A sufficient condition for a toric weak Fano 4-fold to be deformed to a Fano manifold

TL;DR

This paper classifies smooth toric special weak Fano 4-folds and shows that most can be deformed into Fano manifolds, expanding understanding of their structure and deformation properties.

Contribution

It introduces the concept of toric special weak Fano manifolds and provides a complete classification of smooth toric special weak Fano 4-folds.

Findings

Almost all smooth toric special weak Fano 4-folds are deformable to Fano manifolds.

The structure of toric special weak Fano manifolds is characterized by special primitive crepant contractions.

The paper confirms the deformation connection between weak Fano and Fano manifolds in this class.

Abstract

In this paper, we introduce the notion of toric special weak Fano manifolds, which have only special primitive crepant contractions. We study the structure of them, and in particular completely classify smooth toric special weak Fano 4-folds. As a result, we can confirm that almost every smooth toric special weak Fano 4-fold is a weakened Fano manifold, that is, a weak Fano manifold which can be deformed to a Fano manifold.

Tables

Table 1. Table 1. Smooth toric weak del Pezzo surfaces.

notation	$G\left(\Sigma \right)$
${\mathbb{P}}^2$	$w_1=(1,0),w_2=(0,1),w_3=(-1,-1)$
${\mathbb{P}}^1\times {\mathbb{P}}^1$	$w_1=(1,0),w_2=(0,1),w_3=(-1,0),w_4=(0,-1)$
$F_1$	$w_1=(1,0),w_2=(0,1),w_3=(-1,0),w_4=(1,-1)$
$F_2$	$w_1=(1,0),w_2=(1,1),w_3=(-1,0),w_4=(1,-1)$
$S_7$	$w_1=(1,0),w_2=(1,1),w_3=(0,1),w_4=(-1,0),w_5=(0,-1)$
$W_3$	$w_1=(1,0),w_2=(1,1),w_3=(0,1),w_4=(-1,0),w_5=(1,-1)$
$S_6$	$w_1=(1,0),w_2=(1,1),w_3=(0,1),w_4=(-1,0),w_5=(-1,-1),$ $w_6=(0,-1)$
$W_4^1$	$w_1=(1,0),w_2=(1,1),w_3=(0,1),w_4=(-1,0),w_5=(0,-1),$ $w_6=(1,-1)$
$W_4^2$	$w_1=(1,0),w_2=(1,1),w_3=(0,1),w_4=(-1,1),w_5=(-1,0),$ $w_6=(1,-1)$
$W_4^3$	$w_1=(1,0),w_2=(1,1),w_3=(1,2),w_4=(0,1),w_5=(-1,0),$ $w_6=(1,-1)$
$W_5^1$	$w_1=(1,0),w_2=(1,1),w_3=(0,1),w_4=(-1,1),w_5=(-1,0),$ $w_6=(0,-1),w_7=(1,-1)$
$W_5^2$	$w_1=(1,0),w_2=(1,1),w_3=(0,1),w_4=(-1,0),w_5=(0,-1),$ $w_6=(1,-2),w_7=(1,-1)$
$W_6^1$	$w_1=(1,0),w_2=(1,1),w_3=(0,1),w_4=(-1,1),w_5=(-1,0),$ $w_6=(-1,-1),w_7=(0,-1),w_8=(1,-1)$
$W_6^2$	$w_1=(1,0),w_2=(1,1),w_3=(0,1),w_4=(-1,1),w_5=(-1,0),$ $w_6=(0,-1),w_7=(1,-2),w_8=(1,-1)$
$W_6^3$	$w_1=(1,0),w_2=(1,1),w_3=(1,2),w_4=(0,1),w_5=(-1,0),$ $w_6=(0,-1),w_7=(1,-2),w_8=(1,-1)$
$W_7$	$w_1=(1,0),w_2=(1,1),w_3=(0,1),w_4=(-1,1),w_5=(-2,1),$ $w_6=(-1,0),w_7=(0,-1),w_8=(1,-2),w_9=(1,-1)$

Table 2. Table 2. The case where I ⊂ 𝒮 1 𝐼 subscript 𝒮 1 I\subset\mathcal{S}_{1} and φ ​ ( E ) ≅ ℙ 2 𝜑 𝐸 superscript ℙ 2 \varphi(E)\cong{\mathbb{P}}^{2} .

notation	${𝒵}_1$	${𝒵}_2$	${𝒵}_3$	${𝒵}_4$	${𝒵}_5$	${𝒵}_6$
$S\cong$	$F_2$	$W_3$	$W_3$	$W_3$	$W_4^1$	$W_4^1$
$y_1+y_2+y_3=$	$w_2+w_3$	$w_3$	$2w_3$	$w_4+w_5$	$w_3$	$2w_3$

Table 3. Table 3. The case where I ⊂ 𝒮 1 𝐼 subscript 𝒮 1 I\subset\mathcal{S}_{1} and φ ​ ( E ) ≅ ℙ 1 × ℙ 1 𝜑 𝐸 superscript ℙ 1 superscript ℙ 1 \varphi(E)\cong{\mathbb{P}}^{1}\times\mathbb{P}^{1} .

notation	${𝒵}_7$	${𝒵}_8$	${𝒵}_9$
$S\cong$	$W_3$	$W_4^1$	$W_4^1$
$y_1+y_3=$	$w_3$	$w_3$	$w_3$
$y_2+y_4=$	$w_3$	$w_3$	$w_5$

Table 4. Table 4. The case where I ⊂ 𝒮 1 𝐼 subscript 𝒮 1 I\subset\mathcal{S}_{1} and φ ​ ( E ) ≅ F 1 𝜑 𝐸 subscript 𝐹 1 \varphi(E)\cong F_{1} .

notation	${𝒵}_{10}$	${𝒵}_{11}$
$S\cong$	$W_3$	$W_4^1$
$y_1+y_3=$	$w_3$	$w_3$

Table 5. Table 5. The case where I ⊂ 𝒮 1 𝐼 subscript 𝒮 1 I\subset\mathcal{S}_{1} and φ ​ ( E ) ≅ F 2 𝜑 𝐸 subscript 𝐹 2 \varphi(E)\cong F_{2} .

notation	${𝒵}_{12}$	${𝒵}_{13}$
$S\cong$	$W_3$	$W_4^1$
$y_1+y_3=$	$w_3$	$w_3$

notation	${𝒞}_1$	${𝒞}_2$	${𝒞}_5$
$v_3+v_5=$	$v_1$	$v_1$	$v_1$
$v_4+v_6=$	$v_1$	$v_3$	$v_2$

notation	${ℰ}_1$	${ℰ}_2$	${ℰ}_4$
$v_6+v_7=$	$v_1$	$v_2$	$v_3$

special weak Fano

${𝒵}_1$

${𝒵}_2$

${𝒵}_3$

${𝒵}_4$

${𝒵}_5$

${𝒵}_6$

${𝒵}_7$

${𝒵}_8$

${𝒵}_9$

${𝒵}_{10}$

${𝒵}_{11}$

${𝒵}_{12}$

${𝒵}_{13}$

deformed to

$D_9$

$H_4$

$H_1$

$H_{10}$

$K_3$

$K_1$

$Q_1$

$U_1$

$U_8$

$Q_2$

$U_2$

${𝒵}_{19}$

${𝒵}_{24}$

special weak Fano	${𝒵}_{14}$	${𝒵}_{15}$	${𝒵}_{16}$	${𝒵}_{17}$	${𝒵}_{19}$	${𝒵}_{20}$	${𝒵}_{22}$	${𝒵}_{24}$
deformed to	$D_7$	$L_1$	$L_2$	$L_{13}$	$Q_1$	$Q_3$	$Q_{13}$	$U_1$

Equations

$$\mathfrak{f}:\mathcal{X}\to {\Delta }_{ϵ}:=\left\{c\in \mathbb{C}||c|<ϵ\right\}$$

$$\sigma +{\sigma }^{\prime }:=\left\{n+n^{\prime }\in N_{\mathbb{R}}|n\in \sigma \text{mbox}and{n}^{\prime }\in {\sigma }^{\prime }\right\}.$$

$${\operatorname{G}}(\Sigma):=\left\{\mbox{the primitive generators of $1$-dimensional cones in $\Sigma$}\right\}\subset N.$$

$$0\to M\to {\mathbb{Z}}^{G(\Sigma )}\to \mathrm{Pic}(X)\to 0,$$

$$0\leftarrow N\stackrel{\psi }{\leftarrow }{\left({\mathbb{Z}}^{G(\Sigma )}\right)}^{\ast }\leftarrow {\mathrm{A}}^1(X)\leftarrow 0,$$

$$\psi \left((a_x{)}_{x\in G(\Sigma )}\right)=\sum _{x\in G(\Sigma )}{a}_{x}x$$

$${\mathrm{A}}^1(X)\cong \left\{(a_x{)}_{x\in G(\Sigma )}\in {\left({\mathbb{Z}}^{G(\Sigma )}\right)}^{\ast }|\sum _{x\in G(\Sigma )}{a}_{x}x=0\right\}\subset {\left({\mathbb{Z}}^{G(\Sigma )}\right)}^{\ast }.$$

$$x_1+\cdots +x_r=a_1{y}_1+\cdots +a_s{y}_s(a_1,\dots ,a_s\in {\mathbb{Z}}_{>0}).$$

$$\mathrm{NE}(X)=\sum _{P:\text{mbox}primitivecollectionin\Sigma }{\mathbb{R}}_{\ge 0}r(P)\subset {\mathrm{N}}^1(X):={\mathrm{A}}^1(X)\otimes \mathbb{R}.$$

$$C=b_1{C}_1+\cdots +b_l{C}_l$$

$$R={\mathbb{R}}_{\ge 0}r(P)\subset \mathrm{NE}(X).$$

$$⟨x_1,\dots ,\stackrel{\vee }{x_i},\dots ,x_r,y_1,\dots ,y_s⟩+\tau \in \Sigma$$

$$\overline{\Sigma }:=\left\{\sigma \in \Sigma |y_1\notin \sigma \cap G(\Sigma )\right\}\cup$$

$$\left\{⟨x_1,\dots ,x_r⟩+\tau |\tau \in \Sigma ,\left(\tau \cap G(\Sigma )\right)\cap G=\varnothing \text{mbox}and⟨y_1⟩+\tau \in \Sigma \right\}.$$

$$\overline{\Sigma }:=\left\{\sigma \in \Sigma |\{y_1,\dots ,y_s\}\not\subset \sigma \cap G(\Sigma )\right\}\cup$$

$$\left\{⟨x_1,\dots ,x_r,y_1,\dots ,y_s⟩+\tau |\tau \in \Sigma ,\left(\tau \cap G(\Sigma )\right)\cap G=\varnothing \text{mbox}and⟨y_1,\dots ,y_s⟩+\tau \in \Sigma \right\}.$$

$$⟨x_1,\dots ,x_r,y_1,\dots ,y_s⟩\in \overline{\Sigma }.$$

$${\Sigma }_1:=\left\{\sigma \in \Sigma |\sigma \cap G(\Sigma )\subset G_1\right\}\text{mbox}and{\Sigma }_2:=\left\{\sigma \in \Sigma |\sigma \cap G(\Sigma )\subset G_2\right\}.$$

