Towards the classification of Fano 4-folds with $b_2\geq 7$

TL;DR

This paper classifies Fano 4-folds with high Picard number, showing they are either products of surfaces or related to cubic 4-folds, and explores their birational geometry and contractions.

Contribution

It improves classification results for Fano 4-folds with Picard number greater than 6, providing explicit descriptions and new geometric insights.

Findings

For Picard number >9, Fano 4-folds are products of del Pezzo surfaces.

In the range 7 ≤ ρ(X) ≤ 9, non-product cases relate to blow-ups of cubic 4-folds.

Describes the structure of Fano 4-folds with small elementary contractions.

Abstract

We study (smooth, complex) Fano 4-folds X with Picard number rho(X)>6. We show that if rho(X)>9, then X is a product of del Pezzo surfaces, thus improving recent results by the author and by the author and S.A. Secci; the statement is now optimal. In the range rho(X)=7,8,9 we show that if X is not a product of surfaces, and has no small elementary contraction, then it is the blow-up of a cubic 4-fold along a special configuration of planes. When instead rho(X)>6 and X has a small elementary contraction, we study X depending on its fixed prime divisors, giving explicit results on the geometry of X in the framework of birational geometry. In particular for the boundary case rho(X)=9 we show that either X is a product of surfaces, or X belongs to two explicit families, or there is a sequence of flips X-->X' where X' is a smooth projective 4-fold with an elementary contraction onto a 3-fold. In the paper we also give several results on rational contractions of fiber type of Fano 4-folds, and more generally of Mori dream spaces; in particular we use some properties of del Pezzo surfaces over non-closed fields, applied to generic fibers.

Equations

$${\delta }_X:=\max \{\mathrm{codim}{\mathcal{N}}_1(D,X)|D\text{ a prime divisor in }X\}.$$

$$\mathrm{Nef}(X)\subseteq \mathrm{Mov}(X)\subseteq \mathrm{Eff}(X)\subset {\mathcal{N}}^1(X)\text{and}\mathrm{mov}(X)\subseteq \mathrm{NE}(X)\subset {\mathcal{N}}_1(X),$$

$${\mathcal{N}}_1(Z,X):={\iota }_{\ast }({\mathcal{N}}_1(Z))\subseteq {\mathcal{N}}_1(X),$$

$$\dim\mathcal{N}_{1}(Z,X)\leq\rho_{X}-\rho_{Y}+\dim\mathcal{N}_{1}\bigl{(}f(Z),Y\bigr{)}.$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0⟶{\mathcal{O}}_X(-K_X)⟶{\mathcal{O}}_X(-K_X+E)⟶{\mathcal{O}}_X(-K_X+E)\otimes {\mathcal{O}}_E⟶0.$$

$$0⟶f_{\ast }{\mathcal{O}}_X(-K_X)⟶{\mathcal{O}}_Y(-K_Y)⟶{\mathcal{O}}_Y(-K_Y)\otimes {\mathcal{O}}_S⟶0,$$

$$d_f:={\rho }_X-\dim {\tau }_f.$$

$${\tau }_f=⟨[D]|\text{D is effective and SuppD does not dominate Y}⟩.$$

$$f^{-1}(Y_0)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\eta }_{g^{\prime \prime }}:={\mathcal{N}}_1(F_g,X{)}^{\perp }\cap \mathrm{Mov}(X)={\tau }_{g^{\prime \prime }}\cap \mathrm{Mov}(X)$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X\stackrel{{\alpha }_i}{\text{dasharrow}}{W}_i\stackrel{h_i}{\to }Y,$$

$$4\le {\rho }_S=1+r\text{ and }r\le \sum _{i=1}^r{d}_i=K_{S_0}^2-K_S^2\le K_{S_0}^2-1,$$

$$S_0⟵S_1⟶S_0^{\prime }$$

$$Z\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X\stackrel{\xi }{\text{dasharrow}}\tilde{X}\stackrel{\sigma }{⟶}Y$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathrm{Mov}(X{)}^{\vee }=⟨[C_D]{⟩}_{D\text{ fixed}}+\mathrm{mov}(X),$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\hat{X}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\dim {\mathcal{N}}_1(E,Y)\ge \dim {\mathcal{N}}_1(E_{\tilde{X}},\tilde{X})-1=\dim {\mathcal{N}}_1(E_X,X)-1\ge {\rho }_X-2={\rho }_Y-1,$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathcal{N}}^1(\tilde{W})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\tilde{X}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\tilde{X}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\tilde{X}={\tilde{X}}_1\stackrel{{\sigma }_1}{\text{dasharrow}}{\tilde{X}}_2\stackrel{{\sigma }_2}{\text{dasharrow}}\cdots \stackrel{{\sigma }_{r-1}}{\text{dasharrow}}{\tilde{X}}_r=\hat{X}.$$

$$\tilde{X}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

In the paper we also give several results on rational contractions of fiber type of Fano $4$-folds, and more generally of Mori dream spaces; in particular we use some properties of del Pezzo surfaces over non-closed fields, applied to generic fibers.

1. Introduction

Since the classification of Fano $3$-folds in the 80’s, there has been a lot of interest in the study of higher dimensional Fano varieties, starting from dimension $4$. With the introduction of the Lefschetz defect (see below), the author has developped a program to study (smooth, complex) Fano $4$-folds $X$ with large second Betti number $b_{2}(X)$ via birational geometry, in the framework of the Minimal Model Program. Let us recall that, since $X$ is Fano, we have $b_{2}(X)=\rho_{X}$ where $\rho_{X}$ is the Picard number of $X$.

In this paper we focus on the explicit study and classification of Fano $4$-folds $X$ with Picard number $\rho_{X}\geq 7$. Our first result is the following.

Theorem 1.1** (Cor. 10.5).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}>9$. Then $X\cong S_{1}\times S_{2}$, with $S_{i}$ del Pezzo surfaces.

Let us note that the statement above is proven in [Cas24] for $\rho>12$, and in [CS24, Cor. 1.4] for $\rho=12$, therefore the theorem is new for the cases $\rho=10$ and $\rho=11$. It is now optimal, as we know one family of Fano $4$-folds with $\rho=9$ that are not products of surfaces.

We recall that, so far, there are only 5 known families of Fano $4$-folds with $\rho\in\{7,8,9\}$ that are not products. Three such families, with $\rho=7,8,9$, are given by the Fano models of the blow-up of $\mathbb{P}^{4}$ at $6,7,8$ general points respectively (see §11.1), and two additional families with $\rho=7$ are constructed in [CS24, Prop. 1.9] (see §11.2). Some candidates are also given in [ibid., Questions 7.6 and 7.25]; in this range of Picard numbers we expect few families.

Besides Th. 1.1, in this paper we perform a detailed study of Fano $4$-folds $X\not\cong S_{1}\times S_{2}$ with $\rho_{X}\in\{7,8,9\}$; we obtain many explicit results on the geometry of $X$, including some classification results, and new candidates (see Questions 11.2, 11.5, 11.6). In particular we give a complete characterization of the case where $X$ does not have small elementary contractions, as follows.

Theorem 1.2** (Th. 8.1).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and without small elementary contractions.

Then $\rho_{X}\leq 9$ and there are a cubic $4$-fold $Z\subset\mathbb{P}^{5}$ with at most isolated ordinary double points, and $s=\rho_{X}-1$ distinct planes $A_{1},\dotsc,A_{s}\subset Z$ intersecting pairwise in a point, such that $X$ is obtained from $Z$ by blowing-up one plane $A_{i}$ and successively the transforms of the other ones.111The resulting $4$-fold does not depend on the order of the blow-ups, see Lemma 2.10.

We do not know whether a Fano $4$-fold as in the statement above does exist, see §11.7 and Question 11.6, so on one hand we get more candidate families of Fano $4$-fold as blow-ups of cubic $4$-folds, on the other hand it may very well be that this case does not happen. Let us note that Th. 1.2 builds on the study in [Cas24], where under the same assumptions, the bound $\rho_{X}\leq 12$ was proven. We refer the reader to 8.4 for an overview of the proof of Th. 1.2.

Consider now the remaining case of a Fano $4$-fold $X\not\cong S_{1}\times S_{2}$, with $\rho_{X}\geq 7$, and having small elementary contractions. Here we rely on the classification of fixed prime divisors222A prime divisor $D\subset X$ is fixed if $D=\operatorname{Bs}|mD|$ for all $m\in\mathbb{Z}_{>0}$. $D\subset X$ from [Cas13a] (see p. 2.3 and Section 5). Given such $D$, there are a sequence of flips $X\dasharrow X^{\prime}$ with $X^{\prime}$ smooth, and an elementary divisorial contraction $\sigma\colon X^{\prime}\to Y$, such that $D$ is the transform of $\operatorname{Exc}(\sigma)$, and $Y$ is Fano. Moreover we have only four possibilities for $\sigma$: it is the blow-up of a smooth point, or of a smooth irreducible curve, or of an isolated, locally factorial, terminal singularity with $\operatorname{Exc}(\sigma)$ a quadric, or finally of an irreducible surface. We call $D$ of type $(3,0)^{\text{\it sm}}$, $(3,1)^{\text{\it sm}}$, $(3,0)^{Q}$, or $(3,2)$ accordingly.

When $X$ has a small elementary contraction, it must contain some fixed prime divisor $D$ of type $(3,0)^{\text{\it sm}}$, $(3,1)^{\text{\it sm}}$, or $(3,0)^{Q}$ (Rem. 9.1). We study $X$ depending on the type of fixed prime divisors that it contains; in particular in the first case, when $D$ is of type $(3,0)^{\text{\it sm}}$, we show the following.

Theorem 1.3** (Th. 9.2).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, having a fixed prime divisor of type $(3,0)^{\text{\it sm}}$. Then one of the following holds:

1. $(i)$

$\rho_{X}\leq 9$* and $X$ has an elementary rational contraction onto a $3$-fold;333A rational contraction of $X$ is a rational map $f\colon X\dasharrow Y$ that factors as a sequence of flips $X\dasharrow X^{\prime}$ followed by a contraction $f^{\prime}\colon X^{\prime}\to Y$, namely a morphism such that $f^{\prime}_{*}\mathcal{O}_{X^{\prime}}=\mathcal{O}_{Y}$ with $Y$ projective, see §2; $f$ is elementary if $\rho_{X}-\rho_{Y}=1$.* 2. $(ii)$

$\rho_{X}=7$, $h^{0}(X,-K_{X})\leq 15$, and there is a sequence of flips $X\dasharrow X^{\prime}$ such that $X^{\prime}=\operatorname{Bl}_{6\mskip 1.0mu\text{\it pts}}Y$, $Y$ a smooth Fano $4$-fold with $\rho_{Y}=1$.

Th. 1.3 refines the study in [Cas17], where under the same assumptions, the bound $\rho_{X}\leq 12$ was proven. We refer the reader to 9.3 for an overview of the proof of Th. 1.3.

The remaining possibility is that of a Fano $4$-fold $X$ with $\rho_{X}\geq 7$, having a fixed prime divisor of type $(3,1)^{\text{\it sm}}$ or $(3,0)^{Q}$, and no fixed prime divisor of type $(3,0)^{\text{\it sm}}$; we study this case by refining the results in [Cas22]. This allows to complete the proof of Th. 1.1, and also to give a partial result on the case of Picard number $\rho=9$, as follows.

Theorem 1.4** (Cor. 10.5).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}=9$. Then one of the following holds:

1. $(i)$

$X\cong S_{1}\times S_{2}$, with $S_{i}$ del Pezzo surfaces; 2. $(ii)$

$X$* has an elementary rational contraction onto a $3$-fold;* 3. $(iii)$

$X$* is the blow-up of $W$ along a normal surface $S$, where $W$ is the Fano model of $\operatorname{Bl}_{7\mskip 1.0mu\text{\it pts}}\mathbb{P}^{4}$, and $S\subset W$ is the transform of a cubic scroll or a cone over a twisted cubic in $\mathbb{P}^{4}$, containing the blown-up points;* 4. $(iv)$

$X$* is a blow-up of a cubic $4$-fold as in Th. 1.2.*

We note that our only example of Fano $4$-fold $X\not\cong S_{1}\times S_{2}$ with $\rho_{X}=9$ has an elementary rational contraction onto a $3$-fold (see §11.1), while we do not know whether cases $(iii)$ and $(iv)$ do happen (see [CS24, §7.2] for $(iii)$, and §11.7 for $(iv)$).

Our general approach is to study Fano $4$-folds via their contractions and rational contractions, and there are a few tools and results that we use systematically and are crucial in proving the previous theorems; let us introduce them.

Rational contractions of fiber type. We recall that a rational contraction is a rational map $f\colon X\dasharrow Y$ that factors as a sequence of flips $X\dasharrow X^{\prime}$ followed by a contraction $f^{\prime}\colon X^{\prime}\to Y$, namely a morphism such that $f^{\prime}_{*}\mathcal{O}_{X^{\prime}}=\mathcal{O}_{Y}$, with $Y$ projective; $f$ is of fiber type when $\dim Y<\dim X$. We study in detail rational contractions of fiber type $f\colon X\dasharrow Y$ of Fano $4$-folds (more generally of Mori dream spaces), see also [Cas08, Cas20].

The case where $\dim Y=3$ is treated in [CS24], where a sharp bound on $\rho_{X}$ is given, together with a partial classification for the range $\rho_{X}\geq 7$, as follows.

Theorem 1.5** ([CS24], Theorems 1.3, 1.10, and 1.11).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, having a rational contraction onto a $3$-fold, and not isomorphic to a product of surfaces. Then $\rho_{X}\leq 9$, and one of the following holds:

1. $(i)$

$X$* has an elementary rational contraction onto a $3$-fold;* 2. $(ii)$

$X$* is the blow-up of $W$ along a normal surface $S$, where $W$ is the Fano model of $\operatorname{Bl}_{\rho_{\scriptscriptstyle X}-2\mskip 2.0mu\text{\it pts}}\mathbb{P}^{4}$, and $S\subset W$ is the transform of a surface $A\subset\mathbb{P}^{4}$ containing the blown-up points. The surface $A$ is either a cubic scroll, or a cone over a twisted cubic, or a sextic K3 surface with $A_{1}$ or $A_{2}$ singularities at the blown-up points; if $A$ is a K3 surface then $\rho_{X}=7$.*

We note that, in the setting of Th. 1.5, the case where we do not yet have a classification is when $X$ has an elementary rational contraction onto a $3$-fold.

Consider now a rational contraction of fiber type $f\colon X\dasharrow Y$ with $\dim Y\in\{1,2\}$. We can choose a factorization of $f$ as $X\stackrel{{\scriptstyle\varphi}}{{\dasharrow}}X^{\prime}\stackrel{{\scriptstyle f^{\prime}}}{{\to}}Y$ where $\varphi$ is a sequence of flips, $X^{\prime}$ is smooth, and $f^{\prime}$ is a $K$-negative contraction of fiber type. Let $F\subset X^{\prime}$ be a general fiber; then $F$ is smooth and is either a del Pezzo surface, or a Fano $3$-fold.

Let $\mathcal{N}_{1}(X^{\prime})$ be the vector space of one-cycles in $X^{\prime}$, with $\mathbb{R}$-coefficients, modulo numerical equivalence, and let us consider $\mathcal{N}_{1}(F,X^{\prime}):=\iota_{*}(\mathcal{N}_{1}(F))\subset\mathcal{N}_{1}(X^{\prime})$, where $\iota\colon F\hookrightarrow X^{\prime}$ is the inclusion. Then $d_{f}:=\dim\mathcal{N}_{1}(F,X^{\prime})$ is an invariant of $f$ and does not depend on the choice of the resolution $f^{\prime}$ (Def.-Rem. 3.2). Moreover $d_{f}$ is equal to the Picard number of the generic fiber $X_{\eta}$ of $f^{\prime}$, which is a smooth Fano variety over the non-closed field $K=\mathbb{C}(Y)$ of rational functions on $Y$ (Lemma 3.5).

When $d_{f}$ is large enough, we show that $f$ factors through a rational contraction onto a $3$-fold, as follows.

Theorem 1.6** (Th. 4.1).**

Let $X$ be a smooth Fano $4$-fold and $f\colon X\dasharrow Y$ a non-trivial 444Namely $Y\neq\{\text{\it pt}\}$. rational contraction of fiber type. If $d_{f}\geq 5$, then $f$ can be factored as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}Z\stackrel{{\scriptstyle h}}{{\to}}Y$, where $\dim Z=3$.

