\Gamma\cap U$ is an irreducible and reduced affine curve. It is also straightforward to check that

[TABLE]

This shows that $\Gamma$ is an irreducible and reduced curve.

Let $\varphi\colon(E\subset Y)\to(\mathsf{p}\in X)$ be the Kawamata blowup and let $\tilde{\Delta}$ be the proper transform via $\varphi$ of a divisor or a curve on $X$. We show that $\tilde{D}_{p}\cap\tilde{D}_{q}\cap E$ does not contain a curve. Consider the initial weight

[TABLE]

We set $f_{i}=F_{i}^{\mathbf{w}_{\operatorname{in}}}(p,q,1,s,t,u,v,w)$. We have

[TABLE]

Since $E$ is isomorphic to the subvariety

[TABLE]

it is straightforward to check that $\tilde{D}_{p}\cap\tilde{D}_{q}\cap E$ consists of finite set of points (in fact, $2$ points). Thus we have $\tilde{D}_{p}\cdot\tilde{D}_{q}=\tilde{\Gamma}$ since $D_{p}\cdot D_{q}=\Gamma$.

We have

[TABLE]

for some integer $e\geq 6$ and hence

[TABLE]

By [20, Lemma 2.18], $\mathsf{p}$ is not a maximal center. ∎

4.2. $\#282$ by $\mathsf{C}_{2}$ format

Let $X$ be a codimension $4$ prime Fano $3$-fold of numerical type $\#282$ constructed in $\mathsf{G}_{2}^{(4)}$ format. Then, by [10, Example 5.5], $X$ is defined by the following polynomials in $\mathbb{P}(1_{p},6_{q},6_{r},7_{s},8_{t},9_{u},10_{v},11_{w})$.

[TABLE]

Here $P_{12},Q_{10},R_{8},S_{6}\in\mathbb{C}[p,q,r,s,t,u,v,w]$ are homogeneous polynomials of the indicated degree.

Theorem 4.2**.**

Let $X$ be a quasi-smooth prime codimension $4$ Fano $3$-fold of numerical type $\#282$ constructed by $\mathsf{C}_{2}$ format. We assume that $q\in S_{6}$. Then $X$ is birationally superrigid.

Proof.

By Proposition 3.3, it remains to exclude the singular point $\mathsf{p}$ of type $\frac{1}{6}(1,1,5)$ as a maximal center.

Replacing $q$, we may assume that $S_{6}=q$. The singular point $\mathsf{p}$ corresponds to the solution of the equation

[TABLE]

and thus $\mathsf{p}=\mathsf{p}_{r}$. We set $\mathcal{C}=\{p,q\}$ and $\Pi=\Pi(\mathcal{C})$.

We will show that $\Gamma:=\Pi\cap X$ is an irreducible and reduced curve. We have $\Pi_{X}(\{p,q,r,s\})=\emptyset$ (see the proof of Proposition 3.3). Hence $\Gamma\cap(s=0)=\Pi_{X}(\{p,q,s\})$ does not contain a curve and it remains to show that $\Gamma\cap U_{s}$ is irreducible and reduced, where $U_{s}:=(s\neq 0)\subset\mathbb{P}$ is the open subset. We can write

[TABLE]

for some $\lambda,\mu,\nu\in\mathbb{C}$, and we have $S_{6}|_{\Pi}=0$. We see that $t^{2}$ appears in $F_{1}$ (resp. $v^{2}$ appears in either $F_{6}$ or $F_{7}$) with non-zero coefficient since $\mathsf{p}_{t}\notin X$ (resp. $\mathsf{p}_{v}\notin X$), which implies that $\mu,\neq 0$. Note that $F_{i}|_{\Pi}=F_{i}|_{\Pi}(r,s,t,u,v,w)$ is a polynomial in variables $r,s,t,u,v,w$ and we set $f_{i}=F_{i}|_{\Pi}(r,1,t,u,v,w)$. Let $C\subset\mathbb{A}^{5}_{r,t,u,v,w}$ be the affine scheme defined by the equations

[TABLE]

Then $\Gamma\cap U_{s}$ is isomorphic to the quotient of $C$ by the natural $\mathbb{Z}/7\mathbb{Z}$-action. We have

[TABLE]

By the equations $f_{1}=0$, $f_{2}=0$ and $f_{4}=0$, we have

[TABLE]

By eliminating the variables $u,v,w$ and cleaning up the equations, $C$ is isomorphic to the hypersurface in $\mathbb{A}^{2}_{r,t}$ defined by

[TABLE]

which is an irreducible and reduced curve since $\mu\nu\neq 0$, and so is $\Gamma\cap U_{s}$. Thus $\Gamma$ is an irreducible and reduced curve.