$$\left|{\Sigma }_1\right|:=\bigcup _{\sigma \in {\Sigma }_1}\sigma$$

$$\left\{\tau \in \Sigma |\left(\tau \cap G(\Sigma )\right)\cap \{x,x_1,x_2\}=\varnothing ,\text{mbox}while\exists \sigma \in \Sigma \text{mbox}s.t.\tau \prec \sigma \text{mbox}andx\in \sigma \cap G(\Sigma )\right\},$$

$$r(P)=a_{1}r(P_1)+\cdots +a_{n}r(P_n),$$

$$x_1+x_2=2x\text{mbox}and{y}_1+y_2=2y,$$

$$x_1+x_2+2y=2x+y_1+y_2⟺x_2+2y=2x+y_2.$$

$$m(y_1)=m(y_2)=m(y)=-1\text{mbox}andm(x_1)=m(x_2)=0.$$

$$m(x_2)=m(2x-x_1)=2m(x)-m(x_1)=2m(y_1)-m(x_1)=-2<0,$$

$$m(z_1)+m(z_2)=m(x)+m(y)=-2$$

$$m(z_1)=m(z_2)=-1.$$

$$\left\{u\in N_{\mathbb{R}}|m(u)=-1\right\}\subset N_{\mathbb{R}}$$

$$⟨x,y,x_1,y_1⟩,⟨x,y,x_1,y_2⟩,⟨x,y,x_2,y_1⟩,⟨x,y,x_2,y_2⟩\in \Sigma .$$

$$m_1(x_1)=m_1(x_2)=m_1(x)=m_2(y_1)=m_2(y_2)=m_2(y)=-1,$$

Full text

1. 1 Introduction
2. 2 Preliminaries
3. 3 Toric special weak Fano manifolds
4. 4 The classification of smooth toric special weak Fano 4-folds
5. 5 Deformations of smooth toric special weak Fano 4-folds

1. Introduction

The deformation theory of Hirzebruch surfaces are well-known. The deformation type of the Hirzebruch surface $F_{a}$ is determined by the parity of its degree $a$. In particular, the weak del Pezzo surface $F_{2}$ is deformed to the del Pezzo surface $F_{0}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$, and this phenomenon is very interesting. As a generalization of this result, Minagawa studied the deformations of primitive crepant contractions on a smooth weak Fano $3$-fold in [Mi1], and introduced the following notion of weakened Fano manifolds in [Mi2]:

Definition 1.1**.**

Let $X$ be a smooth weak Fano $d$-fold over the complex number field $\mathbb{C}$. Then, $X$ is a weakened Fano manifold if the following hold:

1. (1)

$X$ is not a Fano manifold. 2. (2)

There exists a one-parameter deformation

[TABLE]

for sufficiently small $\epsilon>0$ such that the central fiber $\mathcal{X}_{0}:=\mathfrak{f}^{-1}(0)$ is isomorphic to $X$, while any general fiber $\mathcal{X}_{t}:=\mathfrak{f}^{-1}(t)$ is a Fano manifold for any $t\in\Delta_{\epsilon}\setminus\{0\}$.

For $d=2$, every smooth weak del Pezzo surface is always a weakened del Pezzo surface if it is not a del Pezzo surface. Smooth toric weakened Fano $3$-folds were classified in [Sa2] by using the characterization of smooth weakened Fano $3$-folds in [Mi1], which says that a smooth weak Fano $3$-fold $X$ is a weakened Fano manifold if and only if every primitive crepant contraction on $X$ is of $(0,2)$-type, and $X$ is not a Fano manifold. In this paper, based on this characterization theorem and the deformation theory of toric manifolds in [LaMe], we define smooth toric special weak Fano $d$-folds (see Definitions 3.2 and 3.3). These varieties have common geometric properties (see Propositions 3.10 and 3.11). As an application, we classify all the smooth toric special weak Fano $4$-folds in Section 4 as follows:

Theorem 1.2** (Theorem 4.9).**

There exist exactly $26$ smooth toric special weak Fano $4$-folds which do not decompose into a direct product of lower-dimensional varieties.

Moreover, by using the deformation families of toric manifolds constructed in [LaMe], we can see almost every smooth toric special weak Fano $4$-fold is deformation equivalent to a Fano manifold as follows:

Theorem 1.3** (Theorem 5.3).**

Let $X$ be a smooth toric special weak Fano $4$-fold. In all but five cases, $X$ is deformation equivalent to a toric Fano manifold. In particular, $X$ is a smooth toric weakened Fano $4$-fold with the exception of two unknown cases.

Thus, the condition to be smooth toric special weak Fano $4$-folds is also a sufficient condition for smooth toric weakened Fano $4$-folds under some assumption. This result seems to be the first step toward the study of the deformation theory of primitive crepant contractions on higher-dimensional smooth weak Fano varieties.

This paper is organized as follows: In Section 2, we collect some basic results of toric geometry such as primitive collections and relations, toric Mori theory, toric bundles and so on. In Section 3, we introduce the concept of smooth toric special weak Fano varieties, which are the main objects in this paper, and study their structure. In Section 4, we completely classify all the smooth toric special weak Fano $4$-folds, and investigate their deformations in Section 5.

Acknowledgments**.**

The author was partially supported by JSPS KAKENHI Grant Number JP18K03262. He would like to thank Professor Hirokazu Nasu, who kindly answered his questions about deformation theory. He would also like to thank Professors Osamu Fujino and Yukari Ito, who gave him useful comments about the descriptions of Proposition 3.5 and Lemma 3.6. Finally, he thanks the referee very much for many useful comments and suggestions.

2. Preliminaries

In this section, we prepare basic results in the toric geometry and the toric Mori theory. For the details, please see [CoLiSc], [Fl] and [O] for toric geometry, and see [FjSa], [Ma] and [R] for toric Mori theory. Also refer [B1], [B2], [Ca] and [Sa1] for the concepts of primitive collections and relations, which are very useful for our theory. We will work over the complex number field $\mathbb{C}$ throughout this paper.

Let $N:=\mathbb{Z}^{d}$, $M:={\operatorname{Hom}}_{\mathbb{Z}}(N,\mathbb{Z})$, $N_{\mathbb{R}}:=N\otimes_{\mathbb{Z}}\mathbb{R}$ and $M_{\mathbb{R}}:=M\otimes_{\mathbb{Z}}\mathbb{R}$. $\langle n_{1},\ldots,n_{r}\rangle$ stands for the cone generated by $n_{1},\ldots,n_{r}\in N_{\mathbb{R}}$. For any two cones $\sigma,\sigma^{\prime}\subset N_{\mathbb{R}}$, put

[TABLE]

For a fan $\Sigma$ in $N$, we denote by $X=X_{\Sigma}$ the toric $d$-fold associated to $\Sigma$. For a smooth complete fan $\Sigma$, put

[TABLE]

For any $x\in{\operatorname{G}}(\Sigma)$, there is the corresponding torus invariant divisor $D_{x}$ on $X$. We have the following fundamental exact sequence

[TABLE]

where we regard $\mathbb{Z}^{{\operatorname{G}}(\Sigma)}$ as the group $\sum_{x\in{\operatorname{G}}(\Sigma)}\mathbb{Z}D_{x}$ of torus invariant divisors on $X$, naturally. In its dual exact sequence

[TABLE]

where ${\rm A}^{1}(X)$ is the group of numerical $1$-cycles on $X$, we have

[TABLE]

for $(a_{x})_{x\in{\operatorname{G}}(\Sigma)}\in\left(\mathbb{Z}^{{\operatorname{G}}(\Sigma)}\right)^{*}$. Therefore, we obtain the following isomorphism of abelian groups:

[TABLE]

Thus, we can regard a linear relation among elements in ${\operatorname{G}}(\Sigma)$ as a numerical $1$-cycle on $X$. In particular, the following relation is important for our theory:

Definition 2.1** ([B1, Definitions 2.6, 2.7 and 2.8]).**

Let $X=X_{\Sigma}$ be a smooth complete toric variety. We call a nonempty subset $P\subset{\operatorname{G}}(\Sigma)$ a primitive collection in $\Sigma$ if $P$ does not generate a cone in $\Sigma$, while any proper subset of $P$ generates a cone in $\Sigma$.

For a primitive collection $P=\{x_{1},\ldots,x_{r}\}$, there exists the unique cone $\sigma(P)\in\Sigma$ such that $x_{1}+\cdots+x_{r}$ is contained in the relative interior of $\sigma(P)$. Put $\sigma(P)\cap{\operatorname{G}}(\Sigma)=\{y_{1},\ldots,y_{s}\}$. Then, we have a linear relation

[TABLE]

We call this relation the primitive relation of $P$. Thus, by the above argument, we obtain the numerical $1$-cycle $r(P)\in{\rm A}^{1}(X)$ for any primitive collection $P\subset{\operatorname{G}}(\Sigma)$. We put $\deg P:=r-(a_{1}+\cdots+a_{s})$. We remark that $\deg P$ is the intersection number $\left(-K_{X}\cdot r(P)\right)$.

Primitive collections and relations are very important in toric Mori theory as follows:

Theorem 2.2** ([B1, Theorem 2.15]).**

Let $X=X_{\Sigma}$ be a smooth projective toric variety. Then, the Kleiman-Mori cone ${\rm NE}(X)$ of $X$ is generated by the $1$-cycles associated to primitive relations. Namely,

[TABLE]

When $r(P)$ is contained in an extremal ray of ${\rm NE}(X)$, we call $P$ an extremal primitive collection.

Remark 2.3**.**

It is well-known that for a smooth projective toric variety $X=X_{\Sigma}$, ${\rm NE}(X)$ is generated by torus invariant curves on $X$. More strongly, for any curve $C\subset X$, there exist torus invariant curves $C_{1},\ldots,C_{l}$ and positive integers $b_{1},\ldots,b_{l}\in\mathbb{Z}_{>0}$ such that

[TABLE]

in ${\rm N}^{1}(X)$. The proof is contained in the Japanese version of [O] (also see [SaSu, Proposition 2.3]).

For an extremal primitive collection $P\subset{\operatorname{G}}(\Sigma)$, the associated extremal contraction can be described by using its primitive relation. Let $x_{1}+\cdots+x_{r}=a_{1}y_{1}+\cdots+a_{s}y_{s}$ be its primitive relation, where $G:=\{x_{1},\ldots,x_{r},y_{1},\ldots,y_{s}\}\subset{\operatorname{G}}(\Sigma)$ and $a_{1},\ldots,a_{s}\in\mathbb{Z}_{>0}$, and $\varphi_{R}:X\to\overline{X}$ the extremal contraction associated to the extremal ray

[TABLE]

The following is the key to constructing $\varphi_{R}$. Also, this proposition is essential for our theory in this paper:

Proposition 2.4** ([Ca, Theorem 2.2]).**

For any cone $\tau\in\Sigma$ such that $\left(\tau\cap{\operatorname{G}}(\Sigma)\right)\cap G=\emptyset$ and $\langle y_{1},\ldots,y_{s}\rangle+\tau\in\Sigma$, we have

[TABLE]

for any $1\leq i\leq r$.