To prove this, we use different strategies depending on the dimension of the base $Y$. When $\dim Y=2$, we consider the generic fiber $X_{\eta}$ of $f^{\prime}\colon X^{\prime}\to Y$ (notation as above), which is a smooth del Pezzo surface over the field $K=\mathbb{C}(Y)$. When $d_{f}=\rho_{X_{\eta}}\geq 5$, there is a morphism $X_{\eta}\to C$ onto a curve over $K$ (Lemma 4.2), and we use results on spreading out to get the statement (see Lemmas 3.6 and 3.14, and Section 4).

Instead, when $Y\cong\mathbb{P}^{1}$, a general fiber $F\subset X^{\prime}$ of $f^{\prime}$ is a smooth, complex Fano $3$-fold with $\rho_{F}\geq d_{f}=\dim\mathcal{N}_{1}(F,X^{\prime})$. When $d_{f}\geq 5$, we use the monodromy action on $\mathcal{N}^{1}(F)$ and $\operatorname{Nef}(F)$ to extend a contraction $F\to S$ with $\dim S=2$ to a relative contraction over some open subset of $Y=\mathbb{P}^{1}$, see Prop. 4.4.

Using the results from [CS24] on rational contractions onto $3$-folds, we deduce from Th. 1.6 the following result.

Theorem 1.7** (Th. 4.5).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and $f\colon X\dasharrow Y$ a non-trivial rational contraction of fiber type. Then $d_{f}\leq 4$.

The bound $d_{f}\leq 4$ is sharp, see the discussion after Th. 4.5.

Theorem 1.7 is very much related to how the faces of the cone of effective divisors can intersect the cone of movable divisors. More precisely, let us consider $\mathcal{N}^{1}(X)$, the vector space of divisors with $\mathbb{R}$-coefficients up to numerical equivalence, and in it the cones $\operatorname{Eff}(X)$ and $\operatorname{Mov}(X)$ respectively of effective and movable divisors (see Section 2); we have $\operatorname{Mov}(X)\subset\operatorname{Eff}(X)$, and since $X$ is Fano, both cones are rational polyhedral. If $\tau$ is a face of $\operatorname{Eff}(X)$, we say that $\tau$ is a movable face if $\tau\cap\operatorname{Mov}(X)\neq\{0\}$, otherwise we say that $\tau$ is a fixed face (Def. 2.4).

Given a non-trivial rational contraction of fiber type $f\colon X\dasharrow Y$, we have that $\tau_{f}:=\langle[D]\in\mathcal{N}^{1}(X)\,|D$ is effective and $\operatorname{Supp}D$ does not dominate $Y\rangle$ is a movable face of $\operatorname{Eff}(X)$, of dimension $\rho_{X}-d_{f}$ (Def.-Rem. 3.2, Rem. 3.4). Conversely, given a proper, movable face $\tau$ of $\operatorname{Eff}(X)$, there exists a non-trivial rational contraction of fiber type $f\colon X\dasharrow Y$ with $d_{f}\geq\rho_{X}-\dim\tau$ (see Rem. 3.11). This gives the following.

Corollary 1.8** (Cor. 4.10).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces. If $\tau$ is a movable face of $\operatorname{Eff}(X)$, then $\dim\tau\geq\rho_{X}-4$. In particular the cone $\operatorname{Eff}(X)$ is generated by classes of fixed prime divisors.

The Lefschetz defect. Another essential tool is the Lefschetz defect, an invariant of $X$ defined as follows. For any prime divisor $\iota\colon D\hookrightarrow X$, set $\mathcal{N}_{1}(D,X):=\iota_{*}(\mathcal{N}_{1}(D))\subset\mathcal{N}_{1}(X)$. Then we define the Lefschetz defect of $X$ as

[TABLE]

We refer the reader to [Cas12, CRS22] for results on the Lefschetz defect of Fano varieties of arbitrary dimension; in dimension $4$ we have the following.

Theorem 1.9** ([Cas12], Cor. 1.3, [CS24], Th. 1.8).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces. Then $\delta_{X}\leq 1$.

Fixed prime divisors of type $(3,2)$. A fixed prime divisor of type $(3,2)$ is the exceptional divisor $E$ of an elementary divisorial contraction $\sigma\colon X\to Y$ such that $\dim\sigma(E)=2$ (see Th.-Def. 5.1). If $X\not\cong S_{1}\times S_{2}$ and $\rho_{X}\geq 7$, we have $\delta_{X}\leq 1$ by Th. 1.9, therefore $\mathcal{N}_{1}(E,X)$ has codimension at most one in $\mathcal{N}_{1}(X)$. We show the following.

Theorem 1.10** (Th. 7.1).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and $E\subset X$ a fixed prime divisor of type $(3,2)$ such that $\mathcal{N}_{1}(E,X)\subsetneq\mathcal{N}_{1}(X)$. Then $\rho_{X}\leq 9$ and $X$ has an elementary rational contraction onto a $3$-fold.

This improves the bound $\rho_{X}\leq 12$ shown in [Cas17, Prop. 5.32]. Besides the bound $\rho_{X}\leq 9$, in Th. 7.1 we give several geometric properties of $X$, that could lead to an explicit classification of this case.

We refer the reader to 7.3 for an overview of the proof of Th. 1.10. In particular, we need a generalization of Theorems 1.6 and 1.7 in the case of rational contractions $f\colon X\dasharrow S$ with $\dim S=2$ that are quasi-elementary, namely with $d_{f}=\rho_{X}-\rho_{S}$ (see Def. 3.8), as follows.

Theorem 1.11** (Th. 6.1).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and $f\colon X\dasharrow S$ a quasi-elementary rational contraction onto a surface. Then $\rho_{X}-\rho_{S}\leq 3$, and if $\rho_{X}-\rho_{S}>1$, then $f$ factors as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}Y\stackrel{{\scriptstyle h}}{{\dasharrow}}S$ where $\dim Y=3$.

The bound $\rho_{X}-\rho_{S}\leq 3$ follows easily from the previous results, while the proof that, in the non-elementary case, $f$ factors through a $3$-fold, is rather long (see Section 6). We work by contradiction: by assuming that the statement does not hold, we get an explicit description of $X$ as a sequence of flips and blow-ups from $\mathbb{P}^{2}\times\mathbb{P}^{2}$; finally we show that this construction never yields a Fano $4$-fold. See 6.5 for a more detailed overview of the proof.

Let us conclude this Introduction by noting that the condition $\rho\geq 7$ arises naturally and is optimal for many of these results, like Th. 1.2, Th. 1.5, Th. 1.7, Cor. 1.8, Th. 1.9, and Th. 1.11; on the other hand Fano $4$-folds with $\rho\leq 6$ exhibit different behaviours, as we show in Examples 4.6, 4.7, and 4.11, and in §11.4.

The structure of the paper is as follows. In Section 2 we fix the notation and terminology, and we recall some preliminary results needed in the sequel.

Section 3 contains several preliminary notions and results on regular and rational contractions of fiber type $f\colon X\dasharrow Y$ of Mori dream spaces. In particular we define quasi-elementary and special rational contractions of fiber type and recall their main properties, and we introduce the invariant $d_{f}$ mentioned above.

In Section 4 we apply the previous results to Fano $4$-folds, and we prove Theorems 1.6 and 1.7 on rational contractions of fiber type, and Cor. 1.8 on movable faces of the cone of effective divisors.

In Section 5 we recall the classification of fixed prime divisors in Fano $4$-folds $X$ with $\rho_{X}\geq 7$ in four types, and give many related properties that are used in the sequel.

In Section 6 we prove Th. 1.11 on quasi-elementary rational contractions onto a surface. In Section 7 we prove Th. 1.10 on fixed prime divisors of type $(3,2)$, and give some applications needed in the sequel.

In Section 8 we prove Th. 1.2, that characterizes Fano $4$-folds with no small contractions (with $\rho\geq 7$, and not products) as suitable blow-ups of cubic $4$-folds.

In Section 9 we prove Th. 1.3 on Fano $4$-folds with a fixed prime divisor of type $(3,0)^{\text{\it sm}}$. Finally in Section 10 we consider Fano $4$-folds with a fixed prime divisor of type $(3,1)^{\text{\it sm}}$ or $(3,0)^{Q}$, and prove Theorems 1.1 and 1.4; we also give a partial result on the case $\rho=8$ (Cor. 10.6).

In Section 11 we collect several examples. We first recall the known examples of Fano $4$-folds that are not products, with $\rho\geq 7$ (§11.1 and §11.2), and then give other relevant explicit examples; in some cases we do not know whether these $4$-folds are actually Fano for large values of the Picard number.

Acknowledgements. I had the idea of using the properties of del Pezzo surfaces over non-closed fields at the minicourse “Quotients of groups of birational transformations” by Susanna Zimmermann, at the V Latin American School of Algebraic Geometry (V ELGA) in Brazil, August 2024; I thank her and the organisers of the school.

I am also grateful to Brendan Hassett for explaining me that there exist families of cubic $4$-folds containing a configuration of planes as in Th. 1.2.

The author has been partially supported by PRIN 2022L34E7W “Moduli spaces and birational geometry”, and is a member of GNSAGA, INdAM.

Contents

1. 1 Introduction
2. 2 Notation and preliminaries
3. 3 Preliminaries on rational contractions of fiber type
4. 4 Factoring rational contractions of fiber type of Fano $4$-folds through $3$-folds
5. 5 Preliminaries on fixed prime divisors of Fano $4$-folds
6. 6 Quasi-elementary rational contractions onto surfaces
7. 7 Fixed divisors of type $(3,2)$
8. 8 Fano $4$-folds with no small contractions and blow-ups of cubic $4$-folds
9. 9 Fano $4$-folds with a divisor of type $(3,0)^{\text{\it sm}}$
10. 10 Fano $4$-folds with a divisor of type $(3,1)^{\text{\it sm}}$ or $(3,0)^{Q}$
11. 11 Examples

2. Notation and preliminaries

If $\mathcal{N}$ is a finite-dimensional real vector space and $a_{1},\dotsc,a_{r}\in\mathcal{N}$, $\langle a_{1},\dotsc,a_{r}\rangle$ denotes the convex cone in $\mathcal{N}$ generated by $a_{1},\dotsc,a_{r}$. Moreover, for every $a\neq 0$, $a^{\perp}$ is the hyperplane orthogonal to $a$ in the dual vector space $\mathcal{N}^{*}$. If $\alpha\in\mathcal{N}^{*}$ and $\tau:=\langle a\rangle$, we write $\alpha\cdot\tau>0$ (respectively $\alpha\cdot\tau<0$, $\alpha\cdot\tau=0$) if $\alpha\cdot a>0$ (respectively $\alpha\cdot a<0$, $\alpha\cdot a=0$). A facet of a cone is a face of codimension one. If $\mathcal{C}\subset\mathcal{N}$ is a convex polyhedral cone, we denote by $\mathcal{C}^{\vee}\subset\mathcal{N}^{*}$ its dual cone.

Lemma 2.1**.**

Let $\sigma$ be a convex polyhedral cone in a finite dimensional real vector space $\mathcal{N}$. Let $\tau$ be a one-dimensional face of $\sigma$, and let $\alpha\in\mathcal{N}^{*}$ be such that $\alpha\cdot\tau<0$ and $\alpha\cdot\tau^{\prime}\geq 0$ for every one-dimensional face $\tau^{\prime}\neq\tau$ of $\sigma$. If $\eta$ is a face of $\sigma$ such that $\eta\subset\ker\alpha$, then $\tau+\eta$ is a face of $\sigma$.

Proof.

This is proved in [Cas20, Lemma 4.7] when $\dim\eta=1$; the same proof applies to the general case. ∎

The index of a locally factorial Fano variety $X$ is the divisibility of $-K_{X}$ in $\operatorname{Pic}(X)$.

For the standard terminology and properties in birational geometry and about Mori dream spaces, we refer the reader to [KM98, HK00], and we recall that smooth Fano varieties are Mori dream spaces by [BCHM10, Cor. 1.3.2].

A contraction is a projective, surjective morphism with connected fibers, between normal quasi-projective varieties. It can be birational or of fiber type. A contraction $f\colon X\to Y$ is $K$-negative if, for some $m\in\mathbb{N}$, $-mK_{X}$ is Cartier and $f$-ample.

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space. As usual we denote by $\mathcal{N}_{1}(X)$ the real vector space of numerical equivalence classes of one-cycles in $X$ with real coefficients, and by $\mathcal{N}^{1}(X)$ the dual vector space of numerical equivalence classes of $\mathbb{R}$-divisors in $X$. If $D$ is a divisor and $C$ a curve in $X$, we denote by $[D]\in\mathcal{N}^{1}(X)$ and $[C]\in\mathcal{N}_{1}(X)$ their numerical equivalence classes, and we set $D^{\perp}:=[D]^{\perp}\subset\mathcal{N}_{1}(X)$ and $C^{\perp}:=[C]^{\perp}\subset\mathcal{N}^{1}(X)$.

A movable divisor is an effective divisor $D$ such that the stable base locus of the linear system $|D|$ has codimension $\geq 2$. We will consider the usual cones of divisors and of curves:

[TABLE]

where all the notations are standard except maybe $\operatorname{mov}(X)$, which is the convex cone generated by classes of curves moving in a family covering $X$. Since $X$ is a Mori dream space, all these cones are rational polyhedral. An extremal ray $R$ of $\operatorname{NE}(X)$ is a one-dimensional face of this cone. Given a divisor $D$ on $X$, we say that $R$ is $D$-negative (respectively $D$-positive, $D$-trivial) if $D\cdot R<0$ (respectively $D\cdot R>0$, $D\cdot R=0$). We also write $\operatorname{Locus}(R)$ for the union in $X$ of all curves with class in $R$.

Given a contraction $f\colon X\to Y$, we set $\operatorname{NE}(f):=(\ker f_{*})\cap\operatorname{NE}(X)$, a face of $\operatorname{NE}(X)$ of dimension $\rho_{X}-\rho_{Y}$. The contraction $f$ is elementary if $\rho_{X}-\rho_{Y}=1$. An elementary contraction can be divisorial, small, or of fiber type.

An elementary contraction $f$ is of type $(a,b)$ if $\dim\operatorname{Exc}(f)=a$ and $\dim f(\operatorname{Exc}(f))=b$. If $n=\dim X$, we say that $f$ is of type $(n-1,b)^{\text{\it sm}}$ if $f$ is the blow-up of a smooth, $b$-dimensional irreducible subvariety $A\subset Y_{\text{\it reg}}$. Finally when $n=4$ we say that $f$ is of type $(3,0)^{Q}$ if $\operatorname{Exc}(f)$ is an irreducible $3$-dimensional quadric with $\mathcal{N}_{\operatorname{Exc}(f)/X}\cong\mathcal{O}_{Q}(-1)$.

A flip is the flip of a small elementary contraction. Given a divisor $D$ on $X$, a $D$-negative (respectively $D$-positive, $D$-trivial) flip is the flip of a small elementary contraction $f\colon X\to Y$ such that $D\cdot\operatorname{NE}(f)<0$ (respectively $D\cdot\operatorname{NE}(f)>0$, $D\cdot\operatorname{NE}(f)=0$). We do not assume that contractions or flips are $K$-negative, unless specified.

A SQM (small $\mathbb{Q}$-factorial modification) is a birational map $\varphi\colon X\dasharrow X^{\prime}$ such that $X^{\prime}$ is again projective, normal, and $\mathbb{Q}$-factorial, and $\varphi$ is an isomophism in codimension one; since $X$ is a Mori dream space, $\varphi$ always factors as a finite sequence of flips.

A rational contraction is a rational map $f\colon X\dasharrow Y$ that factors as $X\stackrel{{\scriptstyle\text{SQM}}}{{\dasharrow}}X^{\prime}\stackrel{{\scriptstyle f^{\prime}}}{{\to}}Y$, where $f^{\prime}$ is a contraction; $f$ can be birational or of fiber type, and we say that $f$ is non-trivial if $Y\neq\{\text{\it pt}\}$. We call $f^{\prime}$ a resolution of $f$. We say that $f$ is elementary if $\rho_{X}-\rho_{Y}=1$; if $f$ is elementary and birational, then it can be divisorial or small, depending whether $\operatorname{Exc}(f^{\prime})$ is a prime divisor or $\operatorname{codim}\operatorname{Exc}(f^{\prime})>1$.

There is a fan in $\mathcal{N}^{1}(X)$, called the Mori chamber decomposition and denoted by $\operatorname{MCD}(X)$, supported on $\operatorname{Mov}(X)$, whose cones are in bijection with the set of rational contractions of $X$ (up to isomorphism of the target); the cone corresponding to $f\colon X\dasharrow Y$ is $f^{*}(\operatorname{Nef}(Y))$.