Let $\varphi\colon(E\subset Y)\to(\mathsf{p}\in X)$ be the Kawamata blowup. We have $e:=\operatorname{ord}_{E}(D_{q})\geq 6/6$ and $\operatorname{ord}_{E}(D_{p})=1/6$ so that we have

[TABLE]

We show that $\tilde{D}_{q}\cap\tilde{D}_{p}\cap E$ does not contain a curve. The Kawamata blowup $\varphi$ is realized as the weighted blowup at $\mathsf{p}$ with the weight

[TABLE]

We have

[TABLE]

where we define $h:=Q_{10}^{\mathbf{w}_{\operatorname{in}}}-\mu v$. Note that $h$ is a linear combination of $up,tp^{2},sp^{3},rp^{4}$ and thus $h$ is divisible by $p$. It follows that $E$ is isomorphic to the subscheme in $\mathbb{P}(1_{p},6_{q},1_{s},2_{t},3_{u},4_{v},5_{w})$ defined by the equations

[TABLE]

It is now straightforward to check that $\tilde{D}_{q}\cap\tilde{D}_{p}\cap E=(p=q=0)\cap E$ is a finite set of points (in fact, it consists of $2$ points). This shows that $\tilde{D}_{q}\cdot\tilde{D}_{p}=\tilde{\Gamma}$ since $D_{q}\cdot D_{p}=\Gamma$. We have

[TABLE]

since $e\geq 6$. By [20, Lemma 2.18], $\mathsf{p}$ is not a maximal center. ∎

5. On further problems

5.1. Prime Fano $3$-folds with no projection centers

We further investigate birational superrigidity of prime Fano $3$-folds of codimension $c$ with no projection centers for $5\leq c\leq 9$. There are only a few such candidates, which can be summarized as follows.

• •

In codimension $c\in\{5,7,8\}$, there is a unique candidate and it corresponds to smooth prime Fano $3$-folds of degree $2c+2$. All of these Fano $3$-folds are rational (see [15, Corollary 4.3.5 or §12.2]) and are not birationally superrigid.

• •

In codimension $6$, there are $2$ candidates; one candidate corresponds to smooth prime Fano $3$-folds of degree $14$ which are birational to smooth cubic $3$-folds (see [25], [13]) and are not birationally superrigid, and the existence is not known for the other candidate which is $\#78$ in the database.

• •

In codimension $9$, there is a unique candidate of smooth prime Fano $3$-folds of degree $20$. However, according to the classification of smooth Fano $3$-folds there is no such Fano $3$-fold (see e.g. [25, Theorem 0.1]).

It follows that, in codimension up to $9$, $\#78$ is the only remaining unknown case for birational superrigidity (of general members).

Question 5.1**.**

Do there exist prime Fano $3$-folds which correspond to $\#78$? If yes, then is a (general) such Fano $3$-fold birationally superrigid?

In codimension $10$ and higher, there are a lot of candidates of Fano $3$-folds with no projection centers. We expect that many of them are non-existence cases and that there are only a few birationally superrigid Fano $3$-folds in higher codimensions.

Question 5.2**.**

Is there a numerical type (in other words, Graded Ring Database ID) $\#\mathsf{i}$ in codimension greater than $9$ such that a (general) quasi-smooth prime Fano $3$-fold of numerical type $\#\mathsf{i}$ is birationally superrigid?

5.2. Classification of birationally superrigid Fano $3$-folds

There are many difficulties in the complete classification of birationally superrigid Fano $3$-folds. For example, we need to consider Fano $3$-folds which are not necessarily quasi-smooth or not necessarily prime, and also we need to understand subtle behaviors of birational superrigidity in a family, etc.

Question 5.3**.**

Is there a birationally superrigid Fano $3$-fold which is either of Fano index greater than $1$ or has a non-quotient singularity?

Remark 5.4**.**

By recent developments [23], [24], [18], it has been known that there exist birationally superrigid Fano varieties which have non-quotient singularities (see [23], [24], [18]) at least in very high dimensions. On the other hand, only a little is known for Fano varieties of index greater than $1$ (cf. [22]) and there is no example of birationally superrigid Fano varieties of index greater than $1$.

We concentrate on quasi-smooth prime Fano $3$-folds. Even in that case, it is necessary to consider those with a projection center, which are not treated in this paper. Let $X$ be a general quasi-smooth prime Fano $3$-fold of codimension $c$. Then the following are known.

• •

When $c=1$, $X$ is birationally superrigid if and only if $X$ does not admit a type $\mathrm{I}$ projection center (see [14], [9], [7]).

• •

When $c=2,3$, $X$ is birationally superrigid if and only if $X$ is singular and admits no projection center (see [16], [19], [2], [1]).

With these evidences, we expect the following.

Conjecture 5.5**.**

Let $X$ be a general quasi-smooth prime Fano $3$-fold of codimension at least $2$. Then $X$ is birationally superrigid if and only if $X$ is singular and admits no projection centers.