$\varphi_{R}$ is constructed as follows. We divide into three cases:

1. (1)

$\varphi_{R}$ is a Fano contraction $\Longleftrightarrow$ $s=0$. Let $\mathcal{N}\subset N_{\mathbb{R}}$ be the subspace spanned by $\{x_{1},\ldots,x_{r}\}$. Then, $\varphi_{R}$ is induced by the surjective morphism $N_{\mathbb{R}}\to N_{\mathbb{R}}/\mathcal{N}$. $X$ has a $\mathbb{P}^{r-1}$-bundle structure over $\overline{X}$. 2. (2)

$\varphi_{R}$ is a divisorial contraction $\Longleftrightarrow$ $s=1$. Let

[TABLE]

[TABLE]

Then, $\overline{\Sigma}$ is the fan associated to $\overline{X}$. $\varphi_{R}$ contracts the torus invariant divisor corresponding to the cone $\langle y_{1}\rangle\in\Sigma$ to the $(d-r)$-dimensional torus invariant subvariety corresponding to the cone $\langle x_{1},\ldots,x_{r}\rangle\in\overline{\Sigma}$. 3. (3)

$\varphi_{R}$ is a small contraction $\Longleftrightarrow$ $s\geq 2$. Let

[TABLE]

[TABLE]

Then, $\overline{\Sigma}$ is the fan associated to $\overline{X}$. $\varphi_{R}$ contracts the $(d-s)$-dimensional torus invariant subvariety corresponding to the cone $\langle y_{1},\ldots,y_{s}\rangle\in\Sigma$ to the $(d-r-s+1)$-dimensional torus invariant subvariety corresponding to the cone

[TABLE]

$\overline{X}$ has a non-$\mathbb{Q}$-factorial singularity.

Moreover, the following hold: $R$ is $K_{X}$-negative if and only if $\deg P>0$, while $\varphi_{R}$ is a primitive crepant contraction if and only if $\deg P=0$.

The following is a characterization of a toric bundle structure using primitive collections: Let $X=X_{\Sigma}$ be a smooth projective toric $d$-fold. Suppose that there exist nonempty subsets $G_{1}\subset{\operatorname{G}}(\Sigma)$ and $G_{2}\subset{\operatorname{G}}(\Sigma)$ such that $G_{1}\cap G_{2}=\emptyset$, $G_{1}\cup G_{2}={\operatorname{G}}(\Sigma)$ and for any primitive collection $P\subset{\operatorname{G}}(\Sigma)$, either $P\subset G_{1}$ or $P\subset G_{2}$ holds. This condition means that $\Sigma$ is equivalent to the fan of a direct product as a simplicial complex (see e.g. [CoLiSc, Proposition 3.1.14]). Put

[TABLE]

Then, for some $d_{1},d_{2}\in\mathbb{Z}_{>0}$ such that $d=d_{1}+d_{2}$, $\Sigma_{1}\subset\Sigma$ is a subfan whose every maximal cone is $d_{1}$-dimensional, while $\Sigma_{2}\subset\Sigma$ is a subfan whose every maximal cone is $d_{2}$-dimensional. Suppose further that the support

[TABLE]

of $\Sigma_{1}$ is a $d_{1}$-dimensional subspace in $N_{\mathbb{R}}$. Then, $X$ has an $X_{1}$-bundle structure induced by the surjective morphism $N_{\mathbb{R}}\to N_{\mathbb{R}}/|\Sigma_{1}|$, where $X_{1}$ is a smooth projective toric $d_{1}$-fold associated to the $d_{1}$-dimensional fan $\Sigma_{1}$ in $|\Sigma_{1}|$.

In order to test the above condition, the following easy lemma is useful:

Lemma 2.5**.**

$|\Sigma_{1}|\subset N_{\mathbb{R}}$* is a $d_{1}$-dimensional subspace if and only if for any primitive collection $P\subset G_{1}$, we have $\sigma(P)\cap{\operatorname{G}}(\Sigma)\subset G_{1}$.*

3. Toric special weak Fano manifolds

Fano manifolds and weak Fano manifolds are fundamental objects in birational geometry, and are simply defined as follows:

Definition 3.1**.**

Let $X=X_{\Sigma}$ be a smooth projective toric $d$-fold. We call $X$ a toric Fano (resp. weak Fano) manifold if its anticanonical divisor $-K_{X}$ is ample (resp. nef). By Definition 2.1 and Theorem 2.2, we have the following equivalence: $X$ is a Fano (resp. weak Fano) manifold if and only if $\deg P>0$ (resp. $\deg P\geq 0$) for any primitive collection $P$ in $\Sigma$.

We introduce the following subclass of toric weak Fano manifolds in order to settle the characterization of toric weakened Fano manifolds.

Definition 3.2**.**

Let $X=X_{\Sigma}$ be a smooth toric weak Fano $d$-fold, and $\varphi:X\to\overline{X}$ be a primitive crepant contraction. We call $\varphi$ a special primitive crepant contraction if the following hold:

1. (1)

$\varphi$ corresponds to an extremal primitive relation $x_{1}+x_{2}=2x$, where $x_{1},x_{2},x\in{\operatorname{G}}(\Sigma)$. Namely, $\varphi|_{E}:E\to\varphi(E)$ has a $\mathbb{P}^{1}$-bundle structure, where $E$ is the exceptional divisor of $\varphi$ and $\varphi(E)$ is a $(d-2)$-dimensional torus invariant submanifold in $\overline{X}$. 2. (2)

$\varphi(E)$ is a smooth toric weak Fano $(d-2)$-fold. 3. (3)

Let $\Sigma_{\varphi}\subset\Sigma$ be the $(d-2)$-dimensional subfan

[TABLE]

where $\tau\prec\sigma$ means that $\tau$ is a face of $\sigma$. Then, there exists $m\in M$ such that $m(v)\geq-1$ for any $v\in{\operatorname{G}}(\Sigma)$, $m(x)=m(x_{1})=m(x_{2})=-1$ and $m(y)=0$ for any $y\in{\operatorname{G}}(\Sigma_{\varphi})$.

We frequently use the following claim, though this is an easy consequence of the definition of $\Sigma_{\varphi}$:

Claim**.**

$\langle x\rangle+\tau\in\Sigma$* for any $\tau\in\Sigma_{\varphi}$.*

Definition 3.3**.**

Let $X=X_{\Sigma}$ be a smooth toric weak Fano $d$-fold. We call $X$ a toric special weak Fano $d$-fold if $X$ is not a Fano manifold, and every primitive crepant contraction $\varphi:X\to\overline{X}$ on $X$ is special.

Remark 3.4**.**

The notion of special primitive crepant contractions is a generalization of $3$-dimensional $(0,2)$-type contractions studied by Minagawa (see [Mi1] and [Sa2]). In particular, a smooth toric weak Fano $3$-fold is special if and only if it is a toric weakened Fano $3$-fold.

In the rest of this section, we study the structure of toric special weak Fano manifolds.

The following proposition is well-known for experts of Minimal Model Program. We describe a sketch of the proof for the reader’s convenience.

Proposition 3.5**.**

Let $X$ be a quasi-projective canonical toric $d$-fold. If there exist two projective crepant toric morphisms $\varphi_{1}:X_{1}\to X$ and $\varphi_{2}:X_{2}\to X$ such that $X_{1}$ and $X_{2}$ are $\mathbb{Q}$-factorial and terminal, then $X_{1}$ and $X_{2}$ can be obtained from each other by a finite succession of flops.

Sketch of the proof.

Since $\varphi_{1}$ and $\varphi_{2}$ are crepant, $X_{1}$ and $X_{2}$ are isomorphic in codimension one. So, we can do the same argument in the proof of [Fj, Proposition 5.7] by using relative toric Mori theory developed in [FjSa]. ∎

Lemma 3.6**.**

A smooth toric special weak Fano $d$-fold $X=X_{\Sigma}$ is uniquely determined only by ${\operatorname{G}}(\Sigma)$.

Proof.

The anticanonical model $\Phi_{\left|-K_{X}\right|}(X)$ is a Gorenstein toric Fano variety and it is uniquely determined by ${\operatorname{G}}(\Sigma)$. Every primitive crepant contraction of $X$ is special, that is, $X$ has no flopping contraction. Therefore, Proposition 3.5 tells us that the crepant resolution of $\Phi_{|-K_{X}|}(X)$ is also unique. ∎

Remark 3.7**.**

Lemma 3.6 does not hold for a general smooth toric weak Fano variety even in dimension $3$, because the Ordinary Double Point has two different crepant resolutions.

Proposition 3.8**.**

Let $X=X_{\Sigma}$ be a smooth toric special weak Fano $d$-fold. Then, for any primitive collection $P=\{z_{1},z_{2}\}\subset{\operatorname{G}}(\Sigma)$ such that $z_{1}+z_{2}\neq 0$, there exists $z\in{\operatorname{G}}(\Sigma)$ such that $z$ is contained in the relative interior of $\langle z_{1},z_{2}\rangle$.

Proof.

If $\deg P=2$ then $\dim\sigma(P)=0$, while if $\deg P=1$ then $\dim\sigma(P)=1$. Thus, we may assume that $\deg P=0$ and $\dim\sigma(P)=2$, that is, $\sigma(P)=\langle x,y\rangle$ for $x,y\in{\operatorname{G}}(\Sigma)$ and the primitive relation for $P$ is $z_{1}+z_{2}=x+y$.

Since every primitive crepant contraction is special, $P$ is not extremal. Therefore, there exist extremal primitive collections $P_{1},\ldots,P_{n}\subset{\operatorname{G}}(\Sigma)$ such that

[TABLE]

where $n\geq 2$ and $a_{1},\ldots,a_{n}>0$. Since $\deg P=0$, we have $\deg P_{1}=\cdots=\deg P_{n}=0$. This means that for any $1\leq i\leq n$, the extremal contraction associated to $P_{i}$ must be a special primitive crepant contraction. In particular, there exist two extremal primitive relations

[TABLE]

where $\{x_{1},x_{2},y_{1},y_{2}\}\subset{\operatorname{G}}(\Sigma)$. We remark that $\langle x,y\rangle=\sigma(P)\in\Sigma$. Since some of elements $x,y,x_{1},x_{2},y_{1},y_{2}$ may coincide, we have to consider the following four cases:

1. (1)

$x_{1}=y_{1}$. 2. (2)

$x=y_{1}$. 3. (3)

$x=y_{1}$ and $y=x_{2}$. 4. (4)

$x,y,x_{1},x_{2},y_{1},y_{2}$ are distinct elements.

(1) The case where $x_{1}=y_{1}$. Since $x_{1}+x_{2}=2x$ and $y_{1}+y_{2}=2y$ are extremal, Proposition 2.4 tells us that $\langle x,y_{2}\rangle,\langle x_{2},y\rangle\in\Sigma$. On the other hand, the relations $x_{1}+x_{2}=2x$, $y_{1}+y_{2}=2y$ and $x_{1}=y_{1}$ imply that

[TABLE]

The last relation means that either $\{x_{2},y\}$ or $\{x,y_{2}\}$ does not generate a cone in $\Sigma$. This is a contradiction.