Remark 2.2**.**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space, and $f\colon X\dasharrow Y$, $g\colon X\dasharrow Z$ two rational contractions. Then there exists a contraction $h\colon Z\to Y$ such that $f=h\circ g$ if and only if $f^{*}(\operatorname{Nef}(Y))\subset g^{*}(\operatorname{Nef}(Z))$.

A fixed prime divisor is a prime divisor $D$ which is the stable base locus of the linear system $|D|$; then $[D]\in\mathcal{N}^{1}(X)$ generates a one-dimensional face of $\operatorname{Eff}(X)$.

Remark 2.3** ([Cas13a], Rem. 2.19).**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space. The fixed prime divisors in $X$ are the exceptional divisors of divisorial elementary rational contractions of $X$.

Definition 2.4**.**

Let $\tau$ be a face of $\operatorname{Eff}(X)$. We say that $\tau$ is a fixed face if $\tau\cap\operatorname{Mov}(X)=\{0\}$, otherwise we say that $\tau$ is a movable face.

Definition 2.5**.**

We say that two fixed prime divisors $D,E\subset X$ are adjacent if $\langle[D],[E]\rangle$ is a fixed face of $\operatorname{Eff}(X)$.

Let $Z\subset X$ be a closed subset. We set

[TABLE]

where $\iota\colon Z\hookrightarrow X$ is the inclusion.

Lemma 2.6**.**

Let $f\colon X\to Y$ be a contraction and $Z\subset X$ a closed subset. Then:

[TABLE]

Proof.

The pushforward $f_{*}\colon\mathcal{N}_{1}(X)\to\mathcal{N}_{1}(Y)$ is a surjective linear map with kernel of dimension $\rho_{X}-\rho_{Y}$, and $f_{*}(\mathcal{N}_{1}(Z,X))=\mathcal{N}_{1}(f(Z),Y)$. ∎

Remark 2.7**.**

Let $Z\subset X$ a closed subset and $D\subset X$ a prime divisor such that $Z\cap D=\emptyset$. Then $\mathcal{N}_{1}(Z,X)\subset D^{\perp}\subsetneq\mathcal{N}_{1}(X)$.

Indeed for every curve $C\subset Z$ we have $D\cdot C=0$, hence $[C]\in D^{\perp}$.

An ordinary double point is an isolated singularity $x_{0}$ of an $n$-dimensional variety $X$ that is locally analytically isomorphic to the vertex of the cone over a smooth $(n-1)$-dimensional quadric.

Remark 2.8**.**

Let $X$ be a quasi-projective $4$-fold with an ordinary double point at $x_{0}$ and $\operatorname{Sing}(X)=\{x_{0}\}$. Let $f\colon X^{\prime}\to X$ be the blow-up of $X$ at $x_{0}$. Then $X^{\prime}$ is smooth, $\operatorname{Exc}(f)$ is a smooth quadric, and $f$ is a divisorial, $K$-negative elementary contraction of type $(3,0)^{Q}$; in particular $x_{0}$ is a locally factorial, terminal singularity.

Let $A\subset X$ be a smooth irreducible surface containing $x_{0}$, and let $g\colon X^{\prime\prime}\to X$ be the blow-up of $A$. Then $X^{\prime\prime}$ is smooth and $g$ is a divisorial, $K$-negative elementary contraction of type $(3,2)$; we have $g^{-1}(x_{0})\cong\mathbb{P}^{2}$ while the other non-trivial fibers of $g$ are $\mathbb{P}^{1}$’s.

By a family of curves in a projective variety $X$ we mean an irreducible closed subset $V\subset\operatorname{Chow}(X)$ such that for general $v\in V$, the corresponding cycle $C_{v}$ of $X$ is an integral curve; then for every $v\in V$ the cycle $C_{v}$ is a connected curve in $X$. The numerical equivalence class $[C_{v}]\in\mathcal{N}_{1}(X)$ does not depend on $v\in V$, and we denote it by $[V]$. The anticanonical degree of the family is $-K_{X}\cdot[V]$. We say that $V$ is a family of rational curves if for general $v\in V$ the curve $C_{v}$ is rational. We denote by $\operatorname{Locus}(V)$ the union of the curves $C_{v}$ for $v\in V$, and we say that $V$ is covering if $\operatorname{Locus}(V)=X$.

We work over the field of complex numbers, but we will occasionally also consider varieties defined over non-closed fields of characteristic zero, as follows. Let $X$ be a complex projective variety and $f\colon X\to Y$ a contraction of fiber type. The generic fiber $X_{\eta}$ of $f$ is the fiber over the generic point $\eta$ of $Y$, and it is a projective variety over the field $K=\mathbb{C}(\eta)=\mathbb{C}(Y)$ of rational functions on $Y$. We will be interested in the case where $X_{\eta}$ is a smooth del Pezzo surface; we refer the reader to [Has09] for properties of del Pezzo surfaces over non-closed fields.

Preliminaries on Fano $4$-folds. We finally recall some preliminary results on contractions and rational contractions of $4$-folds and Fano $4$-folds.

Lemma 2.9** ([AW98a], Th. on p. 256; [AW98b], Prop. 2.1).**

Let $X$ be a smooth projective $4$-fold and $f\colon X\to Y$ a $K$-negative elementary contraction of type $(3,2)$. Then $Y$ has at most isolated ordinary double points, in particular $Y$ has locally factorial, terminal singularities, and $S:=f(\operatorname{Exc}(f))$ has isolated singularities. If $y_{0}\in S$ is a singular point for $Y$ or for $S$, then $\dim f^{-1}(y_{0})=2$. Moreover $f$ is the blow-up of $Y$ along $S$.555Namely the blow-up of $Y$ along the ideal sheaf $\mathcal{I}_{S/Y}$; $S$ is a reduced closed subscheme of $Y$ that may be singular.

Lemma 2.10**.**

Let $Y$ be a smooth, irreducible, quasi-projective $4$-fold, and $S_{1},S_{2}\subset Y$ two smooth, irreducible surfaces intersecting transversally in one point $y_{0}\in Y$. Then there is a commutative diagram:

[TABLE]

where $\sigma_{i}\colon X_{i}\to Y$ is the blow-up of $S_{i}$, and $\tau_{i}\colon X\to X_{j}$ is the blow-up of the transform $\widetilde{S}_{i}\subset X_{j}$ of $S_{i}$, where $\{i,j\}=\{1,2\}$. The composition $f\colon X\to Y$ is a $K$-negative birational contraction with $f(\operatorname{Exc}(f))=S_{1}\cup S_{2}$, $f^{-1}(y)\cong\mathbb{P}^{1}$ for every $y\in(S_{1}\cup S_{2})\smallsetminus\{y_{0}\}$, and $f^{-1}(y_{0})\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$.

Proof.

Let us consider the blow-up $\sigma_{1}\colon X_{1}\to Y$ of $S_{1}$. The surface $S_{2}$ is blown-up at $y_{0}$, and its transform $\widetilde{S}_{2}\subset X_{1}$ intersects the exceptional divisor $\operatorname{Exc}(\sigma_{1})$ along a smooth irreducible curve $\Gamma$, which is the exceptional curve of the blow-up $\sigma_{1|\widetilde{S}_{2}}\colon\widetilde{S}_{2}\to S_{2}$. We also have $\mathcal{N}_{\widetilde{S}_{2}/X_{1}}=(\sigma_{1|\widetilde{S}_{2}})^{*}\mathcal{N}_{S_{2}/Y}$, therefore $(\mathcal{N}_{\widetilde{S}_{2}/X_{1}})_{|\Gamma}\cong\mathcal{O}_{\mathbb{P}^{1}}^{\oplus 2}$.

Now let $\tau_{2}\colon X\to X_{1}$ be the blow-up of $\widetilde{S}_{2}$, and $f:=\tau_{2}\circ\sigma_{1}\colon X\to Y$. We have $f^{-1}(y)\cong\mathbb{P}^{1}$ and $-K_{X}\cdot f^{-1}(y)=1$ for every $y\in(S_{1}\cup S_{2})\smallsetminus\{y_{0}\}$.

Moreover $f^{-1}(y_{0})=(\tau_{2})^{-1}(\Gamma)=\mathbb{P}_{\Gamma}(\mathcal{N}_{\widetilde{S}_{2}/X_{1}}^{\vee})_{|\Gamma}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$, and $\operatorname{Exc}(\tau_{2})_{|f^{-1}(y_{0})}\cong\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(-1,0)$. If $C\subset f^{-1}(y_{0})$ is a curve $\{\text{\it pt}\}\times\mathbb{P}^{1}$ with $\tau_{2}(C)=\Gamma$, then $\operatorname{Exc}(\tau_{2})\cdot C=0$ and hence $-K_{X}\cdot C=-K_{Y}\cdot\Gamma=1$, so that $-K_{X|f^{-1}(y_{0})}\cong\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,1)$. In particular we see that $f$ is $K$-negative, and $\operatorname{NE}(f)=\operatorname{NE}(\tau_{2})+\mathbb{R}_{\geq 0}[C]$. It is not difficult to see that $C$ is numerically equivalent to the transform of a general fiber of $\sigma_{1|\operatorname{Exc}(\sigma_{1})}$, and that the contraction of the extremal ray $\mathbb{R}_{\geq 0}[C]$ is $\tau_{1}\colon X\to X_{2}$ as in the statement. ∎

Lemma 2.11**.**

Let $X$ be a projective $4$-fold with locally factorial and terminal singularities, and $f\colon X\to Y$ a $K$-negative elementary contraction of type $(3,2)$; set $E:=\operatorname{Exc}(f)$ and $S:=f(E)\subset Y$. Then $\chi(X,-K_{X})=\chi(Y,-K_{Y})-\chi(S,-K_{Y|S})$.

Proof.

We note that $Y$ still has locally factorial and terminal singularities.

We have $R^{i}f_{*}\mathcal{O}_{X}(-K_{X})=0$ for every $i>0$ [KM98, Cor. 2.68], thus $\chi(X,-K_{X})=\chi(Y,f_{*}\mathcal{O}_{X}(-K_{X}))$. Consider the exact sequence on $X$:

[TABLE]

We have $-K_{X}+E=f^{*}(-K_{Y})$, thus using $f_{*}\mathcal{O}_{X}=\mathcal{O}_{Y}$, $R^{1}f_{*}\mathcal{O}_{X}(-K_{X})=0$, and the projection formula, via $f_{*}$ we get

[TABLE]

therefore $\chi(Y,f_{*}\mathcal{O}_{X}(-K_{X}))=\chi(Y,-K_{Y})-\chi(S,-K_{Y|S})$. ∎

Theorem 2.12** ([Kaw00], Th. 5.2; [Flo13]; Th. 1.2).**

Let $X$ be a smooth Fano $4$-fold. Then $h^{0}(X,-K_{X})\geq 2$.

Let $X$ be a normal projective $4$-fold. An exceptional plane is a surface $L\subset X_{\text{\it reg}}$ such that $L\cong\mathbb{P}^{2}$ and $\mathcal{N}_{L/X}\cong\mathcal{O}_{\mathbb{P}^{2}}(-1)^{\oplus 2}$; we denote by $C_{L}\subset L$ a curve corresponding to a line in $\mathbb{P}^{2}$. An exceptional line is a smooth rational curve $\ell\subset X_{\text{\it reg}}$ such that $\mathcal{N}_{\ell/X}\cong\mathcal{O}_{\mathbb{P}^{1}}(-1)^{\oplus 3}$. Note that $K_{X}\cdot C_{L}=-1$ while $K_{X}\cdot\ell=1$.

Lemma 2.13** ([Kaw89], Th. 1.1).**

Let $X$ be a smooth projective $4$-fold and $f\colon X\to Y$ a small elementary $K$-negative contraction. Then $\operatorname{Exc}(f)$ is a finite disjoint union of exceptional planes.

Lemma 2.14** ([Cas13a], Rem. 3.6).**

Let $X$ be a smooth Fano $4$-fold and $\varphi\colon X\dasharrow X^{\prime}$ a SQM. We have the following:

1. $(a)$

$X^{\prime}$* is smooth, $X\smallsetminus\operatorname{dom}(\varphi)=L_{1}\cup\cdots\cup L_{r}$ where $L_{i}$ are pairwise disjoint exceptional planes, and $X^{\prime}\smallsetminus\operatorname{dom}(\varphi^{-1})=\ell_{1}\cup\cdots\cup\ell_{r}$ where $\ell_{i}$ are pairwise disjoint exceptional lines. * 2. $(b)$

Let $C\subset X^{\prime}$ be an irreducible curve, different from $\ell_{1},\dotsc,\ell_{r}$, and intersecting $\ell_{1}\cup\cdots\cup\ell_{r}$ in $s\geq 0$ points; then $-K_{X^{\prime}}\cdot C\geq 1+s$. 3. $(c)$

If $-K_{X^{\prime}}\cdot C=1$, then $C\cap(\ell_{1}\cup\cdots\cup\ell_{r})=\emptyset$.

Lemma 2.15** ([Cas13a], Rem. 3.7).**

Let $X$ be a smooth Fano $4$-fold and $f\colon X\dasharrow Y$ a rational contraction. Then there exists a resolution $f^{\prime}\colon X^{\prime}\to Y$ of $f$ such that $f^{\prime}$ is $K$-negative.

Lemma 2.16**.**

Let $X$ be a smooth Fano $4$-fold and $g\colon X\to Z$ a contraction of fiber type. Then $\rho_{Z}\leq\delta_{X}+2$.

Proof.

The proof of [Cas24, Lemma 2.6] gives the statement. ∎

3. Preliminaries on rational contractions of fiber type

In this section we give many preliminary results on regular and rational contractions of fiber type of Mori dream spaces, that will be used throughout the paper.

Lemma 3.1** ([Cas13a], Lemma 2.21).**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space, $f\colon X\to Y$ a contraction of fiber type, and set $F_{y}:=f^{-1}(y)$.

Then there is a non-empty open subset $Y_{0}\subset Y$ such that the linear subspace $\mathcal{N}_{1}(F_{y},X)\subset\mathcal{N}_{1}(X)$ is constant for $y\in Y_{0}$. Moreover, for $y\in Y_{0}$, $\mathcal{N}_{1}(F_{y},X)^{\perp}\cap\operatorname{Eff}(X)$ is a face of $\operatorname{Eff}(X)$, has dimension $\dim(\mathcal{N}_{1}(F_{y},X)^{\perp})=\rho_{X}-\dim\mathcal{N}_{1}(F_{y},X)$, and is the smallest face of $\operatorname{Eff}(X)$ containing $f^{*}(\operatorname{Eff}(Y))$.

Definition - Remark 3.2** ($\bm{\tau_{f},d_{f}}$).**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space, and $f\colon X\dasharrow Y$ a rational contraction of fiber type. We denote by $\tau_{f}$ the minimal face of $\operatorname{Eff}(X)$ containing $f^{*}(\operatorname{Eff}(Y))$,666Equivalently, $\tau_{f}$ is the minimal face of $\operatorname{Eff}(X)$ containing $f^{*}(\operatorname{Nef}(Y))$. and we set:

[TABLE]

We will use the following properties:

1. $\bullet$

for every resolution $f^{\prime}\colon X^{\prime}\to Y$ of $f$, if $F\subset X^{\prime}$ is a general fiber of $f^{\prime}$, we have $\dim\mathcal{N}_{1}(F,X^{\prime})=d_{f}$ (see Lemma 3.1); 2. $\bullet$

$d_{f}\leq\rho_{X}-\rho_{Y}$, in particular $d_{f}=1$ when $f$ is elementary; 3. $\bullet$

$d_{f}=1$ when $\dim X-\dim Y=1$; 4. $\bullet$

$\tau_{f}\cap\operatorname{Mov}(X)$ is a face of $\operatorname{Mov}(X)$ (because $\operatorname{Mov}(X)\subset\operatorname{Eff}(X)$) and contains $f^{*}(\operatorname{Nef}(Y))$, therefore $\dim(\tau_{f}\cap\operatorname{Mov}(X))\geq\rho_{Y}$, and $\tau_{f}$ is a movable face of $\operatorname{Eff}(X)$ if $Y\neq\text{\it pt}$; 5. $\bullet$

whenever $f$ is regular, or more generally regular and proper on some non-empty open subset of $X$, if $F\subset X$ is a general fiber, then $\tau_{f}=\mathcal{N}_{1}(F,X)^{\perp}\cap\operatorname{Eff}(X)$. Indeed if $\varphi\colon X\dasharrow X^{\prime}$ is a SQM such that $f\circ\varphi^{-1}$ is regular, then $F$ must be contained in the open subset where $\varphi$ is an isomorphism, hence $\mathcal{N}_{1}(F,X)\cong\mathcal{N}_{1}(\varphi(F),X^{\prime})$ under the natural isomorphism $\mathcal{N}_{1}(X)\cong\mathcal{N}_{1}(X^{\prime})$ (see [Cas20, Lemma 2.17]).