(2) The case where $x=y_{1}$. By Proposition 2.4, $\langle y,x_{1}\rangle$ and $\langle y,x_{2}\rangle$ are in $\Sigma$ as above. Since $y_{1}+y_{2}=2y$ corresponds to a special primitive crepant contraction, there exists $m\in M$ as in Definition 3.2. For this $m$, we have

[TABLE]

On the other hand,

[TABLE]

a contradiction.

(3) The case where $x=y_{1}$ and $y=x_{2}$. Let $m\in M$ be the element as in Definition 3.2 for the extremal primitive relation $x_{1}+y=2x$.

[TABLE]

implies

[TABLE]

If $z_{1}$ and $z_{2}$ are not contained in the line passing through $x_{1},x,y,y_{2}$, then there must exist $v\in{\operatorname{G}}(\Sigma)\setminus\{x_{1},y,x\}$ such that $m(v)=-1$ and $\langle x,v\rangle\in\Sigma$, since the triangle whose vertices are $x_{1}$, $y$ and $z_{1}$ is contained in the hyperplane

[TABLE]

and $-1$ is the minimum value of $m$ on ${\operatorname{G}}(\Sigma)$. However, this is impossible, because $\langle x,v\rangle\in\Sigma$ implies $m(v)=0$ by the definition of a special primitive crepant contraction. Therefore, $z_{1},z_{2},x_{1},x,y,y_{2}$ are contained in a line. Since $z_{1}+z_{2}=x+y$ and $\langle x,y\rangle\in\Sigma$, in this case, $x$ and $y$ are contained in the relative interior of $\langle z_{1},z_{2}\rangle$.

(4) The case where $x_{1},x_{2},x,y_{1},y_{2},y$ are distinct elements. By applying Proposition 2.4 to the extremal primitive relations $x_{1}+x_{2}=2x$ and $y_{1}+y_{2}=2y$ several times, we can easily see that

[TABLE]

Let $m_{1},m_{2}\in M$ be the elements associated to the extremal primitive relations $x_{1}+x_{2}=2x$ and $y_{1}+y_{2}=2y$ as in Definition 3.2, respectively. We remark that

[TABLE]

while

[TABLE]

Then,

[TABLE]

says that either $m_{1}(z_{1})$ or $m_{1}(z_{2})$ is equal to $-1$. Without loss of generality, we may assume $m_{1}(z_{1})=-1$ and $m_{1}(z_{2})=0$. As in the case of (3), $z_{1}$ has to be contained in the line $L_{1}$ passing through $x_{1},x_{2},x$. Similarly, we obtain $m_{2}(z_{1})+m_{2}(z_{2})=m_{2}(x)+m_{2}(y)=0-1=-1$. Thus, $m_{2}(z_{1})=-1$ or $m_{2}(z_{2})=-1$. If $m_{2}(z_{1})=-1$, then $z_{1}$ has to be contained in the line $L_{2}$ passing through $y_{1},y_{2},y$ as above. Since $L_{1}\cap L_{2}=\emptyset$, this is impossible. Thus, we have $m_{2}(z_{2})=-1$ and $z_{2},y_{1},y_{2},y$ are contained in a line.

Consequently, $z_{1}$ and $z_{2}$ are expressed as follows:

[TABLE]

Since $x,y,x_{1},y_{1}$ are linearly independent, $x,y,z_{1},z_{2}$ are also linearly independent. However, this is a contradiction, because there is a linear relation $z_{1}+z_{2}=x+y$. ∎

Let $X=X_{\Sigma}$ be a smooth toric special weak Fano $d$-fold. Since $X$ is not Fano, there exists at least one special primitive crepant contraction $\varphi:X\to\overline{X}$. So, let $x_{1}+x_{2}=2x$ ($x_{1},x_{2},x\in{\operatorname{G}}(\Sigma)$) be the extremal primitive relation associated to $\varphi$. We use the same notation $E\subset X$, $\varphi(E)$ and $\Sigma_{\varphi}\subset\Sigma$ as in Definition 3.2.

Put

[TABLE]

while put

[TABLE]

Remark 3.9**.**

By Proposition 2.4, we have $\mathbb{R}_{\geq 0}x+\mathbb{R}_{\geq 0}x_{1},\mathbb{R}_{\geq 0}x+\mathbb{R}_{\geq 0}x_{2}\in\Sigma$, while $\mathbb{R}_{\geq 0}x+\sigma\in\Sigma$ for any $\sigma\in\Sigma_{\varphi}$, where $\mathbb{R}_{\geq 0}x+\sigma$ is a $(\dim\sigma+1)$-dimensional cone, because it is simplicial. We remark that $\mathcal{S}_{1}$ is a subspace in $N_{\mathbb{R}}$, but $\mathcal{S}_{2}$ is not necessarily a subspace. Moreover, $\mathcal{S}_{1}\cap\mathcal{S}_{2}=\mathbb{R}x$.

We can show the following:

Proposition 3.10**.**

Put

[TABLE]

Then, either $I\subset\mathcal{S}_{1}$ or $I\subset\mathcal{S}_{2}$ holds.

Proof.

First, we show that $I\subset\mathcal{S}_{1}\cup\mathcal{S}_{2}$. Remark that for any $z\in I$, $\{x,z\}$ is a primitive collection. If $x+z=0$, then $z=-x\in\mathcal{S}_{1}\cap\mathcal{S}_{2}$. So suppose that $x+z\neq 0$. Then, Proposition 3.8 says that there exists $z^{\prime}\in{\operatorname{G}}(\Sigma)$ such that $z^{\prime}$ is contained in the relative interior of $\langle x,z\rangle$. If $z^{\prime}=x_{1}$, then $ax+bz=x_{1}$ for $a,b>0$. So, we have $z=\frac{1}{b}(-ax+x_{1})\in\mathcal{S}_{1}$. The case where $z^{\prime}=x_{2}$ is completely similar. If $z^{\prime}=y$ for some $y\in{\operatorname{G}}(\Sigma_{\varphi})$, then we can show that $z\in\mathcal{S}_{2}$ similarly. So, we may assume $z^{\prime}\in I$. In this case, we can replace $z$ by $z^{\prime}$ and do the same arguments repeatedly. This operation has to finish in finite steps. Therefore, we obtain a finite subset $\{z_{0}:=z,z_{1},\ldots,z_{s}\}\subset{\operatorname{G}}(\Sigma)$ such that $z_{0},\ldots,z_{s-1}\in I$, while $z_{s}\in\{x_{1},x_{2}\}\cup{\operatorname{G}}(\Sigma_{\varphi})$. In any case, $\{z_{0},z_{1},\ldots,z_{s}\}$ is contained in either $\mathcal{S}_{1}$ or $\mathcal{S}_{2}$.

Next, we show that $I\subset\mathcal{S}_{1}$ or $I\subset\mathcal{S}_{2}$. Suppose that there exist $z_{1}\in\mathcal{S}_{1}\cap I$ and $z_{2}\in\mathcal{S}_{2}\cap I$ such that $z_{1}\neq-x$ and $z_{2}\neq-x$. As mentioned above, without loss of generality, we may assume that $x_{1}$ is contained in the relative interior of $\langle x,z_{1}\rangle$, while for some $y\in{\operatorname{G}}(\Sigma_{\varphi})$, $y$ is contained in the relative interior of $\langle x,z_{2}\rangle$. Thus, we obtain

[TABLE]

Then,

[TABLE]

means that either $\{z_{1},y\}$ or $\{x_{1},z_{2}\}$ is a primitive collection because the intersection of the relative interiors of $\langle z_{1},y\rangle$ and $\langle x_{1},z_{2}\rangle$ is one-dimensional. Suppose that $\{z_{1},y\}$ is a primitive collection, that is, the relative interior of $\{z_{1},y\}$ contains $w\in{\operatorname{G}}(\Sigma)$ by Proposition 3.8. We remark that $z_{1}\in\mathbb{R}x+\mathbb{R}_{\geq 0}x_{1}\subset\mathcal{S}_{1}$, while there exists a $(d-2)$-dimensional cone $\sigma\in\Sigma_{\varphi}$ such that $y\in\mathbb{R}x+\sigma\in\mathcal{S}_{2}$. This means $w\not\in\mathcal{S}_{1}\cup\mathcal{S}_{2}$, a contradiction. The other case is similar. ∎

For the case $I\subset\mathcal{S}_{1}$, more precisely, we obtain the following.

Proposition 3.11**.**

*If $I\subset\mathcal{S}_{1}$, then $X$ is isomorphic to a smooth toric weak del Pezzo surface bundle over $\varphi(E)$. *

Proof.

First, we remark that ${\operatorname{G}}(\Sigma)\cap\mathcal{S}_{1}=\{x,x_{1},x_{2}\}\cup I$. For any $y\in{\operatorname{G}}(\Sigma_{\varphi})$, we have $m(y)=0$ by the definition, so there must exist an element $z\in{\operatorname{G}}(\Sigma)\cap\mathcal{S}_{1}$ such that $m(z)>0$. More precisely, since $X$ is a smooth weak Fano variety, an easy observation tells us that ${\operatorname{G}}(\Sigma)\cap\mathcal{S}_{1}$ must contain $z_{0}:=-x$, $z_{1}:=-x_{2}$ or $z_{2}:=-x_{1}$.

Suppose $z_{0}\in{\operatorname{G}}(\Sigma)\cap\mathcal{S}_{1}$. Proposition 3.8 says that $\{z_{0},x_{1}\}$ generates a $2$-dimensional cone in $\Sigma$ or there exists an element in ${\operatorname{G}}(\Sigma)$ in the relative interior of $\langle z_{0},x_{1}\rangle$. $\{z_{0},x_{2}\}$ is similar. By repeating this argument, we obtain the $2$-dimensional nonsingular complete fan $\Sigma^{\prime}$ in the subspace $\mathcal{S}_{1}\subset N_{\mathbb{R}}$ such that ${\operatorname{G}}(\Sigma^{\prime})=\{x,x_{1},x_{2}\}\cup I$. The toric surface $S$ associated to $\Sigma^{\prime}$ is a weak del Pezzo surface. Then, one can construct a fan $\Sigma^{+}$ such that ${\operatorname{G}}(\Sigma^{+})={\operatorname{G}}(\Sigma)$ and the associated toric variety is an $S$-bundle over $\varphi(E)$ (see Lemma 2.5). Then, Lemma 3.6 tells us that $X^{+}\cong X$.

So, suppose $z_{0}\not\in{\operatorname{G}}(\Sigma)\cap\mathcal{S}_{1}$. In this case, we may assume that $z_{1}\in{\operatorname{G}}(\Sigma)\cap\mathcal{S}_{1}$ without loss of generality. $z_{3}:=\frac{1}{2}(x_{1}+z_{1})$ has to be contained in ${\operatorname{G}}(\Sigma)\cap\mathcal{S}_{1}$. Since ${\operatorname{G}}(\Sigma)\setminus\mathcal{S}_{1}={\operatorname{G}}(\Sigma_{\varphi})$, $x_{1}+z_{1}=2z_{3}$ has to be an extremal primitive relation such that the associated primitive crepant contraction $\varphi^{\prime}$ is special. $\varphi^{\prime}$ contracts its exceptional divisor to $\varphi(E)$. Therefore, there exists $m^{\prime}\in M$ such that $m^{\prime}(x_{1})=m^{\prime}(z_{1})=m^{\prime}(z_{3})=-1$, while $m^{\prime}(y)=0$ for any $y\in{\operatorname{G}}(\Sigma_{\varphi})$. Thus, $\Sigma_{\varphi}$ is contained in the $(d-2)$-dimensional subspace in $N_{\mathbb{R}}$ defined by $m=0$ and $m^{\prime}=0$. So, ${\operatorname{G}}(\Sigma)\cap\mathcal{S}_{1}$ must contain an element other than $x,x_{1},x_{2},z_{1},z_{3}$. Similarly as the above case, we obtain the $2$-dimensional nonsingular complete fan $\Sigma^{\prime}$ in $\mathcal{S}_{1}$. In this case, $X$ is isomorphic to the direct product of $S=S_{\Sigma^{\prime}}$ and $\varphi(E)$. ∎

4. The classification of smooth toric special weak Fano 4-folds

In this section, we classify all the smooth toric special weak Fano $4$-folds using the results in the previous section. We omit the trivial case, that is, the case where a smooth toric special weak Fano $4$-fold $X$ is isomorphic to a direct product of lower-dimensional subvarieties. In this case, $X$ is trivially a smooth toric weakened Fano $4$-fold. We use the same notation as in Proposition 3.10 such as $x_{1}+x_{2}=2x$, $m$, $\mathcal{S}_{1}$, $\mathcal{S}_{2}$, $I$, $\varphi(E)$ and $\Sigma_{\varphi}$.

For the classification, we describe the list of smooth toric weak del Pezzo surfaces in [Sa2] here. We should remark that this list is equivalent to the one of Gorenstein toric del Pezzo surfaces (see [K]). Here, we use row vectors $w_{i}$ to describe elements in $N$. In this classification list, we denote by $F_{a}$ the Hirzebruch surface of degree $a$, while we denote by $S_{n}$ the del Pezzo surface of degree $n$.

We give the figures for $\mathbb{P}^{2}$, $\mathbb{P}^{1}\times\mathbb{P}^{1}$, $F_{1}$, $F_{2}$, $W_{3}$ and $W_{4}^{1}$ for the reader’s convenience. These varieties are recurring throughout our classification.

(I) the case where $I\subset\mathcal{S}_{1}$.

Let $S$ be the smooth toric weak del Pezzo surface in $X$ corresponding to the fan in $\mathcal{S}_{1}$. In this case, $X$ is an $S$-bundle over $\varphi(E)$ (see Proposition 3.11). $\varphi(E)$ is also a smooth toric weak del Pezzo surface.

Remark 4.1**.**

In this case, for any primitive collection $P\subset{\operatorname{G}}(\Sigma)$, we have either $P\subset{\operatorname{G}}(\Sigma_{\varphi})$ or $P\subset{\operatorname{G}}(\Sigma)\setminus{\operatorname{G}}(\Sigma_{\varphi})$. Moreover, the primitive collections in ${\operatorname{G}}(\Sigma_{\varphi})$ correspond to $\varphi(E)$, while the primitive collections in ${\operatorname{G}}(\Sigma)\setminus{\operatorname{G}}(\Sigma_{\varphi})$ correspond to $S$. For example, let $X=X_{\Sigma}$ be the smooth toric special weak Fano $4$-fold of type $\mathcal{Z}_{10}$ below. Then, the primitive relations of $\Sigma$ are

[TABLE]

[TABLE]

where ${\operatorname{G}}(\Sigma)=\{y_{1},y_{2},y_{3},y_{4},w_{1},w_{2},w_{3},w_{4},w_{5}\}$ and ${\operatorname{G}}(\Sigma_{\varphi})=\{y_{1},y_{2},y_{3},y_{4}\}$. $\varphi(E)\cong F_{1}$, while $S\cong W_{3}$. The primitive relations of the $2$-dimensional fan $\Sigma_{S}$ associated to $S$ are

[TABLE]

where ${\operatorname{G}}(\Sigma_{S})={\operatorname{G}}(\Sigma)\setminus{\operatorname{G}}(\Sigma_{\varphi})=\{w_{1},w_{2},w_{3},w_{4},w_{5}\}$. On the other hand, by omitting $w_{3}$ from the above primitive relations, we obtain the primitive relations

[TABLE]

of the $2$-dimensional fan $\Sigma^{\prime}$ associated to $\varphi(E)$, where ${\operatorname{G}}(\Sigma^{\prime})={\operatorname{G}}(\Sigma_{\varphi})=\{y_{1},y_{2},y_{3},y_{4}\}$.

By Lemma 2.5, $X$ is not isomorphic to the direct product $S\times\varphi(E)$ if and only if there exists a primitive collection $P\subset{\operatorname{G}}(\Sigma_{\varphi})$ such that $\sigma(P)\cap{\operatorname{G}}(\Sigma)\not\subset{\operatorname{G}}(\Sigma_{\varphi})$.

Suppose that $\rho(\varphi(E))\geq 2$, where $\rho(\varphi(E))=\#{\operatorname{G}}(\Sigma_{\varphi})$ stands for the Picard number of $\varphi(E)$. In this case, every primitive collection of $\Sigma_{\varphi}$ contains exactly two elements. For $y\in N$, we denote by $\overline{y}$ the image of $y$ by the projection $N_{\mathbb{R}}\to N_{\mathbb{R}}/\mathcal{S}_{1}$, and we denote by $\overline{\Sigma}_{\varphi}$ the natural fan in $N_{\mathbb{R}}/\mathcal{S}_{1}$ associated to $\varphi(E)$. Let $P=\{y_{1},y_{2}\}\subset{\operatorname{G}}(\Sigma_{\varphi})$ be a primitive collection, and assume that $\sigma(P)\cap{\operatorname{G}}(\Sigma)\not\subset{\operatorname{G}}(\Sigma_{\varphi})$. We remark that $\{\overline{y}_{1},\overline{y}_{2}\}$ is also a primitive collection of $\overline{\Sigma}_{\varphi}$. If the primitive relation of $\{\overline{y}_{1},\overline{y}_{2}\}$ is either

[TABLE]

where $\overline{y}_{3},\overline{y}_{4}\in{\operatorname{G}}(\overline{\Sigma}_{\varphi})$, then the primitive relation of $P$ is also either

[TABLE]

because $X$ is a weak Fano manifold, that is, $\deg P\geq 0$. This contradicts the assumption $\sigma(P)\cap{\operatorname{G}}(\Sigma)\not\subset{\operatorname{G}}(\Sigma_{\varphi})$. Next, suppose that the primitive relation of $\{\overline{y}_{1},\overline{y}_{2}\}$ is

[TABLE]

We may assume this primitive relation is extremal, that is, $\langle\overline{y}_{1},\overline{y}_{3}\rangle$ and $\langle\overline{y}_{3},\overline{y}_{2}\rangle$ are $2$-dimensional cones in $\overline{\Sigma}_{\varphi}$. Then, for some element $z\not\in{\operatorname{G}}(\Sigma_{\varphi})$, the primitive relation of $P$ has to be

[TABLE]

and $\{y_{1},y_{2},y_{3},z\}$ spans a $3$-dimensional subspace in $N_{\mathbb{R}}$. On the other hand, there exists an element in ${\operatorname{G}}(\Sigma)$ in the relative interior of $\langle y_{1},y_{2}\rangle$ by Proposition 3.8. This is impossible. So, the remaining case is that the primitive relation of $\{\overline{y}_{1},\overline{y}_{2}\}$ is

[TABLE]

Let us show that $\{\overline{y}_{1},\overline{y}_{2}\}$ is extremal. If $\{\overline{y}_{1},\overline{y}_{2}\}$ is not extremal, then there exist extremal primitive collection $P_{1},\ldots,P_{l}$ of $\overline{\Sigma}_{\varphi}$ $(l\geq 2)$ such that $r(\{\overline{y}_{1},\overline{y}_{2}\})=a_{1}r(P_{1})+\cdots+a_{l}r(P_{l})$ for $a_{1},\ldots,a_{l}\in\mathbb{Z}_{>0}$ (see Remark 2.3). Obviously, $\deg P_{i}\leq 1$ for any $i$. This means that the primitive relation of $P_{i}$ in $\Sigma$ is same as the one in $\overline{\Sigma}_{\varphi}$ for any $i$ by the above argument. This is impossible because the primitive relation of $\{\overline{y}_{1},\overline{y}_{2}\}$ in $\Sigma$ and the one in $\overline{\Sigma}_{\varphi}$ are distinct. Thus, we obtain the following:

Proposition 4.2**.**

Let $X$ be a smooth toric special weak Fano $4$-fold which is isomorphic to an $S$-bundle over $\varphi(E)$, where $S$ and $\varphi(E)$ are smooth toric weak del Pezzo surfaces, and $\rho(\varphi(E))\geq 2$. If $X\not\cong S\times\varphi(E)$, then $\varphi(E)$ has a Fano contraction, that is, a $\mathbb{P}^{1}$-bundle structure.

On the other hand, for the possiblities for the fiber $S$, the following holds. This fact makes our classification easier.

Proposition 4.3**.**

If $X$ is not isomorphic to the direct product $S\times\varphi(E)$, then $S$ is one of the following: $F_{2}$, $W_{3}$ and $W_{4}^{1}$.

Proposition 4.3 is a consequence of the following lemma and Table $1$. Remark that $S$ admits at least one crepant contraction.

Lemma 4.4**.**

Suppose that there exist two special primitive crepant contractions associated to the extremal primitive relations $x_{1}+x_{2}=2x$ and $x^{\prime}_{1}+x^{\prime}_{2}=2x^{\prime}$, where $x_{1},x_{2},x,x^{\prime}_{1},x^{\prime}_{2},x^{\prime}\in{\operatorname{G}}(\Sigma)\cap\mathcal{S}_{1}$ such that associated elements $m,m^{\prime}\in M$ as in Definition 3.2 are linearly independent. Then, $X\cong S\times\varphi(E)$.

Proof.

Put ${\operatorname{G}}(\Sigma_{\varphi})=\{y_{1},\ldots,y_{l}\}$. Then, we have

[TABLE]

This means that $y_{1},\ldots,y_{l}$ are contained in a $2$-dimensional subspace in $N_{\mathbb{R}}$. Namely, $X$ is isomorphic to the direct product of $S$ and $\varphi(E)$. ∎

Thus, the possibilities for $\varphi(E)$ are $\mathbb{P}^{2}$, $\mathbb{P}^{1}\times\mathbb{P}^{1}$, $F_{1}$ and $F_{2}$ by Table $1$, while the possibilities for $S$ are $F_{2}$, $W_{3}$ and $W_{4}^{1}$ by Proposition 4.3. Here, for the reader’s convenience, we explicitly describe the values $m(w_{i})$ for the element $m\in M$ associated to a special primitive crepant contraction (we use the same notation as in Table $1$):

• •

$S\cong F_{2}$: The primitive relation associated to the special primitive crepant contraction is $w_{2}+w_{4}=2w_{1}$. $m(w_{1})=m(w_{2})=m(w_{4})=-1$ and $m(w_{3})=1$.