Lemma 3.3**.**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space, $f\colon X\to Y$ a contraction of fiber type, and $F\subset X$ a general fiber. Let $W\subset\mathcal{N}^{1}(X)$ be the linear subspace of classes of $\mathbb{R}$-divisors whose support does not dominate $Y$. Then $W=\mathcal{N}_{1}(F,X)^{\perp}$.

Proof.

Let $Y_{0}\subset Y$ be an open subset as in Lemma 3.1, so that we can assume that $\mathcal{N}_{1}(F,X)=\mathcal{N}_{1}(F_{y},X)$ for every $y\in Y_{0}$.

Let $D$ be a divisor whose support does not dominate $Y$. Then there is $y\in Y_{0}$ such that $(\operatorname{Supp}D)\cap F_{y}=\emptyset$, hence $D\cdot C=0$ for every curve $C\subset F_{y}$, and $[D]\in\mathcal{N}_{1}(F_{y},X)^{\perp}$. This shows that $W\subseteq\mathcal{N}_{1}(F,X)^{\perp}$.

By Lemma 3.1 $\mathcal{N}_{1}(F,X)^{\perp}\cap\operatorname{Eff}(X)$ is a face of $\operatorname{Eff}(X)$ of dimension equal to $\dim(\mathcal{N}_{1}(F,X)^{\perp})$, therefore $\mathcal{N}_{1}(F,X)^{\perp}$ is generated by the one-dimensional faces of $\operatorname{Eff}(X)$ contained in it. On the other hand every one-dimensional face of $\operatorname{Eff}(X)$ is generated by the class of some prime divisor, and we conclude that $\mathcal{N}_{1}(F,X)^{\perp}$ can be generated by classes of prime divisors.

Now let $D\subset X$ be a prime divisor such that $[D]\in\mathcal{N}_{1}(F,X)^{\perp}$. Then $D$ cannot dominate $Y$, otherwise we could choose $y\in Y_{0}$ such that $F_{y}\not\subset D$ and $F_{y}\cap D\neq\emptyset$. Thus we could find a curve $C\subset F_{y}$ such that $D\cdot C>0$, a contradiction. Therefore $[D]\in W$, and we conclude that $\mathcal{N}_{1}(F,X)^{\perp}\subseteq W$ and finally $\mathcal{N}_{1}(F,X)^{\perp}=W$. ∎

Remark 3.4**.**

In the setting of Def.-Rem. 3.2, it follows from Lemmas 3.1 and 3.3 that

[TABLE]

Lemma 3.5**.**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space, $f\colon X\to Y$ a contraction of fiber type, and $F\subset X$ a general fiber. Let also $\eta$ be the generic point of $Y$ and $X_{\eta}$ the generic fiber of $f$. Then $\rho_{X_{\eta}}=\dim\mathcal{N}_{1}(F,X)=d_{f}$.

Proof.

Recall that since $X$ is a Mori dream space, $\operatorname{Pic}(X)$ is finitely generated, and $\mathcal{N}^{1}(X)\cong\operatorname{Pic}(X)\otimes\mathbb{R}$.

Consider the restriction $r\colon\operatorname{Pic}(X)\to\operatorname{Pic}(X_{\eta})$ and the induced homomorphism $r_{\mathbb{R}}\colon\operatorname{Pic}(X)\otimes\mathbb{R}\to\operatorname{Pic}(X_{\eta})\otimes\mathbb{R}$. Since $X$ is $\mathbb{Q}$-factorial, $r_{\mathbb{R}}$ is surjective. Moreover $\ker r_{\mathbb{R}}$ coincides with the linear subspace $W\subset\mathcal{N}^{1}(X)$ of classes of $\mathbb{R}$-divisors whose support does not dominate $Y$. Therefore by Lemma 3.3 we have $\dim\ker r_{\mathbb{R}}=\rho_{X}-\dim\mathcal{N}_{1}(F,X)$, which yields the statement. ∎

Lemma 3.6**.**

Let $X$ be a normal quasi-projective variety and $f\colon X\to Y$ a contraction of fiber type. Let $\eta$ be the generic point of $Y$, $X_{\eta}$ the generic fiber of $f$, $Z_{\eta}$ a normal projective variety over $K=\mathbb{C}(\eta)$, and $g_{\eta}\colon X_{\eta}\to Z_{\eta}$ a projective morphism over $K$ such that $(g_{\eta})_{*}\mathcal{O}_{X_{\eta}}=\mathcal{O}_{Z_{\eta}}$.

Then there exists an open subset $Y_{0}$ of $\,Y$ such that $f_{|f^{-1}(Y_{0})}\colon f^{-1}(Y_{0})\to Y_{0}$ factors as the composition of two contractions $f^{-1}(Y_{0})\to Z_{0}\to Y_{0}$ that extend $X_{\eta}\to Z_{\eta}\to\eta$.

Proof.

We apply the results on spreading out schemes and morphisms. By [Poo17, Th. 3.2.1(i) and (ii)], there exist an open subset $Y_{0}$ of $Y$ and a quasi-projective variety $Z_{0}$, with a surjective projective morphism $h\colon Z_{0}\to Y_{0}$, that extend $Z_{\eta}\to\eta$.

Consider the normalization $\nu\colon Z_{0}^{\nu}\to Z_{0}$; restricting to the generic fibers we get $\nu_{\eta}\colon(Z_{0}^{\nu})_{\eta}\to Z_{\eta}$ which is the normalization of $Z_{\eta}$. Since $Z_{\eta}$ is normal, $\nu$ is an isomorphism on the generic fibers, and up to composing with $\nu$ we can assume that $Z_{0}$ is normal.

Note that $(g_{\eta})_{*}\mathcal{O}_{X_{\eta}}=\mathcal{O}_{Z_{\eta}}$ implies that every fiber of $g_{\eta}$ is geometrically connected [Liu02, Ch. 5, Th. 3.15 and Cor. 3.17]. Then by [Poo17, Th. 3.2.1(iii) and (iv)], up to shrinking $Y_{0}$, we can assume that there exists a projective morphism $g\colon f^{-1}(Y_{0})\to Z_{0}$, with connected fibers, extending $g_{\eta}$, so that $f_{|f^{-1}(Y_{0})}=h\circ g$. ∎

Lemma 3.7**.**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space. Let $X_{0}\subset X$ be an open subset and $f_{0}\colon X_{0}\to Y_{0}$ a contraction. Then there exists a rational contraction $f\colon X\dasharrow Y$ such that $Y$ contains $Y_{0}$ as an open subset and $f_{|X_{0}}=f_{0}$.

Proof.

Let $Y_{0}\subset\mathbb{P}^{N}$ be an embedding as locally closed subset. Then $f_{0}$ gives a rational map $F_{0}\colon X\dasharrow\mathbb{P}^{N}$ with $B:=X\smallsetminus\operatorname{dom}(F_{0})$ closed subset of codimension $\geq 2$.

Let $H\subset\mathbb{P}^{N}$ be a general hyperplane and consider the divisor $F_{0}^{*}(H)$ on $X\smallsetminus B$. Then $F_{0}^{*}(H)$ extends uniquely to a Weil divisor on $X$, that we still denote by $F_{0}^{*}(H)$. Since $X$ is $\mathbb{Q}$-factorial, there exists $m\in\mathbb{Z}_{>0}$ such that $D:=mF_{0}^{*}(H)$ is Cartier; then $D$ is also movable, because its base locus is contained in $B$. Moreover, since $X$ is a Mori dream space, up to taking a possibly larger $m$ we can assume that the complete linear system $|D|$ defines a rational map $F\colon X\dasharrow\mathbb{P}^{M}=\mathbb{P}(H^{0}(X,D))$ such that, if $Y:=\overline{F(\operatorname{dom}(F))}\subset\mathbb{P}^{M}$, the induced map $f\colon X\dasharrow Y$ is a rational contraction. Then $Y$ contains $Y_{0}$ as an open subset and $f_{|X_{0}}=f_{0}$. ∎

We will work with two particular types of rational contractions of fiber type, namely “quasi-elementary” and “special” contractions; let us recall the definition and a few properties, and refer the reader to [Cas13a, §2.2] and [Cas20, §2] respectively for more details.

Definition 3.8**.**

Let $f\colon X\to Y$ be a contraction of fiber type.

We say that $f$ is quasi-elementary if $\dim\mathcal{N}_{1}(F,X)=\rho_{X}-\rho_{Y}$ for every fiber $F\subset X$ of $f$, namely if $d_{f}=\rho_{X}-\rho_{Y}$.

We say that $f$ is special if $Y$ is $\mathbb{Q}$-factorial and $\operatorname{codim}f(D)\leq 1$ for every prime divisor $D\subset X$.

An elementary contraction of fiber type is always quasi-elementary. A quasi-elementary contraction is always special, by the following.

Lemma 3.9** ([Cas20], Prop. 2.2).**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space and $f\colon X\to Y$ a contraction of fiber type. Then $f$ is quasi-elementary if and only if $Y$ is $\mathbb{Q}$-factorial and $f^{*}(B)$ is an irreducible (possibly non-reduced) divisor for every prime divisor $B\subset Y$.

If $Y$ is a curve, then every contraction of fiber type $f\colon X\to Y$ is special. If $Y$ is a surface, then $f$ is special if and only if it is equidimensional [Cas20, Lemma 2.7].

Consider now a rational contraction of fiber type $f\colon X\dasharrow Y$. We say that $f$ is quasi-elementary, respectively special, if a resolution of $f$ is quasi-elementary, respectively special; this does not depend on the choice of the resolution.

Lemma 3.10**.**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space and $f\colon X\dasharrow Y$ a rational contraction. The following are equivalent:

1. $(i)$

$f$* is of fiber type and special;* 2. $(ii)$

$f$* is of fiber type and $\dim(\tau_{f}\cap\operatorname{Mov}(X))=\rho_{Y}$ (see Def.-Rem. 3.2);* 3. $(iii)$

there exists a proper face $\eta_{f}$ of $\operatorname{Mov}(X)$ such that $f^{*}(\operatorname{Nef}(Y))\subset\eta_{f}$, $\dim\eta_{f}=\rho_{Y}$, and $\eta_{f}=\tau\cap\operatorname{Mov}(X)$ for some face $\tau$ of $\operatorname{Eff}(X)$.

If these conditions hold and moreover $f$ is regular with general fiber $F\subset X$, then $\eta_{f}=(\mathcal{N}_{1}(F,X))^{\perp}\cap\operatorname{Mov}(X)$.

Proof.

The equivalence of $(i)$ and $(ii)$ is [Cas22, Lemma 5.3].

Assume $(ii)$. Note that $\eta_{f}:=\tau_{f}\cap\operatorname{Mov}(X)$ is a face of $\operatorname{Mov}(X)$ because $\operatorname{Mov}(X)\subset\operatorname{Eff}(X)$, it is proper because $\tau_{f}\subsetneq\operatorname{Eff}(X)$, it contains $f^{*}(\operatorname{Nef}(Y))$, and $\dim\eta_{f}=\rho_{Y}$, hence we get $(iii)$.

Conversely, assume $(iii)$. Since $\eta_{f}\subsetneq\operatorname{Mov}(X)$, we have $\tau\subsetneq\operatorname{Eff}(X)$, hence $f^{*}(\operatorname{Nef}(Y))\subset\partial\operatorname{Eff}(X)$, and $f$ is of fiber type. By definition $\tau_{f}$ is the minimal face of $\operatorname{Eff}(X)$ containing $f^{*}(\operatorname{Nef}(Y))$, therefore $\tau_{f}\subset\tau$, and $\tau_{f}\cap\operatorname{Mov}(X)=\tau\cap\operatorname{Mov}(X)=\eta_{f}$ has dimension $\rho_{Y}$, and we get $(ii)$.

Finally, when $f$ is regular, $\eta_{f}=\tau_{f}\cap\operatorname{Mov}(X)=(\mathcal{N}_{1}(F,X))^{\perp}\cap\operatorname{Mov}(X)$ by Lemma 3.1. ∎

Remark 3.11**.**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space, and $\tau$ a proper, movable face of $\operatorname{Eff}(X)$. Then there exists a special rational contraction of fiber type $f\colon X\dasharrow Y$ with $\eta_{f}=\tau\cap\operatorname{Mov}(X)$ and $\tau_{f}\subset\tau$, hence $\rho_{Y}=\dim(\tau\cap\operatorname{Mov}(X))$ and $d_{f}=\rho_{X}-\dim\tau_{f}\geq\rho_{X}-\dim\tau$ (notation as in Lemma 3.10 and Def.-Rem. 3.2).

Indeed it is enough to choose a cone $\eta_{0}\in\operatorname{MCD}(X)$ such that $\eta_{0}\subset\tau\cap\operatorname{Mov}(X)$ and $\dim\eta_{0}=\dim(\tau\cap\operatorname{Mov}(X))$; then the rational contraction $f\colon X\dasharrow Y$ such that $f^{*}(\operatorname{Nef}(Y))=\eta_{0}$ has the desired properties, by Lemma 3.10.

Proposition 3.12** ([Cas20], Prop. 2.13).**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space and $f\colon X\dasharrow Y$ a rational contraction of fiber type. Then $f$ can be factored as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}Z\stackrel{{\scriptstyle h}}{{\to}}Y$ where $h$ is birational and $g$ is a special rational contraction of fiber type.

Lemma 3.13**.**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space and $f\colon X\dasharrow Y$ a rational contraction of fiber type that is regular and proper on some non-empty open subset of $X$.

Then $f$ can be factored as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}Z\stackrel{{\scriptstyle h}}{{\to}}Y$ where $h$ is birational and $g$ is a special rational contraction of fiber type that is regular and proper on some non-empty open subset of $X$. In particular $f$ and $g$ have isomorphic general fibers.

Proof.

This follows from the same proof as [Cas20, Prop. 2.13]. ∎

Lemma 3.14**.**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space and $f\colon X\to Y$ a contraction of fiber type. Suppose that there exist a non-empty open subset $Y_{0}\subset Y$ and a contraction of fiber type $g_{0}\colon f^{-1}(Y_{0})\to Z_{0}$ such that $f_{|f^{-1}(Y_{0})}$ factors through $g_{0}$:

[TABLE]

Then $f$ can be factored as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}Z\stackrel{{\scriptstyle h}}{{\to}}Y$, where $h$ is a contraction, and $g$ is a special rational contraction that coincides with $g_{0}$ on $g_{0}^{-1}(U)$ for some non-empty open subset $U$ of $Z_{0}$.

Proof.

By Lemma 3.7 there exists a rational contraction $g^{\prime}\colon X\dasharrow Z^{\prime}$ that restricts to $g_{0}$ on $f^{-1}(Y_{0})$, in particular $g^{\prime}$ is of fiber type and is regular and proper on $f^{-1}(Y_{0})$.

Now we apply Lemma 3.13 to $g^{\prime}$, and deduce that $g^{\prime}$ factors as $X\stackrel{{\scriptstyle g^{\prime\prime}}}{{\dasharrow}}Z^{\prime\prime}\stackrel{{\scriptstyle\alpha}}{{\to}}Z^{\prime}$ where $\alpha$ is birational and $g^{\prime\prime}$ is a special rational contraction of fiber type that is regular and proper on some non-empty open subset of $X$. In particular $g^{\prime\prime}$ has the same general fiber as $g_{0}$.

Let us consider $F_{f},F_{g}\subset X$ general fibers of $f$ and $g_{0}$ respectively; we have $F_{g}\subset F_{f}$, so $\mathcal{N}_{1}(F_{g},X)\subset\mathcal{N}_{1}(F_{f},X)$, and hence $\mathcal{N}_{1}(F_{f},X)^{\perp}\subset\mathcal{N}_{1}(F_{g},X)^{\perp}\subset\mathcal{N}^{1}(X)$.

Let us set:

[TABLE]

(see Def.-Rem. 3.2), so that $\eta_{g^{\prime\prime}}$ is a face of $\operatorname{Mov}(X)$, and it has dimension $\rho_{Z^{\prime\prime}}$ because $g^{\prime\prime}$ is special, by Lemma 3.10.