• •

$S\cong W_{3}$: The primitive relation associated to the special primitive crepant contraction is $w_{2}+w_{5}=2w_{1}$. $m(w_{1})=m(w_{2})=m(w_{5})=-1$, $m(w_{3})=0$ and $m(w_{4})=1$.

• •

$S\cong W_{4}^{1}$: The primitive relation associated to the special primitive crepant contraction is $w_{2}+w_{6}=2w_{1}$. $m(w_{1})=m(w_{2})=m(w_{6})=-1$, $m(w_{3})=m(w_{5})=0$ and $m(w_{4})=1$.

$\varphi(E)\cong\mathbb{P}^{2}$ Let ${\operatorname{G}}(\Sigma_{\varphi})=\{y_{1},y_{2},y_{3}\}$. In this case, $\{y_{1},y_{2},y_{3}\}$ is the unique primitive collection in $\Sigma_{\varphi}$, and we have the primitive relation

[TABLE]

in $\overline{\Sigma}_{\varphi}$. Obviously, $\{y_{1},y_{2},y_{3}\}$ is an extremal primitive collection. So, the possibilities for its primitive relation are

[TABLE]

for $z_{1},z_{2}\in{\operatorname{G}}(\Sigma)\setminus{\operatorname{G}}(\Sigma_{\varphi})$ and $a=1,2$ because the associated extremal ray has to be $K_{X}$-negative (otherwise, this primitive crepant contraction would be non-special). For the first case, $am(z_{1})=m(y_{1}+y_{2}+y_{3})=m(y_{1})+m(y_{2})+m(y_{3})=0$. For the second case, similarly, we have $m(z_{1})+m(z_{2})=0$. However, since $m(z_{1})$ and $m(z_{2})$ are at least $-1$, we have three cases $(m(z_{1}),m(z_{2}))=(0,0)$, $(m(z_{1}),m(z_{2}))=(-1,1)$ and $(m(z_{1}),m(z_{2}))=(1,-1)$. The first case does not occur because $\langle z_{1},z_{2}\rangle\in\Sigma$.

By using the above arguments and Table $1$, we can easily classify all the smooth toric special weak Fano $4$-folds as follows. In the list, we use the notation in Table $1$ for $S$. $X$ is determined by giving the primitive relation for $\{y_{1},y_{2},y_{3}\}$.

$\varphi(E)\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$, $F_{1}$ or $F_{2}$ For any extremal primitive relation $\overline{y}_{1}+\overline{y}_{2}=0$ in $\overline{\Sigma}_{\varphi}$, $\{y_{1},y_{2}\}$ is also extremal, because $\{y_{1},y_{2}\}\cap P=\emptyset$ for any primitive collection $P\neq\{y_{1},y_{2}\}$ of $\Sigma$. So, if the associated contraction morphism is crepant, then it has to be special. Thus, the possibilities for the primitive relation of $\{y_{1},y_{2}\}$ are

[TABLE]

where $z\in{\operatorname{G}}(\Sigma)\setminus{\operatorname{G}}(\Sigma_{\varphi})$ and $m(z)=0$. In the second case, there exists $m^{\prime}\in M$ associated to this primitive relation. For $z_{1},z_{2}\not\in{\operatorname{G}}(\Sigma_{\varphi})$ such that $\langle z_{1},z\rangle,\langle z,z_{2}\rangle\in\Sigma$, we have $m^{\prime}(z_{1})=m^{\prime}(z_{2})=0$. Since $z$, $z_{1}$ and $z_{2}$ are contained in a $2$-dimensional subspace, $z_{1}$ and $z_{2}$ have to be centrally symmetric, that is, $z_{1}+z_{2}=0$. However, this phenomenon does not occur (see the classification below).

By using these results, we can do the classification easily. The classification lists are as follows.

Let $\varphi(E)\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$, and $\overline{y}_{1}+\overline{y}_{3}=0,\overline{y}_{2}+\overline{y}_{4}=0$ the primitive relations for $\varphi(E)$. Then, $X$ is determined by giving the primitive relations for $\{y_{1},y_{3}\}$ and $\{y_{2},y_{4}\}$. We remark that if either $y_{1}+y_{3}=0$ or $y_{2}+y_{4}=0$ holds, then $X$ is isomorphic to $\mathbb{P}^{1}\times V$ for a smooth toric special weak Fano $3$-fold $V$. As before, we omit this trivial case.

Let $\varphi(E)\cong F_{1}$, and $\overline{y}_{1}+\overline{y}_{3}=0,\overline{y}_{2}+\overline{y}_{4}=\overline{y}_{1}$ the primitive relations for $\varphi(E)$. Then, $X$ is determined by giving the primitive relation for $\{y_{1},y_{3}\}$.

Let $\varphi(E)\cong F_{2}$, and $\overline{y}_{1}+\overline{y}_{3}=0,\overline{y}_{2}+\overline{y}_{4}=2\overline{y}_{1}$ the primitive relations for $\varphi(E)$. Then, $X$ is determined by giving the primitive relation for $\{y_{1},y_{3}\}$.

Remark 4.5**.**

$\mathcal{Z}_{12}$ and $\mathcal{Z}_{13}$ have two special primitive crepant contractions.

(II) the case where $I\subset\mathcal{S}_{2}$.

First, we remark that for any $v\in{\operatorname{G}}(\Sigma)$,

[TABLE]

because if $v\in I$, then $v\in\mathbb{R}_{\geq 0}(-x)+\sigma\subset\mathcal{S}_{2}$ for some $\sigma\in\Sigma_{\varphi}$ by the definition of $\mathcal{S}_{2}$.

Suppose that $\rho(\varphi(E))\geq 2$. Then, for any primitive collection $\{y_{1},y_{2}\}\subset{\operatorname{G}}(\Sigma_{\varphi})$, an element $z\in\sigma(\{y_{1},y_{2}\})\cap{\operatorname{G}}(\Sigma)$ has to have $m(z)=0$ by Proposition 3.8. Namely, $z\in{\operatorname{G}}(\Sigma_{\varphi})$. So, $|\Sigma_{\varphi}|$ is a subspace in $N_{\mathbb{R}}$, and then $\mathcal{S}_{2}$ is also a subspace in $N_{\mathbb{R}}$. Thus, we obtain the following.

Proposition 4.6**.**

Let $X$ be a smooth toric special weak Fano $4$-fold, and suppose that $I\subset\mathcal{S}_{2}$. If $\rho(\varphi(E))\geq 2$, then $X$ is isomorphic to a $V$-bundle over $\mathbb{P}^{1}$, where $V$ is the $3$-dimensional subvariety corresponding to the complete fan in $\mathcal{S}_{2}$.

Remark 4.7**.**

In Proposition 4.6, $V$ has to be either a smooth toric Fano $3$-fold or a smooth toric special weak Fano $3$-fold. In the case of dimension $3$, it was shown by [Mi1] and [Mi2] that a smooth toric weak Fano $3$-fold is special if and only if it is a weakened Fano manifold (also see [Sa2]). Moreover, if $V$ is a direct product of $\mathbb{P}^{1}$ and a smooth toric weak del Pezzo surface, then $X$ is isomorphic to either a direct product of $\mathbb{P}^{1}$ and a smooth toric special weak Fano $3$-fold or a direct product of $F_{2}$ and a smooth toric weak del Pezzo surface. We therefore omit these trivial cases.

Now, we start the classification. First we consider the case where $X$ is isomorphic to a $V$-bundle over $\mathbb{P}^{1}$. $V$ have a toric fibration

[TABLE]

associated to the map $\mathcal{S}_{2}\to\mathcal{S}_{2}/|\Sigma_{\varphi}|$ such that $V_{t}:=f^{-1}(t)\cong\varphi(E)$ for any $t\in\mathbb{C}$. The fiber $V_{0}$ is the torus invariant prime divisor associated to the cone $\mathbb{R}_{\geq 0}x$ in the subfan $\Sigma_{V}$ corresponding to $V$, while any irreducible component of $V_{\infty}$ is associated to an element in

[TABLE]

If $V_{\infty}$ is also isomorphic to $\varphi(E)$, that is,

[TABLE]

then $V$ is a $\varphi(E)$-bundle over $\mathbb{P}^{1}$.

The case where $V$ is a smooth toric Fano $3$-fold. We use the notation in the classification list of smooth toric Fano $3$-folds in [B2, Propositions 2.5.5, 2.5.6, 2.5.8 and 2.5.9]. Also, we use the notation ${\operatorname{G}}(\Sigma_{V})=\{v_{1},v_{2},\ldots\}$, thus we have ${\operatorname{G}}(\Sigma)=\{x_{1},x_{2},v_{1},v_{2},\ldots,\}$. We remark that $|\Sigma_{V}|=\mathcal{S}_{2}$. Trivially, $V\cong\mathbb{P}^{3}$ is impossible.

• •

$V$ is of type $\mathcal{B}$. $\mathcal{B}_{4}$ is a direct product. For $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$, there is no subvariety which corresponds to $\varphi(E)$. So, the remaining case is $\mathcal{B}_{3}$. The primitive relations of $\mathcal{B}_{3}$ are

[TABLE]

where ${\operatorname{G}}(\Sigma_{V})=\{v_{1},v_{2},v_{3},v_{4},v_{5}\}$. $f:V\to\mathbb{P}^{1}$ is induced by

[TABLE]

So, the possibilities for $x$ are $v_{1}$ and $v_{2}$. By adding one more primitive relation

[TABLE]

where ${\operatorname{G}}(\Sigma)=\{x_{1},x_{2}\}\cup{\operatorname{G}}(\Sigma_{V})$, we obtain a new smooth toric special weak Fano $4$-fold $\mathcal{Z}_{14}$. The last primitive relation corresponds to a special primitive crepant contraction. $\varphi(E)\cong\mathbb{P}^{2}$ and $V$ is a $\varphi(E)$-bundle over $\mathbb{P}^{1}$. In the case where $x=v_{2}$, we obtain $\mathcal{Z}_{14}$, too.