We note that $\eta_{g^{\prime\prime}}$ is a union of cones of $\operatorname{MCD}(X)$ of dimension $\rho_{Z^{\prime\prime}}$, one being $(g^{\prime\prime})^{*}(\operatorname{Nef}(Z^{\prime\prime}))$.

The cone $f^{*}(\operatorname{Nef}(Y))$ belongs to the fan $\operatorname{MCD}(X)$ and is contained both in $\operatorname{Mov}(X)$ and in $\mathcal{N}_{1}(F_{f},X)^{\perp}$, therefore in $\eta_{g^{\prime\prime}}$. Hence there exists a cone $\eta\in\operatorname{MCD}(X)$ of dimension $\rho_{Z^{\prime\prime}}$ such that $f^{*}(\operatorname{Nef}(Y))\subseteq\eta\subseteq\eta_{g^{\prime\prime}}$.

Let $g\colon X\dasharrow Z$ be the rational contraction of fiber type such that $\eta=g^{*}(\operatorname{Nef}(Z))$. Then $\rho_{Z}=\rho_{Z^{\prime\prime}}$ and there is a SQM $\psi\colon Z^{\prime\prime}\dasharrow Z$ such that $g=\psi\circ g^{\prime\prime}$ (see [Oka16, Th. 1.2]), in particular $g$ and $g^{\prime\prime}$ have the same general fiber $F_{g}$. Then $g$ is special again by Lemma 3.10, and it coincides with $g_{0}$ on $g_{0}^{-1}(U)$ for some non-empty open subset $U$ of $Z_{0}$.

Finally $f^{*}(\operatorname{Nef}(Y))\subset g^{*}(\operatorname{Nef}(Z))$, therefore there exists a contraction $h\colon Z\to Y$ such that $f=h\circ g$ (see Rem. 2.2). ∎

Lemma 3.15**.**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space, and $f\colon X\dasharrow Y$ a rational contraction of fiber type that factors as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}Z\stackrel{{\scriptstyle h}}{{\dasharrow}}Y$ where $g$ and $h$ are rational contractions, $h$ is of fiber type, and $Z$ is $\mathbb{Q}$-factorial.

If $f$ is special (respectively, quasi-elementary), then $h$ is special (respectively, quasi-elementary).

Proof.

Up to composing with SQM’s, we can assume that $f$, $g$, and $h$ are regular.

Suppose that $f$ is special; in particular $Y$ is $\mathbb{Q}$-factorial. Let $D\subset Z$ be a prime divisor such that $h(D)\subsetneq Y$. Then $g^{-1}(D)$ has pure codimension one in $X$ and $g(g^{-1}(D))=D$, so there is an irreducible component $D_{X}$ of $g^{-1}(D)$ such that $g(D_{X})=D$. Since $f$ is special, $h(D)=f(D_{X})$ must be a prime divisor in $Y$, so that $h$ is special.

Suppose now that $f$ is quasi-elementary. We use Lemma 3.9: first of all $Y$ is $\mathbb{Q}$-factorial. Let $B\subset Y$ be a prime divisor. Then $f^{*}(B)=g^{*}(h^{*}(B))$ is irreducible (possibly non-reduced), therefore $h^{*}(B)$ must have a unique irreducible component, and $h$ is quasi-elementary too. ∎

Lemma 3.16**.**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space, and $f\colon X\dasharrow Y$ a rational contraction of fiber type. We have the following:

1. $(a)$

$f$* factors as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}Z\stackrel{{\scriptstyle h}}{{\to}}Y$ where $h$ is a contraction and $g$ is an elementary rational contraction, either divisorial or of fiber type;* 2. $(b)$

if $f$ is special, then $h$ is of fiber type or an isomorphism; 3. $(c)$

when $g$ is divisorial:

1. $\bullet$

if $f$ is special, then $\operatorname{Exc}(g)$ dominates $Y$ or a prime divisor in $Y$; 2. $\bullet$

if $f$ is quasi-elementary, then $\operatorname{Exc}(g)$ dominates $Y$.

Proof.

Consider the cone $f^{*}(\operatorname{Nef}(Y))\in\operatorname{MCD}(X)$. Since $f$ is of fiber type, $f^{*}(\operatorname{Nef}(Y))$ is contained in the boundary of $\operatorname{Eff}(X)$, hence also in the boundary of $\operatorname{Mov}(X)$.

Let us consider a facet $\eta$ of $\operatorname{Mov}(X)$ containing $f^{*}(\operatorname{Nef}(Y))$, and $\eta_{0}\in\operatorname{MCD}(X)$ of dimension $\rho_{X}-1$ such that $f^{*}(\operatorname{Nef}(Y))\subset\eta_{0}\subset\eta$. Then $\eta_{0}=g^{*}(\operatorname{Nef}(Z))$ for an elementary rational contraction $g\colon X\dasharrow Z$ and we have a factorization:

[TABLE]

where $h$ is a contraction (see Rem. 2.2). By construction $\eta_{0}$ is contained in the boundary of $\operatorname{Mov}(X)$, therefore $g$ cannot be small. This gives $(a)$, and $(b)$ follows from [Cas20, Prop. 2.10].

For $(c)$, suppose that $g$ is divisorial. The first statement follows from the definition of special contraction. For the second statement, up to composing with a SQM we can assume that $g$ is regular; set $E:=\operatorname{Exc}(g)\subset X$, and let $F\subset X$ be a general fiber of $f$. If $f(E)\subsetneq Y$, then $F\cap E=\emptyset$, thus $\mathcal{N}_{1}(F,X)\subset E^{\perp}$ (see Rem. 2.7). We also have $E\cdot\operatorname{NE}(g)<0$, thus $\operatorname{NE}(g)\not\subset E^{\perp}$, in particular $\operatorname{NE}(g)\not\subset\mathcal{N}_{1}(F,X)$. On the other hand $\operatorname{NE}(g)\subset\ker f_{*}$, and we conclude that $\mathcal{N}_{1}(F,X)\subsetneq\ker f_{*}$ and $f$ cannot be quasi-elementary. ∎

Remark 3.17**.**

In the setting of Lemma 3.16, it follows from the proof that the choices for $g$ correspond to the facets of $\operatorname{Mov}(X)$ that contain $f^{*}(\operatorname{Nef}(Y))$. More precisely, given a facet $\eta$ of $\operatorname{Mov}(X)$ containing $f^{*}(\operatorname{Nef}(Y))$, we costruct $g\colon X\dasharrow Z$ as in the statement, with moreover $\eta=\operatorname{Mov}(X)\cap g^{*}(\mathcal{N}^{1}(Z))$. Then $g$ is of fiber type if and only if $\eta$ is contained in the boundary of $\operatorname{Eff}(X)$, otherwise $g$ is divisorial.

Lemma 3.18**.**

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space of dimension $n$, and $f\colon X\dasharrow Y$ a quasi-elementary rational contraction with $\dim Y=n-2$ and $\rho_{X}-\rho_{Y}=2$. Then one of the following holds:

1. $(i)$

$f$* factors as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}Z\stackrel{{\scriptstyle h}}{{\to}}Y$ where $\dim Z=n-1$ and $g$ and $h$ are elementary of fiber type;* 2. $(ii)$

$f$* admits two factorizations $X\stackrel{{\scriptstyle\alpha_{i}}}{{\dasharrow}}W_{i}\stackrel{{\scriptstyle h_{i}}}{{\to}}Y$ for $i=1,2$, where $\alpha_{i}$ is elementary and divisorial with exceptional locus dominating $Y$, $h_{i}$ is elementary of fiber type, and $\alpha_{1}^{*}(\mathcal{N}^{1}(W_{1}))\neq\alpha_{2}^{*}(\mathcal{N}^{1}(W_{2}))$.*

Proof.

We note that since $f$ is quasi-elementary, the cone $f^{*}(\operatorname{Nef}(Y))$ has dimension $\rho_{X}-2$ and is contained in a face of $\operatorname{Mov}(X)$ of dimension $\rho_{X}-2$ (see Lemma 3.10). Therefore $f^{*}(\operatorname{Nef}(Y))$ is contained in exactly two facets $\tau_{1}$ and $\tau_{2}$ of $\operatorname{Mov}(X)$.

If one of these facets is contained in $\partial\operatorname{Eff}(X)$, then by applying Lemma 3.16 and Rem. 3.17 we factor $f$ as in $(i)$. Otherwise, if no $\tau_{i}$ is contained in $\partial\operatorname{Eff}(X)$, for $i=1,2$ again by applying Lemma 3.16 and Rem. 3.17 we get a factorization of $f$ as

[TABLE]

where $\alpha_{i}$ is a divisorial elementary rational contraction with exceptional locus dominating $Y$, and $h_{i}$ is elementary. Moreover $\alpha_{1}^{*}(\mathcal{N}^{1}(W_{1}))\neq\alpha_{2}^{*}(\mathcal{N}^{1}(W_{2}))$ because $\tau_{1}\neq\tau_{2}$. ∎

We conclude this section by recalling two results on special or quasi-elementary rational contractions of Fano $4$-folds, that are needed in the sequel.

Lemma 3.19** ([Cas20], Lemma 4.3).**

Let $X$ be a smooth Fano $4$-fold and $f\colon X\dasharrow S$ a special rational contraction onto a surface. Then $S$ is smooth.

Lemma 3.20** ([Cas22], Lemma 5.9).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $f\colon X\dasharrow S$ a quasi-elementary rational contraction onto a surface. Then $S$ is a smooth del Pezzo surface.

4. Factoring rational contractions of fiber type of Fano $4$-folds through $3$-folds

In this section we consider rational contractions of fiber type $f\colon X\dasharrow Y$ of Fano $4$-folds. When $\dim Y=3$, the results in [CS24] give a good understanding of $X$ and $f$, see Th. 1.5. We consider here the case $\dim Y\in\{1,2\}$. We show that when $d_{f}\geq 5$, then $f$ always factors through a $3$-fold (Th. 1.6 from the Introduction, see Th. 4.1 below). This has in turn important applications on the geometry of Fano $4$-folds with $\rho_{X}\geq 7$, given in Th. 4.5 and Cor. 4.10.

Theorem 4.1**.**

Let $X$ be a smooth Fano $4$-fold and $f\colon X\dasharrow Y$ a non-trivial rational contraction of fiber type.

1. $(a)$

If $d_{f}\geq 5$, then $f$ can be factored as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}Z\stackrel{{\scriptstyle h}}{{\to}}Y$ where $\dim Z=3$, $g$ is a special rational contraction, and $h$ is a contraction of fiber type. 2. $(b)$

If $d_{f}=4$ and $\dim Y=2$, then either the statement in $(a)$ holds, or $f$ is regular with general fiber a del Pezzo surface of degree one.

As explained in the Introduction, for the proof of Th. 4.1 we will use different strategies depending on the dimension of the base $Y$. We recall that $d_{f}=1$ when $\dim Y=3$, therefore $d_{f}\geq 4$ implies that either $\dim Y=2$ or $Y\cong\mathbb{P}^{1}$. For the case $\dim Y=2$, we will consider the generic fiber of a $K$-negative resolution $f^{\prime}$ of $f$, and rely on Lemma 4.2; for the case $Y\cong\mathbb{P}^{1}$, we will use Prop. 4.4, that exploits the monodromy action on the nef cone of a general fiber of $f^{\prime}$.

Lemma 4.2**.**

Let $K$ be a field of characteristic zero and let $S$ be a smooth, projective del Pezzo surface over $K$. If $\rho_{S}\geq 4$, then one of the following holds:

1. $(i)$

there exists a surjective morphism $S\to C$ onto a curve; 2. $(ii)$

$\rho_{S}=4$* and $K_{S}^{2}=1$.*

Proof.

We assume that $S$ has no surjective morphism to a curve, and show $(ii)$.

There exists a birational map $S\to S_{0}$ where $S_{0}$ is a minimal del Pezzo surface over $K$. By our assumptions $S_{0}$ does not have a conic bundle structure, hence $\rho_{S_{0}}=1$ (see [Has09, Th. 3.9]). Among the (finitely many) possible birational maps $S\to S_{0}$ with $\rho_{S_{0}}=1$, let us choose one with $K_{S_{0}}^{2}$ maximal. Note that $K_{S_{0}}^{2}\neq 7$ (see [ibid., Ex. 3.1.2]).

The map $S\to S_{0}$ is the blow-up of $r$ distinct (closed) points, of degrees $d_{1},\dotsc,d_{r}$;777The degree of a closed point $p$ is dimension of its residue field $K(p)$ over the base field $K$. we have

[TABLE]

in particular $r\geq 3$ and $K_{S_{0}}^{2}\geq 4$.

Let $i\in\{1,\dotsc,r\}$ and let us consider $S_{1}:=\operatorname{Bl}_{p_{i}}S_{0}$. Then $S_{1}$ is a del Pezzo surface with $\rho_{S_{1}}=2$, and $S_{1}$ does not have a morphism onto a curve, therefore there is another birational map $S_{1}\to S_{0}^{\prime}$ which blows-up a point of degree $d_{i}^{\prime}$. We have $K_{S_{1}}^{2}=K_{S_{0}}^{2}-d_{i}=K_{S_{0}^{\prime}}^{2}-d_{i}^{\prime}$, and $K_{S_{0}}^{2}\geq K_{S_{0}^{\prime}}^{2}$ by the maximality of $K_{S_{0}}^{2}$, hence $d_{i}\geq d_{i}^{\prime}$.

The birational map

[TABLE]

is called an elementary link of type II, see [Isk96, (2.2.2)], and these links are classified in [ibid., Th. 2.6(ii)]. It follows from this classification and from $d_{i}\geq d_{i}^{\prime}$ that $d_{i}\geq 2$ if $K_{S_{0}}^{2}\in\{4,6,9\}$, and $d_{i}\geq 3$ if $K_{S_{0}}^{2}\in\{5,8\}$. By (4.3) this implies that $K_{S_{0}}^{2}=9$, therefore $(S_{0})_{\bar{K}}\cong\mathbb{P}^{2}_{\bar{K}}$, where $\bar{K}$ is the algebraic closure of $K$ and $(S_{0})_{\bar{K}}=S_{0}\times_{\operatorname{Spec}K}\operatorname{Spec}{\bar{K}}$. Let $\pi\colon\mathbb{P}^{2}_{\bar{K}}\to S_{0}$ be the projection.

We show that there is at most one index $i\in\{1,\dotsc,r\}$ such that $d_{i}=2$. Assume that this is not the case, for instance that $d_{1}=d_{2}=2$. This means that, for $i=1,2$, $\pi^{-1}(p_{i})=\{p_{i1},p_{i2}\}\subset\mathbb{P}^{2}_{\bar{K}}$ is an orbit for the action of the Galois group $\text{Gal}(\bar{K}/K)$. Then the line $\ell_{i}:=\overline{p_{i1}p_{i2}}\subset\mathbb{P}^{2}_{\bar{K}}$ is fixed by the action of the Galois group, hence $\ell_{i}=\pi^{*}(\ell_{i,K})$ where $\ell_{i,K}\subset S_{0}$ is a curve, and $H^{0}(S_{0},\ell_{1,K}+\ell_{2,K})\cong H^{0}(\mathbb{P}^{2}_{\bar{K}},\mathcal{O}_{\mathbb{P}^{2}_{\bar{K}}}(2))$. In the linear system $|\ell_{1,K}+\ell_{2,K}|$, the pencil of curves containing $p_{1}$ and $p_{2}$ defines a map $\operatorname{Bl}_{p_{1},p_{2}}S_{0}\to\mathbb{P}^{1}_{K}$, against our assumptions.

Therefore there is at most one $i$ such that $d_{i}=2$, and $d_{j}\geq 3$ for every $j\neq i$. Again by (4.3) we get $8\leq 2+3(r-1)\leq\sum_{i}d_{i}=9-K_{S}^{2}\leq 8$, and we conclude that $r=3$, $\rho_{S}=4$, $(d_{1},d_{2},d_{3})=(2,3,3)$ up to order, and $K_{S}^{2}=1$. ∎

Proof of Th. 4.1 in the case $\dim Y=2$.

Let $\varphi\colon X\dasharrow X^{\prime}$ be a SQM such that $f^{\prime}:=f\circ\varphi^{-1}\colon X^{\prime}\to Y$ is a $K$-negative resolution of $f$, and $F\subset X^{\prime}$ a general fiber of $f^{\prime}$. Let also $\eta$ be the generic point of $Y$ and $X_{\eta}$ the generic fiber of $f^{\prime}$; then $X_{\eta}$ is a smooth del Pezzo surface over $K=\mathbb{C}(\eta)$, with $K_{X_{\eta}}^{2}=K_{F}^{2}$ (by generic flatness) and $\rho_{X_{\eta}}=\dim\mathcal{N}_{1}(F,X^{\prime})=d_{f}$ (by Lemma 3.5).