• •

$V$ is of type $\mathcal{C}$. $\mathcal{C}_{3}$ and $\mathcal{C}_{4}$ is isomorphic to $\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ and $\mathbb{P}^{1}\times F_{1}$, respectively. The primitive relations of the remaining cases are as follows: $v_{1}+v_{2}=0,$

where ${\operatorname{G}}(\Sigma_{V})=\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6}\}$. One can easily see that there exist exactly two, one and two toric surface bundle structures over $\mathbb{P}^{1}$ for $\mathcal{C}_{1}$, $\mathcal{C}_{2}$ and $\mathcal{C}_{5}$, respectively. They are induced by the following morphisms:

[TABLE]

[TABLE]

Thus, we have three new smooth toric special weak Fano $4$-folds $\mathcal{Z}_{15}$ $(V\cong\mathcal{C}_{1})$, $\mathcal{Z}_{16}$ $(V\cong\mathcal{C}_{2})$ and $\mathcal{Z}_{17}$ $(V\cong\mathcal{C}_{5})$, by adding a primitve relation

[TABLE]

to the above three cases, where ${\operatorname{G}}(\Sigma)=\{x_{1},x_{2}\}\cup{\operatorname{G}}(\Sigma_{V})$. This primitive relation corresponds to a special primitive crepant contraction. In every case, we have $\varphi(E)\cong F_{1}$.

• •

$V$ is of type $\mathcal{D}$. The type $\mathcal{D}_{1}$ is impossible, because there does not exist a subvariety corresponding to $\varphi(E)$. For $V\cong\mathcal{D}_{2}$ whose primitive relations are

[TABLE]

there exists exactly one toric fibration to $\mathbb{P}^{1}$ induced by the morphism

[TABLE]

Since there exist three elements $v_{3}$, $v_{4}$ and $v_{6}$ other than $\{v_{1},v_{2},v_{5}\}$, $V_{\infty}$ is not isomorphic to $\mathbb{P}^{2}$, that is, $f$ is not a bundle structure. Thus, the possibility for $x$ is only $v_{6}$, and we obtain a smooth toric special weak Fano $4$-fold $\mathcal{Z}_{18}$ with one more primitve relation

[TABLE]

where ${\operatorname{G}}(\Sigma)=\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},x_{1},x_{2}\}$. In this case, $\varphi(E)\cong\mathbb{P}^{2}$.

• •

$V$ is of type $\mathcal{E}$. The type $\mathcal{E}$ is a $S_{7}$-bundle over $\mathbb{P}^{1}$. In particular, $\mathcal{E}_{3}$ is isomorphic to $\mathbb{P}^{1}\times S_{7}$. The primitive relations of the remaining cases are as follows: $v_{2}+v_{4}=0,v_{3}+v_{5}=0,v_{1}+v_{3}=v_{2},v_{2}+v_{5}=v_{1},v_{1}+v_{4}=v_{5},$

where ${\operatorname{G}}(\Sigma_{V})=\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\}$.

For $V\cong\mathcal{E}_{1}$, there exists one toric fibration $V\to\mathbb{P}^{1}$ associated to the projection

[TABLE]

In fact, we obtain a smooth special weak Fano $4$-fold by adding the primitive relation $x_{1}+x_{2}=2v_{6}$, where ${\operatorname{G}}(\Sigma)={\operatorname{G}}(\Sigma_{V})\cup\{x_{1},x_{2}\}$. In this case, $\varphi(E)\cong S_{7}$. We denote by $\mathcal{Z}_{19}$ this smooth toric special weak Fano $4$-fold.

For $V\cong\mathcal{E}_{2}$, there exist two toric fibrations $V\to\mathbb{P}^{1}$ associated to the projections

[TABLE]

The first one is the case where $\varphi(E)\cong S_{7}$ and $x_{1}+x_{2}=2v_{6}$, while the second one is the case where $\varphi(E)\cong F_{1}$ and $x_{1}+x_{2}=2v_{3}$, where ${\operatorname{G}}(\Sigma)={\operatorname{G}}(\Sigma_{V})\cup\{x_{1},x_{2}\}$. We denote by $\mathcal{Z}_{20}$ and $\mathcal{Z}_{21}$ these smooth toric special weak Fano $4$-folds, respectively. For $\mathcal{Z}_{21}$, the toric fibration $V\to\mathbb{P}^{1}$ has a singular fiber. Namely, $V$ is not an $F_{1}$-bundle over $\mathbb{P}^{1}$.

For $V\cong\mathcal{E}_{4}$, we have two toric fibrations $V\to\mathbb{P}^{1}$, too. They are associated to the projections

[TABLE]

The first one is the case where $\varphi(E)\cong S_{7}$ and $x_{1}+x_{2}=2v_{6}$, while the second one is the case where $\varphi(E)\cong F_{1}$ and $x_{1}+x_{2}=2v_{4}$, where ${\operatorname{G}}(\Sigma)={\operatorname{G}}(\Sigma_{V})\cup\{x_{1},x_{2}\}$. We denote by $\mathcal{Z}_{22}$ and $\mathcal{Z}_{23}$ these smooth toric special weak Fano $4$-folds, respectively. We remark that for $\mathcal{Z}_{23}$, $V$ is not an $F_{1}$-bundle over $\mathbb{P}^{1}$.

• •

$V$ is of type $\mathcal{F}$. The type $\mathcal{F}$ is a $S_{6}$-bundle over $\mathbb{P}^{1}$. In particular, $\mathcal{F}_{1}$ is isomorphic to $\mathbb{P}^{1}\times S_{6}$. So, let $V\cong\mathcal{F}_{2}$. The primitive relations of $V$ are

[TABLE]

[TABLE]

where ${\operatorname{G}}(\Sigma_{V})=\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7},v_{8}\}$. Thus, we obtain a new smooth toric special weak Fano $4$-fold $\mathcal{Z}_{24}$ by adding the primitive relation $x_{1}+x_{2}=2v_{7}$, where ${\operatorname{G}}(\Sigma)={\operatorname{G}}(\Sigma_{V})\cup\{x_{1},x_{2}\}$. In this case, $\varphi(E)\cong S_{6}$. We remark that there exist other three toric fibrations $V\to\mathbb{P}^{1}$ whose general fibers are isomorphic to $\mathbb{P}^{2}$. However, they are impossible, since there is no smooth fiber that could correspond to $V_{0}$.

The case where $V$ is a smooth toric weakned Fano $3$-fold.

We describe the four toric weakened Fano $3$-folds $X_{3}^{0},X_{4}^{0},X_{4}^{1},X_{5}^{1}$, which are not isomorphic to direct products of lower-dimensional varieties as in [Sa2].

Put

[TABLE]

[TABLE]

Then, we have $V\cong X_{3}^{0}$ if ${\operatorname{G}}(\Sigma_{V})=\{x_{0},x_{+},x_{-},y_{+},y_{-},z_{1}\}$, while $V\cong X_{4}^{0}$ if ${\operatorname{G}}(\Sigma_{V})=\{x_{0},x_{+},x_{-},y_{+},y_{-},z_{1},z_{2}\}$ (see Lemma 3.6).

For $V\cong X_{3}^{0}$, there exists one toric fibration from $V$ to $\mathbb{P}^{1}$ associated to the projection

[TABLE]

However, the smooth toric weak Fano $4$-fold $X$ obtained by adding a primitive relation $x_{1}+x_{2}=2x_{+}$ is not special, because the primitive crepant contraction associated to $x_{+}+x_{-}=2x_{0}$ is not special. In fact, for $m^{\prime}\in M$ associated to $x_{+}+x_{-}=2x_{0}$ as in Definition 3.2, we have $m^{\prime}(x_{+})=m^{\prime}(x_{-})=m^{\prime}(x_{0})=-1$, while $m^{\prime}(y_{+})=m^{\prime}(y_{-})=0$, $m^{\prime}(x_{1})\geq 0$ and $m^{\prime}(x_{2})\geq 0$. This contradicts the primitive relation $x_{1}+x_{2}=2x_{+}$.

For $V\cong X_{4}^{0}$, we have two toric fibrations $V\to\mathbb{P}^{1}$ associated to

[TABLE]

Let $X_{1}$ and $X_{2}$ be smooth toric weak Fano $4$-folds constructed by adding primitive relations $x_{1}+x_{2}=2x_{+}$ and $x_{1}+x_{2}=2y_{-}$, respectively. $X_{1}$ is not special as in the case where $V\cong X_{3}^{0}$. Thus, we obtain one new smooth toric special weak Fano $4$-fold $\mathcal{Z}_{25}:=X_{2}$. In this case, $\varphi(E)\cong F_{2}$.

Put

[TABLE]

[TABLE]

Then, we have $V\cong X_{4}^{1}$ if ${\operatorname{G}}(\Sigma_{V})=\{x_{0},x_{+},x_{-},y_{+},y_{-},z_{1},z_{2}\}$, while $V\cong X_{5}^{1}$ if ${\operatorname{G}}(\Sigma_{V})=\{x_{0},x_{+},x_{-},y_{+},y_{-},z_{1},z_{2},z_{3}\}$ (see Lemma 3.6).

For $V\cong X_{4}^{1}$, there exists a toric fibration $V\to\mathbb{P}^{1}$ associated to the projection

[TABLE]

Thus, we have a smooth toric special weak Fano $4$-fold by adding the primitive relation $x_{1}+x_{2}=2y_{+}$. $X$ is isomorphic to a $W_{3}$-bundle over $F_{2}$ in this case. In fact, this manifold is isomorphic to $\mathcal{Z}_{12}$.

For $V\cong X_{5}^{1}$, there exist exactly two toric fibrations $V\to\mathbb{P}^{1}$ associated to

[TABLE]

[TABLE]

In the former case, $X$ is a $W_{4}^{1}$-bundle over $F_{2}$ and is isomorphic to $\mathcal{Z}_{13}$. In the latter case, we have a new smooth toric special weak Fano $4$-fold $\mathcal{Z}_{26}$ which has one more primitive relation $x_{1}+x_{2}=2z_{2}$. In this case, $\varphi(E)\cong F_{1}$.

Finally, we consider the remaining case, that is, the case where $\varphi(E)\cong\mathbb{P}^{2}$ and $V$ has no $\varphi(E)$-fibration over $\mathbb{P}^{1}$.

In this case, one can easily see that there exist two primitive relations

[TABLE]

where ${\operatorname{G}}(\Sigma_{\varphi})=\{y_{1},y_{2},y_{3}\}$ and $I=\{z_{1},\ldots,z_{l}\}$ for some $1\leq i\leq l$ (remember that ${\operatorname{G}}(\Sigma)=\{x_{1},x_{2},x\}\cup{\operatorname{G}}(\Sigma_{\varphi})\cup I$). We remark that $y_{1}+y_{2}+y_{3}=x+z_{i}$ is impossible, because $\langle x,z_{i}\rangle\not\in{\operatorname{G}}(\Sigma)$. By [B1, Proposition 3.2], we have one more primitive relation

[TABLE]

since $X$ is projective and $\mathbb{R}x$ is the maximum subspace in $S_{2}$. We may assume $j=1$.

By using Proposition 3.8, we can prove the following. The proof is similar to the one of Proposition 3.10. We therefore omit it.