If $d_{f}=\rho_{X_{\eta}}\geq 4$, then we can apply Lemma 4.2. If we are in case $(i)$, there is a surjective morphism $g_{\eta}\colon X_{\eta}\to C_{\eta}$ onto a curve, defined over $K$. By normalizing and taking the Stein factorization, we can assume that $C_{\eta}$ is smooth and $(g_{\eta})_{*}\mathcal{O}_{X_{\eta}}=\mathcal{O}_{C_{\eta}}$.

Then by Lemma 3.6 there are a non-empty open subset $Y_{0}\subset Y$, a quasi-projective $3$-fold $Z_{0}$ with a contraction $Z_{0}\to Y_{0}$ extending $C_{\eta}\to\eta$, and a contraction $g_{0}\colon(f^{\prime})^{-1}(Y_{0})\to Z_{0}$, such that $f^{\prime}_{|(f^{\prime})^{-1}(Y_{0})}$ factors through $g_{0}$. Finally using Lemma 3.14 we get $(a)$.

Suppose now that we are in case $(ii)$ of Lemma 4.2, namely $d_{f}=\rho_{X_{\eta}}=4$ and $K_{X_{\eta}}^{2}=1$. Then $K_{F}^{2}=1$ and $\varphi$ is an isomorphism by [Cas22, Lemma 5.10], so $f$ is regular and we get $(b)$. ∎

Proposition 4.4**.**

Let $X$ be a smooth Fano $4$-fold, $X\dasharrow X^{\prime}$ a SQM, and $f\colon X^{\prime}\to\mathbb{P}^{1}$ a $K$-negative contraction. Let $\iota\colon F\hookrightarrow X^{\prime}$ be a general fiber of $f$.

If $\rho_{F}\geq 6$, or if $\rho_{F}=4,5$ and $\iota_{*}\colon\mathcal{N}_{1}(F)\to\mathcal{N}_{1}(X^{\prime})$ is injective, then $f$ can be factored as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}Z\stackrel{{\scriptstyle h}}{{\to}}\mathbb{P}^{1}$ where $\dim Z=3$, $g$ is a special rational contraction, and $h$ is a contraction.

Proof.

Let $Y_{0}\subseteq\mathbb{P}^{1}$ be an open subset over which $f$ is smooth, and set $X_{0}:=f^{-1}(Y_{0})$ and $F:=f^{-1}(y)$ for $y\in Y_{0}$. Note that $F$ is a smooth Fano $3$-fold. We consider the monodromy action of $\pi_{1}(Y_{0},y)$ on $H^{2}(F,\mathbb{R})=\mathcal{N}^{1}(F)$; the image of the restriction $\mathcal{N}^{1}(X_{0})\to\mathcal{N}^{1}(F)$ is the invariant subspace $\mathcal{N}^{1}(F)^{\pi_{1}(Y_{0},y)}$, see [Voi02, Th. 16.24].

Assume that $\rho_{F}\geq 6$. Then $F\cong\mathbb{P}^{1}\times S$ where $S$ is a del Pezzo surface [IP99, §12.6]; consider the projections $\pi_{1}\colon F\to\mathbb{P}^{1}$ and $\pi_{2}\colon F\to S$.

Set $\tau:=\pi_{1}^{*}\operatorname{Nef}(\mathbb{P}^{1})$ and $\sigma:=\pi_{2}^{*}\operatorname{Nef}(S)$. Then $\sigma$ is a facet of $\operatorname{Nef}(F)$, $\tau$ is a one-dimensional face of $\operatorname{Nef}(F)$, and every other one-dimensional face of $\operatorname{Nef}(F)$ is contained in $\sigma$, because it corresponds to a contraction of $F$ that factors through $\pi_{2}$.

Let $g\in\pi_{1}(Y_{0},y)$. By [Wiś91b, Wiś09], we have $g(\operatorname{Nef}(F))=\operatorname{Nef}(F)$, so that $g$ permutes the facets of $\operatorname{Nef}(F)$; see [CFST16, §2] for an analysis of the monodromy action on the nef cone of $F$. In particular $g(\sigma)=\sigma$ by [CFST16, Th. 2.7], because $\pi_{2}$ is the unique elementary contraction of fiber type of $F$, the other elementary contractions of $F$ being birational. This also implies that $g(\tau)=\tau$; in particular $g$ preserves the linear subspaces $\pi_{1\,}^{*}\mathcal{N}^{1}(\mathbb{P}^{1})$ and $\pi_{2\,}^{*}\mathcal{N}^{1}(S)$ of $\mathcal{N}^{1}(F)$.

We have $K_{F}=\pi_{1}^{*}K_{\mathbb{P}^{1}}+\pi_{2}^{*}K_{S}$, and $g(K_{F})=K_{F}$ because $K_{F}=K_{X_{0}|F}$. Therefore $g$ must fix also $\pi_{2}^{*}K_{S}$, so that $\pi_{2}^{*}(-K_{S})\in\mathcal{N}^{1}(F)^{\pi_{1}(Y_{0},y)}$. We conclude that there exists $H\in\operatorname{Pic}(X_{0})$ such that $H_{|F}=\pi_{2}^{*}(-K_{S})$. It follows from [Wiś91b, Prop. 1.3] that $H$ is relatively nef over $Y_{0}$, hence it induces a contraction $g_{0}\colon X_{0}\to Z_{0}$ such that $f_{|X_{0}}$ factors through $g_{0}$ and $g_{0|F}=\pi_{2}$, so that the general fiber of $g_{0}$ is $\mathbb{P}^{1}$. Finally using Lemma 3.14 we get the statement.

Suppose now that $\rho_{F}=4,5$ and $\iota_{*}\colon\mathcal{N}_{1}(F)\to\mathcal{N}_{1}(X^{\prime})$ is injective. Then the restriction $\mathcal{N}^{1}(X^{\prime})\to\mathcal{N}^{1}(F)$ is surjective, hence $\mathcal{N}^{1}(X_{0})\to\mathcal{N}^{1}(F)$ is surjective too, and the monodromy action is trivial. Thus again by [Wiś91b, Prop. 1.3] we see that every contraction of $F$ extends to a relative contraction of $X_{0}$ over $Y_{0}$.

Since $F$ is a Fano $3$-fold with $\rho_{F}\geq 4$, by [IP99, Theorem on p. 141] it has a conic bundle onto a surface. Then $f_{|X_{0}}$ factors through a contraction $g_{0}\colon X_{0}\to Z_{0}$ with general fiber $\mathbb{P}^{1}$, and we conclude as above by Lemma 3.14. ∎

Proof of Th. 4.1 in the case $Y\cong\mathbb{P}^{1}$.

Let $f^{\prime}\colon X^{\prime}\to\mathbb{P}^{1}$ be a $K$-negative resolution of $f$, and $\iota\colon F\hookrightarrow X^{\prime}$ a general fiber. If $d_{f}=\dim\mathcal{N}_{1}(F,X^{\prime})\geq 5$, then either $\rho_{F}\geq 6$, or $\rho_{F}=5$ and $\iota_{*}\colon\mathcal{N}_{1}(F)\to\mathcal{N}_{1}(F,X^{\prime})$ is an isomorphism. Thus the statement follows from Prop. 4.4. ∎

The following implies Th. 1.7 from the Introduction.

Theorem 4.5**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and $f\colon X\dasharrow Y$ a non-trivial rational contraction of fiber type.

Then $d_{f}\leq 4$. If moreover $d_{f}=4$ and $\dim Y=2$, then $Y\cong\mathbb{P}^{2}$ and $f\colon X\to\mathbb{P}^{2}$ is regular and equidimensional, with general fiber a del Pezzo surface of degree one.

The bound $d_{f}\leq 4$ is sharp: see Ex. 6.2 for an example where $d_{f}=4$, with $Y=\mathbb{P}^{1}$ and $\rho_{X}=7$. Moreover the condition $\rho_{X}\geq 7$ is necessary, as the following examples show.

Example 4.6**.**

Let $X$ be a Fano $4$-fold with $\delta_{X}=3$ and $\rho_{X}=6$ which is not a product of surfaces. Such $4$-folds have been classified in [CRS22, §7 and Corrigendum], there are 9 families. Each such $X$ has a quasi-elementary contraction $f\colon X\to S$ where $S\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ or $S\cong\mathbb{F}_{1}$, thus $d_{f}=\rho_{X}-\rho_{S}=4$.

Example 4.7**.**

Let $X$ be as in Ex. 4.6. By composing $f\colon X\to S$ with a $\mathbb{P}^{1}$-bundle $S\to\mathbb{P}^{1}$, we get a quasi-elementary contraction $f^{\prime}\colon X\to\mathbb{P}^{1}$ with $d_{f^{\prime}}=\rho_{X}-1=5$.

Proof of Th. 4.5.

We note first of all that if $\dim Y=3$, then the general fiber of $f$ is $\mathbb{P}^{1}$ and $d_{f}=1$, therefore we can assume that $\dim Y\in\{1,2\}$.

4.8**.**

Suppose that $f$ factors as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}Z\stackrel{{\scriptstyle h}}{{\to}}Y$ where $\dim Z=3$, $g$ is a special rational contraction, and $h$ is a contraction of fiber type. Then $d_{f}\leq 3$ if $\dim Y=2$ and $d_{f}\leq 4$ if $Y\cong\mathbb{P}^{1}$.

Proof.

We apply the results in [CS24] to $g\colon X\dasharrow Z$. First of all, we have $\delta_{X}\leq 1$ by Th. 1.9, hence $\rho_{X}-\rho_{Z}\leq 2$ by [ibid., Lemma 3.12]. Moreover $Z$ has the following property: if $Z\dasharrow Z^{\prime}$ is a SQM, and $Z^{\prime}\to W$ is a sequence of elementary divisorial contractions, then every contraction of the sequence is of type $(2,0)$. This is shown in [ibid., Lemma 5.9, 5.13, Lemma 5.15] when $\rho_{X}-\rho_{Z}=2$, and in [ibid., proof of Th. 6.3] and [Cas13a, §4.2] when $g$ is elementary.

Let $g^{\prime}\colon X^{\prime}\to Z$ be a resolution of $g$, and note that $h\circ g^{\prime}\colon X^{\prime}\to Y$ is a resolution of $f$. Let $F_{h}\subset Z$ be a general fiber of $h\colon Z\to Y$, so that $F:=(g^{\prime})^{-1}(F_{h})\subset X^{\prime}$ is a general fiber of $h\circ g^{\prime}$. By Def.-Rem. 3.2 and Lemma 2.6 we get $d_{f}=\dim\mathcal{N}_{1}(F,X^{\prime})\leq\rho_{X}-\rho_{Z}+\dim\mathcal{N}_{1}(F_{h},Z)\leq 2+\dim\mathcal{N}_{1}(F_{h},Z)$.

If $\dim Y=2$, then $F_{h}\cong\mathbb{P}^{1}$, hence $\dim\mathcal{N}_{1}(F_{h},Z)=1$ and $d_{f}\leq 3$.

Assume that $Y\cong\mathbb{P}^{1}$. We show that $\dim\mathcal{N}_{1}(F_{h},Z)\leq 2$, which implies that $d_{f}\leq 4$.

By Lemma 3.16 we can factor $h$ as $Z\stackrel{{\scriptstyle\alpha}}{{\dasharrow}}W\stackrel{{\scriptstyle\beta}}{{\to}}\mathbb{P}^{1}$ where $\alpha$ is an elementary rational contraction, either divisorial or of fiber type, and $\beta$ is a contraction.

If $\alpha$ is divisorial, we apply again Lemma 3.16 to $\beta$, and we proceed in this way until we find a factorization of $h$ as:

[TABLE]

where $\alpha^{\prime}$ is a (possibly empty) sequence of elementary divisorial rational contractions, $\beta^{\prime}$ and $\beta^{\prime\prime}$ are elementary of fiber type, $\dim S=2$, and $\beta^{\prime}$ and $\gamma$ are regular.

Up to composing with SQM’s, we can assume that $\beta^{\prime\prime}$ and $\alpha^{\prime}$ are regular (note that $\dim\mathcal{N}_{1}(F_{h},Z)$ does not vary, see Def.-Rem. 3.2). Then $\alpha^{\prime}(F_{h})\subset W^{\prime}$ is a general fiber of $\beta^{\prime}$ or of $\gamma\circ\beta^{\prime\prime}$, and $\dim\mathcal{N}_{1}(\alpha^{\prime}(F_{h}),W^{\prime})\leq 2$ because both contractions $\beta^{\prime}$ and $\beta^{\prime\prime}$ are elementary, and the general fiber of $\gamma$ is $\mathbb{P}^{1}$. Moreover, since $\dim\alpha^{\prime}(\operatorname{Exc}(\alpha^{\prime}))=0$ by the properties of $Z$, we have $\alpha^{\prime}(F_{h})\cap\alpha^{\prime}(\operatorname{Exc}(\alpha^{\prime}))=\emptyset$, hence $\dim\mathcal{N}_{1}(F_{h},Z)=\dim\mathcal{N}_{1}(\alpha^{\prime}(F_{h}),W^{\prime})\leq 2$ (see [Cas20, Lemma 2.17]). ∎

4.9**.**

We consider now the case where $f$ cannot be factored as in 4.8; then we get $d_{f}\leq 4$ by Th. 4.1$(a)$.

Suppose moreover that $d_{f}=4$ and $\dim Y=2$. Then Th. 4.1$(b)$ implies that $f\colon X\to Y$ is regular, with general fiber a del Pezzo surface of degree one.

We show that $f$ is special and $Y\cong\mathbb{P}^{2}$. By Prop. 3.12 we can factor $f$ as $X\stackrel{{\scriptstyle f^{\prime}}}{{\dasharrow}}Y^{\prime}\stackrel{{\scriptstyle\varphi}}{{\to}}Y$ where $f^{\prime}$ is a special rational contraction of fiber type, and $\varphi$ is birational; then $Y^{\prime}$ is a smooth rational surface (Lemma 3.19). If $Y^{\prime}\cong\mathbb{P}^{2}$, then $\varphi$ is an isomorphism, $f$ is special, and we get the statement.

Assume by contradiction that $Y^{\prime}\not\cong\mathbb{P}^{2}$. Then $Y^{\prime}$ is obtained as a blow-up of some Hirzebruch surface, therefore it has a contraction $\psi\colon Y^{\prime}\to\mathbb{P}^{1}$. Consider $\psi\circ f^{\prime}\colon X\dasharrow\mathbb{P}^{1}$. We show that $d_{\psi\circ f^{\prime}}>4$, contradicting the first part of the proof.

Let $\zeta\colon X\dasharrow X^{\prime}$ be a SQM such that $f^{\prime\prime}:=f^{\prime}\circ\zeta^{-1}\colon X^{\prime}\to Y^{\prime}$ is regular.

[TABLE]

Since $\varphi$ is birational, $f^{\prime\prime}$ and $\varphi\circ f^{\prime\prime}$ have the same general fiber, and $d_{f^{\prime\prime}}=d_{\varphi\circ f^{\prime\prime}}=d_{f}=4$; moreover $d_{\psi\circ f^{\prime}}=d_{\psi\circ f^{\prime\prime}}$ (see Def.-Rem. 3.2). If $F_{2}\subset X^{\prime}$ is a general fiber of $\psi\circ f^{\prime\prime}$, and $F_{1}\subset F_{2}\subset X^{\prime}$ a general fiber of $f^{\prime\prime}$, then $d_{\psi\circ f^{\prime\prime}}=\dim\mathcal{N}_{1}(F_{2},X^{\prime})>\dim\mathcal{N}_{1}(F_{1},X^{\prime})=4$. This concludes the proof. ∎

The following implies Cor. 1.8 from the Introduction.

Corollary 4.10**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces. If $\tau$ is a movable face of $\operatorname{Eff}(X)$, then $\dim\tau\geq\rho_{X}-4\geq 3$.

In particular, every face of dimension $1$ or $2$ of $\operatorname{Eff}(X)$ is fixed.

Again, the condition $\rho_{X}\geq 7$ is necessary, as the following example shows.

Example 4.11**.**

Let $X$ be a Fano $4$-fold as in Examples 4.6 and 4.7; we have $\rho_{X}=6$ and $X$ is not a product of surfaces. The cone $\operatorname{Eff}(X)$ has a one-dimensional movable face, given by $(f^{\prime})^{*}\operatorname{Nef}(\mathbb{P}^{1})$ where $f^{\prime}\colon X\to\mathbb{P}^{1}$ is a quasi-elementary contraction as in Ex. 4.7.

Proof of Cor. 4.10.

By Rem. 3.11 there is a non-trivial rational contraction of fiber type $f\colon X\dasharrow Y$ such that $\dim\tau\geq\rho_{X}-d_{f}$. We have $d_{f}\leq 4$ by Th. 4.5, hence $\dim\tau\geq\rho_{X}-4$. ∎

5. Preliminaries on fixed prime divisors of Fano $4$-folds

In this section we recall the classification of fixed prime divisors in Fano $4$-folds $X$ with $\rho_{X}\geq 7$, or $\rho_{X}=6$ and $\delta_{X}\leq 2$, and give many related properties that are used in the sequel.

Theorem - Definition 5.1** **(the type of a fixed prime divisor).

(See [Cas17, Th. 5.1, Cor. 5.2, Lemma 5.25].)*   Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, or $\rho_{X}=6$ and $\delta_{X}\leq 2$, and $D$ a fixed prime divisor in $X$.*

1. $(a)$

There exists a unique diagram:

[TABLE]

where $\xi$ is a SQM, $\sigma$ is a divisorial elementary contraction with exceptional divisor the transform $\widetilde{D}$ of $D$, and $Y$ is Fano (possibly singular); 2. $(b)$

$\sigma$* is of type $(3,0)^{\text{\it sm}}$, $(3,0)^{Q}$, $(3,1)^{\text{\it sm}}$, or $(3,2)$, and we define $D$ to be of type $(3,0)^{\text{\it sm}}$, $(3,0)^{Q}$, $(3,1)^{\text{\it sm}}$, or $(3,2)$, accordingly;* 3. $(c)$

if $D$ is of type $(3,2)$, then $X=\widetilde{X}$. In the other cases $\xi$ factors as a sequence of at least $\rho_{X}-4$ $D$-negative and $K$-negative flips. 4. $(d)$

We define $C_{D}\subset D\subset X$ to be the transform of a general irreducible curve $C_{\widetilde{D}}\subset\widetilde{D}\subset\widetilde{X}$ contracted by $\sigma$, of minimal anticanonical degree. Then $C_{D}\cong\mathbb{P}^{1}$, $D\cdot C_{D}=-1$, and $C_{D}\subset\operatorname{dom}(\xi)$. 5. $(e)$

Given a SQM $\varphi\colon X\dasharrow X^{\prime}$ and a divisorial elementary contraction $\sigma^{\prime}\colon X^{\prime}\to Y^{\prime}$ with $\operatorname{Exc}(\sigma^{\prime})$ the transform of $D$, there is a commutative diagram:

[TABLE]

where $\psi_{\scriptscriptstyle X}$ and $\psi_{\scriptscriptstyle Y}$ are SQM’s, $\widetilde{D}\subset\operatorname{dom}(\psi_{\scriptscriptstyle X})$, and $\sigma(\widetilde{D})\subset\operatorname{dom}(\psi_{\scriptscriptstyle Y})$.

Among the known families of Fano $4$-folds with $\rho_{X}\geq 7$, we have examples of all four types of fixed prime divisors, see §11.1 and §11.2.

Example 5.2**.**

Let $X=S_{1}\times S_{2}$ with $S_{1}$ and $S_{2}$ del Pezzo surfaces, and $\rho_{X}\geq 7$. Then every fixed prime divisor $D\subset X$ is of type $(3,2)$, and $D=C\times S_{2}$ or $S_{1}\times C$ where $C$ is a $(-1)$-curve.

Lemma 5.3** ([Cas17], Rem. 2.17(2) and [Cas13a], Cor. 3.14).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, or $\rho_{X}=6$ and $\delta_{X}\leq 2$. Let $E\subset X$ be a fixed prime divisor of type $(3,2)$, $X\dasharrow X^{\prime}$ a SQM, and $E^{\prime}\subset X^{\prime}$ the transform of $E$. Then $E$ does not contain exceptional planes and $\dim\mathcal{N}_{1}(E,X)=\dim\mathcal{N}_{1}(E^{\prime},X^{\prime})$.

Let $X$ be a smooth Fano $4$-fold, $\varphi\colon X\dasharrow X^{\prime}$ a SQM, and $E\subset X^{\prime}$ a fixed prime divisor. We define the type of $E$ to be the type of its transform $E_{\scriptscriptstyle X}\subset X$, and we define $C_{E}\subset E\subset X^{\prime}$ to be the transform of $C_{E_{\scriptscriptstyle X}}\subset X$.

Remark 5.4**.**

In the above setting, we have $C_{E_{X}}\subset\operatorname{dom}(\varphi)$ and $C_{E}\subset\operatorname{dom}(\varphi^{-1})$.

Indeed $X\smallsetminus\operatorname{dom}(\varphi)$ is a finite union of exceptional planes (Lemma 2.14). If $E$ is of type $(3,2)$, then $E_{X}$ does contain exceptional planes by Lemma 5.3, hence $X\smallsetminus\operatorname{dom}(\varphi)$ intersects $E_{X}$ at most in dimension one, and does not touch $C_{E_{X}}$.

Suppose that $E$ is not of type $(3,2)$, and let $L\subset X\smallsetminus\operatorname{dom}(\varphi)$ be an exceptional plane. If $L\subset E_{X}$, then $C_{E_{X}}\cap L=\emptyset$. If instead $L\not\subset E_{X}$ but $L\cap E_{X}\neq\emptyset$, let us consider the SQM $\xi\colon X\dasharrow\widetilde{X}$ as in Th.-Def. 5.1, and let $\widetilde{E}\subset\widetilde{X}$ be the transform of $E_{X}$. Then $L\not\subset X\smallsetminus\operatorname{dom}(\xi)$ and if $\widetilde{L}\subset\widetilde{X}$ is its transform, we have $\dim(\widetilde{L}\cap\widetilde{E})=1$, and the general curve $C_{\widetilde{E}}$ is disjoint from $\widetilde{L}$, thus again $C_{E_{X}}\cap L=\emptyset$.

Remark 5.5** (the cone $\operatorname{Mov}(X)^{\vee}$).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, or $\rho_{X}=6$ and $\delta_{X}\leq 2$. Consider the cone $\operatorname{Mov}(X)^{\vee}\subset\mathcal{N}_{1}(X)$, dual of the cone of movable divisors, and note that $\operatorname{mov}(X)\subset\operatorname{Mov}(X)^{\vee}\subset\operatorname{NE}(X)$ because dually $\operatorname{Nef}(X)\subset\operatorname{Mov}(X)\subset\operatorname{Eff}(X)$. By [Cas17, Lemma 5.29] we have:

[TABLE]

and every $\langle[C_{D}]\rangle$ is a one-dimensional face of $\operatorname{Mov}(X)^{\vee}$.

This means that $\operatorname{Mov}(X)^{\vee}$ has two types of one-dimensional faces, and dually that $\operatorname{Mov}(X)$ has two types of facets. The one-dimensional face $\langle[C_{D}]\rangle$ corresponds to the facet $\operatorname{Mov}(X)\cap[C_{D}]^{\perp}$ of $\operatorname{Mov}(X)$; the cones of $\operatorname{MCD}(X)$ of dimension $\rho_{X}-1$ contained in this facet correspond precisely to divisorial elementary rational contractions $X\dasharrow Y$ with exceptional divisor $D$.

The second type of one-dimensional faces $\alpha$ of $\operatorname{Mov}(X)^{\vee}$ are those contained in $\operatorname{mov}(X)$, so that $\alpha$ is a common one-dimensional face of the two cones. Then the corresponding facet $\alpha^{\perp}\cap\operatorname{Mov}(X)$ of $\operatorname{Mov}(X)$ is contained in a (movable) facet of $\operatorname{Eff}(X)$, and the cones of $\operatorname{MCD}(X)$ of dimension $\rho_{X}-1$ contained in this facet correspond to elementary rational contractions of fiber type.

Lemma 5.6**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, or $\rho_{X}=6$ and $\delta_{X}\leq 2$. Let $\varphi\colon X\dasharrow X^{\prime}$ be a SQM and $E\subset X^{\prime}$ a fixed prime divisor of type $(3,2)$. Then $[C_{E}]$ generates an extremal ray of $\operatorname{NE}(X^{\prime})$ if and only if $E$ does not meet any exceptional line.

Proof.

If $[C_{E}]$ generates an extremal ray of $\operatorname{NE}(X^{\prime})$, it must be divisorial by Rem. 5.4, and the statement follows from Th.-Def. 5.1 and Lemma 2.14.

Conversely, suppose that $E\subset X^{\prime}$ does not meet exceptional lines. Then $E\subset\operatorname{dom}(\varphi^{-1})$ (Lemma 2.14$(a)$) and $\varphi^{-1}(E)\subset X$ cannot contain exceptional planes (Lemma 5.3), therefore neither can $E$.

Since $E\cdot C_{E}<0$, $\operatorname{NE}(X^{\prime})$ must have an $E$-negative extremal ray $R$; we have $\operatorname{Locus}(R)\subset E$, hence $R$ is birational.

We claim that $R$ cannot be small. Indeed if $R_{0}$ is a small extremal ray of $\operatorname{NE}(X^{\prime})$, and $K_{X^{\prime}}\cdot R_{0}\geq 0$, then $\operatorname{Locus}(R_{0})$ is a finite union of exceptional lines (see Lemma 2.14), hence $\operatorname{Locus}(R_{0})\cap E=\emptyset$. If instead $-K_{X^{\prime}}\cdot R_{0}>0$, then $\operatorname{Locus}(R_{0})$ is a finite union of exceptional planes (see Lemma 2.13), therefore $\operatorname{Locus}(R_{0})\not\subset E$.

We conclude that $R$ is divisorial, and by Th.-Def. 5.1 we must have $[C_{E}]\in R$. ∎

Lemma 5.7**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, or $\rho_{X}=6$ and $\delta_{X}\leq 2$. Let $\varphi\colon X\dasharrow X^{\prime}$ be a SQM, $E\subset X^{\prime}$ a fixed prime divisor of type $(3,2)$, and $f\colon X^{\prime}\to Y$ a $K$-negative contraction such that $f(C_{E})=\{\text{\it pt}\}$. Then $[C_{E}]$ generates an extremal ray of $\operatorname{NE}(X^{\prime})$.

Proof.

Since $[C_{E}]\in\operatorname{NE}(f)$ and $E\cdot C_{E}<0$, there exists an extremal ray $R$ of $\operatorname{NE}(f)$ such that $E\cdot R<0$. Moreover $R$ is $K$-negative, because $f$ is. If $R$ is small, then $\operatorname{Locus}(R)$ is a finite union of exceptional planes (see Lemma 2.13). In particular $\operatorname{Locus}(R)$ must be contained in $\operatorname{dom}(\varphi^{-1})$ (see Lemma 2.14), and the transform of $E$ in $X$ contains an exceptional plane, a contradiction (see Lemma 5.3). Hence $R$ is divisorial, and by Th.-Def. 5.1 we must have $[C_{E}]\in R$. ∎

Lemma 5.8**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, or $\rho_{X}=6$ and $\delta_{X}\leq 2$. Let $\varphi\colon X\dasharrow X^{\prime}$ be a SQM and $f\colon X^{\prime}\to Y$ an elementary contraction of type $(3,2)$; set $A:=f(\operatorname{Exc}(f))\subset Y$.

Then $Y$ can contain finitely many pairwise disjoint exceptional lines, all disjoint from $A$. If $\Gamma\subset Y$ is an irreducible curve that is not an exceptional line, then $-K_{Y}\cdot\Gamma\geq 1$. Suppose moreover that $-K_{Y}\cdot\Gamma=1$. Then $\Gamma$ cannot meet any exceptional line, and if $\Gamma$ meets $A$, then $\Gamma\subset A$.

Furthermore, for every SQM $\psi\colon Y\dasharrow Y^{\prime}$, we have $A\subset\operatorname{dom}(\psi)$.

Proof.

We note that $f$ is $K$-negative (see Lemma 2.14) and $Y$ is locally factorial with at most ordinary double points, contained in $A$ (Lemma 2.9). By Th.-Def. 5.1$(e)$ we have a commutative diagram:

[TABLE]

where $\varphi_{Y}$ is a SQM, $\sigma$ is a divisorial elementary contraction with exceptional divisor the transform of $\operatorname{Exc}(f)$, $\widetilde{Y}$ is Fano, $\operatorname{Exc}(\sigma)\subset\operatorname{dom}(\varphi)$, and $\sigma(\operatorname{Exc}(\sigma))\subset\operatorname{dom}(\varphi_{Y})$. Then the same proof as [CS24, proof of Lemma 5.48] gives the statement. Note that if $\Gamma\subset Y$ is an irreducible curve with $-K_{Y}\cdot\Gamma=1$, $\Gamma\cap A\neq\emptyset$, and $\Gamma\not\subset A$, then the transform $\Gamma^{\prime}\subset X^{\prime}$ has $-K_{X^{\prime}}\cdot\Gamma^{\prime}\leq 0$, thus $\Gamma^{\prime}$ is an exceptional line (Lemma 2.14) and meets $\operatorname{Exc}(f)$, contradicting Lemma 5.6. ∎

Lemma 5.9** ([Cas22], Lemmas 4.12 and 4.13).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $D,E\subset X$ two distinct fixed prime divisors.

1. $(a)$

If $D\cdot C_{E}=0$, then $D$ and $E$ are adjacent. 2. $(b)$

If $E$ is of type $(3,2)$ and $D$ and $E$ are adjacent, then $D\cdot C_{E}=0$.

Lemma 5.10**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, or $\rho_{X}=6$ and $\delta_{X}\leq 2$. Let $D_{1},\dotsc,D_{m}\subset X$ be fixed prime divisors such that $D_{i}\cdot C_{D_{j}}=0$ for $i\neq j$. Then $\langle[D_{1}],\dotsc,[D_{m}]\rangle$ is a fixed face of $\operatorname{Eff}(X)$ of dimension $m$, and there exists a birational (rational) contraction $f\colon X\dasharrow Y$ with exceptional divisors $D_{1},\dotsc,D_{m}$, where $Y$ is $\mathbb{Q}$-factorial with $\rho_{Y}=\rho_{X}-m$.

Proof.

It is easy to see that $[D_{1}],\dotsc,[D_{m}]\in\mathcal{N}^{1}(X)$ are linearly independent. Moreover if $D$ is an effective divisor with $[D]\in\langle[D_{1}],\dotsc,[D_{m}]\rangle$, $D\neq 0$, then $D\cdot C_{D_{i}}<0$ for some $i\in\{1,\dotsc,m\}$, therefore $D$ is not movable, and $\langle[D_{1}],\dotsc,[D_{m}]\rangle\cap\operatorname{Mov}(X)=\{0\}$.

Each $[D_{i}]$ generates a one-dimensional face of $\operatorname{Eff}(X)$. To show that $\langle[D_{1}],\dotsc,[D_{m}]\rangle$ is a face, we proceed by induction on $m$, by applying Lemma 2.1 with the linear maps on $\mathcal{N}^{1}(X)$ given by intersection with $C_{D_{i}}$.

Finally, for the existence of the map $f$, see [Cas22, Lemma 4.2 and its proof]. ∎

Lemma 5.11**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $D$ and $E$ two adjacent fixed prime divisors, of type $(3,0)^{\text{\it sm}}$ and $(3,2)$ respectively. Then $D\cap E=\emptyset$.

Proof.

This follows from Lemma 5.3 and [Cas20, Lemma 4.9]. ∎

Lemma 5.12** ([Cas22], proof of Lemma 7.2).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$. Let $D$ be a fixed prime divisor of type $(3,0)^{Q}$, and $E_{1}$, $E_{2}$ fixed prime divisors of type $(3,2)$, both adjacent to $D$, such that $D\cap E_{i}\neq\emptyset$ for $i=1,2$. Then $E_{1}\cdot C_{E_{2}}=E_{2}\cdot C_{E_{1}}=1$.

Lemma 5.13**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, $\psi\colon X\dasharrow X^{\prime}$ a SQM, and $D,E\subset X^{\prime}$ two adjacent fixed prime divisors, of type $(3,1)^{\text{\it sm}}$ and $(3,2)$ respectively, with $E\cdot C_{D}>0$. Then $E\cdot C_{D}=1$ and $E\cap L=\emptyset$ for every exceptional plane $L\subset D$.

Proof.