Lemma 4.8**.**

$\{z_{1},\ldots,z_{l}\}$* is contained in one of the following $2$-dimensional cones: $\langle z_{1},y_{1}\rangle$, $\langle z_{1},y_{2}\rangle$ or $\langle z_{1},y_{3}\rangle$.*

Without loss of generality, we may assume $\{z_{1},\ldots,z_{l}\}\subset\langle z_{1},y_{1}\rangle$ by Lemma 4.8. Permute $z_{2},\ldots,z_{l}$ in order of proximity to $z_{1}$. Then, $\{z_{k},z_{k+2}\}$ is a primitive collection for $1\leq k\leq l-1$, where put $z_{l+1}:=y_{1}$. If there exists a primive relation

[TABLE]

then we have a primitive crepant contraction associated to this primitive relation, because there are eight $4$-dimensional cones

[TABLE]

[TABLE]

which contain $z_{k+1}$. We can construct a divisorial contraction by removing $z_{k+1}$ by Proposition 2.4. So, this primitive crepant contraction has to be special. By Definition 3.2 (3), there exists $m^{\prime}\in M$ such that $m^{\prime}(x_{1})=m^{\prime}(x_{2})=m^{\prime}(y_{2})=m^{\prime}(y_{3})=0$. However, since $x_{1},x_{2},y_{2},y_{3}$ spans $N_{\mathbb{R}}$, this is a contradiction. Thus, we have primitive relations $z_{k}+z_{k+2}=z_{k+1}$ for $1\leq k\leq l-1$. Suppose that $l\geq 3$. By adding two primitive relations

[TABLE]

we have another primitive relation $z_{1}+z_{4}=0$. This is impossible. So, we have $l\leq 2$.

If $l=1$, then the primitive relations of $\Sigma$ are

[TABLE]

One can easily see that $X$ is an $F_{2}$-bundle over $\mathbb{P}^{2}$ in this case. So, this is also in the case where $I\subset\mathcal{S}_{1}$. In fact, $X$ is isomorphic to $\mathcal{Z}_{1}$.

Suppose that $l=2$. If $i=1$, then we have the extremal primitive relation

[TABLE]

The associated primitive crepant contraction is not special, a contradiction. If $i=2$, then the relation

[TABLE]

implies that either $\{y_{2},y_{3}\}$ or $\{x_{1},z_{1}\}$ is a primitive collection. However, Proposition 3.8 says that $\{x_{1},z_{1}\}$ is not a primitive collection, because ${\operatorname{G}}(\Sigma)\cap\mathcal{S}_{1}=\{x,x_{1},x_{2},z_{1}\}$. So, we also have a contradiction in this case.

Thus, we have no new smooth toric special weak Fano $4$-fold in this case. This finishes the classification.

Theorem 4.9**.**

There exist exactly $26$ smooth toric special weak Fano $4$-folds which do not decompose into a direct product of lower-dimensional varieties.

5. Deformations of smooth toric special weak Fano 4-folds

If a smooth toric special weak Fano $4$-fold $X$ is isomorphic to a direct product of lower-dimensional varieties, then $X$ is obviously a smooth toric weakened Fano $4$-fold. So, in this final section, we deal with the remaining cases. Namely, we show that smooth toric special weak Fano $4$-folds $\mathcal{Z}_{i}$ ($1\leq i\leq 26$) classified in Section 4 are deformation equivalent to Fano manifolds with a few exceptions using the deformation families constructed in [LaMe]. We use the notation in Batyrev’s classification of smooth toric Fano $4$-folds.

As examples, we show that $\mathcal{Z}_{1}$ (the case where $I\subset\mathcal{S}_{1}$) and $\mathcal{Z}_{14}$ (the case where $I\subset\mathcal{S}_{2}$) are deformation equivalent to Fano manifolds. The other cases are almost completely similar. However, we cannot apply this technique for the case where

[TABLE]

for a special primitive crepant contraction on $X$, because the general fiber of the deformation family in this case is not necessarily a toric variety in general (see [Sa2, Section 4]). Thus, we cannot determine whether smooth toric special weak Fano $4$-folds $\mathcal{Z}_{18}$, $\mathcal{Z}_{21}$, $\mathcal{Z}_{23}$, $\mathcal{Z}_{25}$ and $\mathcal{Z}_{26}$ are deformation equivalent to Fano manifolds using our current techniques.

Example 5.1**.**

Let $X=X_{\Sigma}$ be the smooth toric special weak Fano $4$-fold of type $\mathcal{Z}_{1}$. Then, ${\operatorname{G}}(\Sigma)=\{x,x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}\}$, where

[TABLE]

[TABLE]

The primitive relations of $\Sigma$ are

[TABLE]

In this case, $x_{1}+x_{2}=2x$ is an extremal primitive relation associated to a primitive special crepant contraction. $m\in M$ as in Definition 3.2 is the first projection $(a_{1},a_{2},a_{3},a_{4})\mapsto a_{1}$. Put $D_{0},D_{1},D_{2},D_{3},E_{1},E_{2},E_{3}$ be the torus invariant prime divisors corresponding to $x,x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$, respectively. We use the short exact sequence

[TABLE]

to find the degree matrix of $X$, that is, the matrix associated to the map $\psi$. Its kernel is the $4\times 7$ matrix given by the above rays, providing us with the relations:

[TABLE]

in ${\operatorname{Pic}}(X)$. $\{D_{2},D_{3},E_{3}\}$ is a $\mathbb{Z}$-basis for ${\operatorname{Pic}}(X)$, and the other torus invariant prime divisors are expressed as follows:

[TABLE]

With respect to this $\mathbb{Z}$-basis, we have

[TABLE]

[TABLE]

where $[D]$ stands for the element in $\mathbb{Z}^{3}\cong{\operatorname{Pic}}(X)$ corresponding to a torus invariant divisor $D$. Thus, the degree matrix of $X$ is

[TABLE]

where $[D]$ stands for the element in $\mathbb{Z}^{3}\cong{\operatorname{Pic}}(X)$ corresponding to a torus invariant divisor $D$. On the other hand, the combinatorial structure of $\Sigma$ is determined by the primitive collections. Namely, a subset $S\subset{\operatorname{G}}(\Sigma)$ generates a cone in $\Sigma$ if and only if all of the following hold:

[TABLE]

Then, similarly as in Section 4 in [LaMe], there exists a one-parameter deformation $\pi:\ \mathcal{X}\to\mathbb{C}$ whose every fiber is a toric manifold such that $\mathcal{X}_{0}:=\pi^{-1}(0)\cong X$ and the degree matrix of a general fiber $\mathcal{X}_{t}:=\pi^{-1}(t)$ ($t\neq 0$) is obtained by replacing the first column and the fourth column of the one of $X$ by

[TABLE]

[TABLE]

respectively. Here, we should remark that $x_{1}+x_{2}=2x$ corresponds to a primitive special crepant contraction. Therefore, the degree matrix of $\mathcal{X}_{t}$ is

[TABLE]

By adding the second line to the first line, we obtain the matrix

[TABLE]

Then, by comparing this matrix with the degree matrix of $X$, we put

[TABLE]

Then, one can easily construct the fan $\Sigma^{\prime}$ associated to $\mathcal{X}_{t}$ by replacing $x_{2}\in{\operatorname{G}}(\Sigma)$ by $x_{2}^{\prime}$. Namely, ${\operatorname{G}}(\Sigma^{\prime})=\{x,x_{1},x_{2}^{\prime},x_{3},y_{1},y_{2},y_{3}\}$, and the combinatorial structure of $\Sigma^{\prime}$ is completely similar as $\Sigma$, that is, the primitive relations of $\Sigma^{\prime}$ are

[TABLE]

Thus, $\mathcal{X}_{t}$ is a smooth toric Fano $4$-fold of type $D_{9}$. In particular, $X$ is a smooth toric weakened Fano $4$-fold.

Example 5.2**.**

Let $X=X_{\Sigma}$ be the smooth toric special weak Fano $4$-fold of type $\mathcal{Z}_{14}$. Then, ${\operatorname{G}}(\Sigma)=\{x,x_{1},x_{2},y_{1},y_{2},y_{3},z\}$, where

[TABLE]

[TABLE]

The primitive relations of $\Sigma$ are

[TABLE]

Put $D_{0},D_{1},D_{2},E_{1},E_{2},E_{3},F$ be the torus invariant prime divisors corresponding to $x,x_{1},$ $x_{2},y_{1},y_{2},y_{3},z$, respectively. We use the short exact sequence

[TABLE]

to find the degree matrix of $X$, that is, the matrix associated to the map $\psi$. Its kernel is the $4\times 7$ matrix given by the above rays, providing us with the relations:

[TABLE]

in ${\operatorname{Pic}}(X)$. $\{D_{2},E_{3},F\}$ is a $\mathbb{Z}$-basis for ${\operatorname{Pic}}(X)$, and the other torus invariant prime divisors are expressed as follows:

[TABLE]

With respect to this $\mathbb{Z}$-basis, we have

[TABLE]

[TABLE]

Thus, the degree matrix of $X$ is

[TABLE]

There exists a one-parameter deformation $\pi:\ \mathcal{X}\to\mathbb{C}$ such that $\mathcal{X}_{0}\cong X$ and the degree matrix of a general fiber $\mathcal{X}_{t}$ ($t\neq 0$) is obtained by replacing the first column and the last column of the one of $X$ by

[TABLE]

[TABLE]

respectively. Therefore, the degree matrix of $\mathcal{X}_{t}$ is

[TABLE]

By adding the third line to the first line, we obtain the matrix

[TABLE]

Then, by comparing this matrix with the degree matrix of $X$, we put

[TABLE]

Then, one can easily construct the fan $\Sigma^{\prime}$ associated to $\mathcal{X}_{t}$ by replacing $x_{2}\in{\operatorname{G}}(\Sigma)$ by $x_{2}^{\prime}$. Namely, ${\operatorname{G}}(\Sigma^{\prime})=\{x,x_{1},x_{2}^{\prime},y_{1},y_{2},y_{3},z\}$, and the combinatorial structure of $\Sigma^{\prime}$ is completely similar as $\Sigma$, that is, the primitive relations of $\Sigma^{\prime}$ are

[TABLE]

Thus, $\mathcal{X}_{t}$ is a smooth toric Fano $4$-fold of type $D_{7}$. In particular, $X$ is a smooth toric weakened Fano $4$-fold.

Here, we summarize the central fibers $\mathcal{X}_{0}$ and the general fibers $\mathcal{X}_{t}$ for one-parameter deformation families $\mathcal{X}\to\mathbb{C}$ as in Examples 5.1 and 5.2.

By combining with the case of direct products, we obtain the following:

Theorem 5.3**.**

Let $X$ be a smooth toric special weak Fano $4$-fold. If $X$ is not isomorphic to $\mathcal{Z}_{18}$, $\mathcal{Z}_{21}$, $\mathcal{Z}_{23}$, $\mathcal{Z}_{25}$ or $\mathcal{Z}_{26}$, then $X$ is deformation equivalent to a toric Fano manifold. Moreover, if $X$ is not isomorphic to $\mathcal{Z}_{12}$ or $\mathcal{Z}_{13}$, then $X$ is a smooth toric weakened Fano $4$-fold.

Remark 5.4**.**

$\mathcal{Z}_{12}$ and $\mathcal{Z}_{13}$ become toric Fano manifolds after two deformations

[TABLE]

respectively. However, we do not know if they deform to Fano manifolds directly.