Let $D_{X},E_{X}\subset X$ be the transforms of $D,E$ respectively. It is shown in [Cas22, Lemma 4.23] that $E_{X}\cdot C_{D_{X}}=1$ and that $E_{X}$ is disjoint from every exceptional plane contained in $D_{X}$. This implies the statement, indeed $E\cdot C_{D}=E_{X}\cdot C_{D_{X}}$ (see Rem. 5.4), and if $L\subset D$ is an exceptional plane, then $L\subset\operatorname{dom}(\psi^{-1})$ (see Lemma 2.14), and its transform $L_{X}\subset X$ is an exceptional contained in $D_{X}$, therefore $L_{X}\cap E_{X}=\emptyset$, and hence $L\cap E=\emptyset$. ∎

Lemma 5.14**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, $X\dasharrow\widehat{X}$ a SQM, and $\sigma\colon\widehat{X}\to Y$ and $\alpha\colon\widehat{X}\to Z$ divisorial elementary contractions of type $(3,2)$ and $(3,1)^{\text{\it sm}}$ respectively. Set $E:=\operatorname{Exc}(\sigma)$ and $D:=\operatorname{Exc}(\alpha)$, and assume that $D\cdot C_{E}=0$ and $E\cdot C_{D}>0$.

Then $\langle[C_{D}],[C_{E}]\rangle$ is a face of $\operatorname{NE}(\widehat{X})$ and we have a commutative diagram:

[TABLE]

where $\sigma^{\prime}$ is an elementary contraction of type $(3,2)$ with exceptional divisor $\alpha(E)$, and $\alpha^{\prime}$ is the blow-up of a smooth point with exceptional divisor $\sigma(D)$ (see Fig. 5.1). Moreover $D\cong\mathbb{P}_{\mathbb{P}^{1}}(\mathcal{O}^{\oplus 2}\oplus\mathcal{O}(1))$ and $E\cdot C_{D}=1$.

Proof.

Since $D\cdot C_{E}=0$, $D$ and $E$ are adjacent by Lemma 5.9, and $E\cdot C_{D}=1$ by Lemma 5.13.

We note that $D\cdot R\geq 0$ for every extremal ray $R$ of $\operatorname{NE}(\widehat{X})$ different from $\mathbb{R}_{\geq 0}[C_{D}]$; this follows from Th.-Def. 5.1, Lemma 2.14, and [Cas17, Rem. 5.6].

Let $H\in\operatorname{Pic}(\widehat{X})$ be the pullback under $\sigma$ of an ample divisor on $Y$, and set $m:=H\cdot C_{D}>0$. Then $(H+mD)\cdot C_{E}=(H+mD)\cdot C_{D}=0$, and if $R$ is an extremal ray of $\operatorname{NE}(\widehat{X})$ different from $\mathbb{R}_{\geq 0}[C_{D}]$ and $\mathbb{R}_{\geq 0}[C_{E}]$, then $H\cdot R>0$, $D\cdot R\geq 0$, and hence $(H+mD)\cdot R>0$. This shows that $H+mD$ is nef and $(H+mD)^{\perp}\cap\operatorname{NE}(\widehat{X})=\langle[C_{D}],[C_{E}]\rangle$ is a face of $\operatorname{NE}(\widehat{X})$, hence we have a commutative diagram as (5.15) where $\sigma^{\prime}$ and $\alpha^{\prime}$ are divisorial elementary contractions with exceptional divisors the images of $E$ and $D$ respectively. Then it is not difficult to see as in [Cas22, proof of Lemma 4.23] that $\sigma^{\prime}$ is of type $(3,2)$, $\alpha$ is the blow-up of a one-dimensional fiber of $\sigma^{\prime}$, $D\cong\mathbb{P}_{\mathbb{P}^{1}}(\mathcal{O}^{\oplus 2}\oplus\mathcal{O}(1))$, $\sigma(D)\cong\mathbb{P}^{3}$, and $\alpha^{\prime}$ is of type $(3,0)^{\text{\it sm}}$. See also [CS24, Lemma 2.18]. ∎

Lemma 5.16**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $D$ a fixed prime divisor of type $(3,1)^{\text{\it sm}}$. Then there is at most one fixed prime divisor $E$ of type $(3,2)$ which is adjacent to $D$ and such that $E\cdot C_{D}>0$.

Proof.

This follows from [Cas22, Lemma 4.23 and its proof]; see also Lemma 5.14. Let $X\stackrel{{\scriptstyle\xi}}{{\dasharrow}}\widetilde{X}\to Y$ be the contraction of $D$ as in Th.-Def. 5.1$(a)$, and $\widetilde{D},\widetilde{E}\subset\widetilde{X}$ the transforms of $D,E$ respectively. Then $\widetilde{D}$ is isomorphic to the blow-up of $\mathbb{P}^{3}$ along a line, $E\subset\operatorname{dom}(\xi)$ so that $\widetilde{E}\cong E$, $\widetilde{E}\cap\widetilde{D}$ is the exceptional divisor of $\widetilde{D}\to\mathbb{P}^{3}$, and the fibers of the blow-up $\widetilde{D}\to\mathbb{P}^{3}$ are numerically equivalent to $\xi(C_{E})$. Therefore the class $[C_{E}]$ is uniquely determined by $D$. ∎

Lemma 5.17**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$. Let $D$ be a fixed prime divisor of type $(3,1)^{\text{\it sm}}$, and $E_{1}$, $E_{2}$ distinct fixed prime divisors of type $(3,2)$, both adjacent to $D$, such that $D\cap E_{i}\neq\emptyset$, for $i=1,2$. Then one of the following holds:

1. $(i)$

$E_{1}\cdot C_{E_{2}}=E_{2}\cdot C_{E_{1}}=0$; 2. $(ii)$

up to exchanging $E_{1}$ and $E_{2}$, we have $E_{1}\cdot C_{D}=0$ and $E_{2}\cdot C_{D}=E_{2}\cdot C_{E_{1}}=1$.

Proof.

We assume that $(i)$ does not hold, and show $(ii)$. If $E_{1}\cdot C_{E_{2}}=0$, then $E_{1}$ and $E_{2}$ are adjacent by Lemma 5.9, hence $E_{2}\cdot C_{E_{1}}=0$ by the same lemma; similarly if $E_{2}\cdot C_{E_{1}}=0$. Therefore we can assume that $E_{1}\cdot C_{E_{2}}>0$ and $E_{2}\cdot C_{E_{1}}>0$.

By Lemma 5.16 there exists $i\in\{1,2\}$ such that $E_{i}\cdot C_{D}=0$; up to exchanging $E_{1}$ and $E_{2}$, we can suppose that $E_{1}\cdot C_{D}=0$. Note that $D\cdot C_{E_{1}}=0$ because $D$ and $E_{1}$ are adjacent (Lemma 5.9).

Then by [Cas22, Lemma 6.9 and Prop. 6.1] there exists an exceptional plane $L\subset D$ such that $C_{D}\equiv C_{L}+C_{E_{1}}$. By Lemma 5.3 we have $L\not\subset E_{2}$, thus $E_{2}\cdot C_{L}\geq 0$ and $E_{2}\cdot C_{D}\geq E_{2}\cdot C_{E_{1}}>0$. By Lemma 5.13 we have $E_{2}\cdot C_{D}=1$, which implies that $E_{2}\cdot C_{E_{1}}=1$. ∎

Lemma 5.18**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, $D$ a fixed prime divisor of type $(3,0)^{\text{\it sm}}$ or $(3,1)^{\text{\it sm}}$, and $X\stackrel{{\scriptstyle\xi}}{{\dasharrow}}\widetilde{X}\stackrel{{\scriptstyle\sigma}}{{\to}}Y$ the contraction of $D$ as in Th.-Def. 5.1$(a)$. Then $\delta_{Y}\leq 2$.

Proof.

We note first of all that $X$ is not isomorphic to a product of surfaces because it has a fixed prime divisor of type $(3,0)^{\text{\it sm}}$ or $(3,1)^{\text{\it sm}}$, therefore $\delta_{X}\leq 1$ by Th. 1.9. Moreover $Y$ is a smooth Fano $4$-fold.

If by contradiction $\delta_{Y}\geq 3$, then $\operatorname{NE}(Y)$ has an extremal ray $R$ of type $(3,2)$ with $\operatorname{codim}\mathcal{N}_{1}(E,Y)=\delta_{Y}$ where $E=\operatorname{Locus}(R)$ (see [Cas12, Rem. 3.2.6] for the case $\delta_{Y}\geq 4$, and [CRS22, Th. 1.4 and proof of Lemmas 3.2 and 4.5] for the case $\delta_{Y}=3$).

Consider the transforms $E_{X}\subset X$ and $E_{\widetilde{X}}\subset\widetilde{X}$. Let $\Gamma\subset E$ be a general fiber of the contraction of $R$, and let $\Gamma_{X}\subset X$ and $\Gamma_{\widetilde{X}}\subset\widetilde{X}$ be its transforms. By generality we have $\Gamma\cap\sigma(\operatorname{Exc}(\sigma))=\emptyset$, hence $-K_{\widetilde{X}}\cdot\Gamma_{\widetilde{X}}=-K_{Y}\cdot\Gamma=1$. By Lemma 2.14 this implies that $\Gamma_{\widetilde{X}}\subset\operatorname{dom}(\xi^{-1})$, therefore $-K_{X}\cdot\Gamma_{X}=1$ and $E_{X}\cdot\Gamma_{X}=E\cdot\Gamma=-1$. Thus $E_{X}\subset X$ is a non-nef prime divisor covered by rational curves of anticanonical degree one, so that $E_{X}$ is a fixed prime divisor of type $(3,2)$ by [Cas17, Lemma 2.18]. Then using Lemmas 2.6 and 5.3 we get

[TABLE]

a contradiction. ∎

Lemma 5.19**.**

Let $X$ be a Fano $4$-fold with $\rho_{X}\geq 7$ and $\delta_{X}\leq 1$, $X\dasharrow X^{\prime}$ a SQM, and $f\colon X^{\prime}\to S$ a contraction. Assume that $\rho_{S}\geq 3$, and let $\Gamma\subset S_{\text{\it reg}}$ be a $(-1)$-curve. Then there is an exceptional line $\ell\subset X^{\prime}$, whose class is extremal in $\operatorname{NE}(X^{\prime})$, such that $f(\ell)=\Gamma$.

Proof.

Since $\mathbb{R}_{\geq 0}[\Gamma]$ is an extremal ray of $\operatorname{NE}(S)$, there is an extremal ray $R$ of $\operatorname{NE}(X^{\prime})$ such that $f_{*}(R)=\mathbb{R}_{\geq 0}[\Gamma]$, so that the contraction of $R$ must be birational, with fibers of dimension at most one (see [Cas08, §2.5]). If $K_{X^{\prime}}\cdot R\geq 0$, then $R$ contains the class of an exceptional line $\ell\subset X^{\prime}$ (see Lemma 2.14) and we have $f(\ell)=\Gamma$.

We suppose that $K_{X^{\prime}}\cdot R<0$ and show that this gives a contradiction. Then $R$ cannot be small (Lemma 2.13), thus it must be of type $(3,2)$ with locus $D:=f^{-1}(\Gamma)$, and $D$ is a fixed prime divisor of type $(3,2)$ (see Th.-Def. 5.1). By Lemma 2.6 we have $\dim\mathcal{N}_{1}(D,X^{\prime})\leq 1+\rho_{X}-\rho_{S}\leq\rho_{X}-2$. On the other hand, if $D_{X}\subset X$ is the transform of $D$, Lemma 5.3 gives $\dim\mathcal{N}_{1}(D_{X},X)=\dim\mathcal{N}_{1}(D,X^{\prime})$, contradicting $\delta_{X}\leq 1$. ∎

Remark 5.20**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $E,D_{1},\dotsc,D_{m}$ fixed prime divisors such that $E$ is of type $(3,2)$, each $D_{i}$ is of type $(3,1)^{\text{\it sm}}$, and $E\cdot C_{D_{i}}=D_{i}\cdot C_{D_{j}}=0$ for every $i,j=1,\dotsc,m$, $i\neq j$.

By Lemma 5.9 this implies that $D_{i}\cdot C_{E}=0$ for every $i=1,\dotsc,m$, and Lemma 5.10 gives that $\langle[E],[D_{1}],\dotsc,[D_{m}]\rangle$ is a fixed face of $\operatorname{Eff}(X)$, and $E,D_{1},\dotsc,D_{m}$ can be contracted together by a birational (rational) contraction with $\mathbb{Q}$-factorial target. However the geometry of this contraction depends on the order in which we contract the divisors; let us describe this. See also [CS24, Lemma 2.19] for the case $m=1$, and [Cas22, Lemma 6.9, Prop. 6.1, Prop. 6.4].

Let us first describe how the divisors $E,D_{1},\dotsc,D_{m}$ intersect in $X$. For every $i,j\in\{1,\dotsc,m\}$, $i\neq j$, $D_{i}\cap D_{j}$ is either empty or a disjoint union of exceptional planes [Cas22, Lemma 4.13(b)]. Let us also note that $X$ is not a product of surfaces because it has fixed prime divisors of type $(3,1)^{\text{\it sm}}$, thus $\delta_{X}\leq 1$ by Th. 1.9, and $E$ can be disjoint from at most one $D_{i}$ (otherwise, if $E$ is disjoint from $D_{i}$ and $D_{j}$ with $i\neq j$, then $\mathcal{N}_{1}(E,X)\subset D_{i}^{\perp}\cap D_{j}^{\perp}$ by Rem. 2.7, a contradiction). Moreover by [CS24, Lemma 2.17$(ii)$], for every $i=1,\dotsc,m$ such that $D_{i}\cap E\neq\emptyset$, there exists an exceptional plane $L_{i}\subset D_{i}$ such that $D_{i}\cdot C_{L_{i}}=-1$, $E\cdot C_{L_{i}}=1$, $C_{D_{i}}\equiv C_{E}+C_{L_{i}}$, and $E\cap L=\emptyset$ for every exceptional plane $L\subset D_{i}$ such that $C_{L}\not\equiv C_{L_{i}}$; we also have that $E\cap D_{i}$ is a disjoint union of surfaces isomorphic to $\mathbb{F}_{1}$. For $j\neq i$ we get $D_{j}\cdot C_{L_{i}}=0$ and hence $D_{j}\cap L_{i}=\emptyset$, because $D_{j}\cdot C_{L_{0}}<0$ for every exceptional plane $L_{0}\subset D_{j}$.

We can first perform a sequence of $(D_{1}+\cdots+D_{m})$-negative and $E$-trivial flips, and get a SQM $\xi_{1}\colon X\dasharrow\widetilde{X}$. Then $E\subset\operatorname{dom}(\xi_{1})$, so that $[C_{E}]$ is still extremal in $\operatorname{NE}(\widetilde{X})$ (Lemma 5.6), and we have a diagram:

[TABLE]

where $\alpha$ and $\tilde{\alpha}$ are the contractions of $\mathbb{R}_{\geq 0}[C_{E}]$, of type $(3,2)$, and locally isomorphic, $Y$ is Fano with at most isolated, locally factorial, terminal singularities, and $\xi_{Y}$ is a SQM (Th.-Def. 5.1, Lemma 2.9).

In $\widetilde{Y}$ the divisors $D_{1},\dotsc,D_{m}$ are pairwise disjoint, are contained in $\widetilde{Y}_{\text{\it reg}}$, and are the exceptional divisors of $\sigma_{Y}\colon\widetilde{Y}\to W$ blow-up of $m$ pairwise disjoint smooth irreducible curves $\Gamma_{1},\dotsc,\Gamma_{m}\subset W_{\text{\it reg}}$; $W$ has the same singularities as $Y$ and $\widetilde{Y}$ (see Fig. 5.2).

Set $f:=\sigma_{Y}\circ\tilde{\alpha}\colon\widetilde{X}\to W$. Then $f$ is a resolution of the birational map $X\dasharrow W$, with exceptional divisors $E,D_{1},\dotsc,D_{m}$. In $\widetilde{X}$ the relative cone of $f$ is $\operatorname{NE}(f)=\langle[C_{E}],[C_{L_{1}}],\dotsc,[C_{L_{m}}]\rangle$, and $C_{D_{i}}\equiv C_{E}+C_{L_{i}}$ for every $i$,888Or $\operatorname{NE}(f)=\langle[C_{E}],[C_{L_{1}}],\dotsc,[C_{L_{m-1}}],[C_{D_{m}}]\rangle$ if $E\cap D_{m}=\emptyset$ in $X$. while in $\widetilde{Y}$ we have $\operatorname{NE}(\sigma_{Y})=\langle[C_{D_{1}}],\dotsc,[C_{D_{m}}]\rangle$.

Moreover $W$ is Fano. To see this, consider $f^{*}(-K_{W})=-K_{\widetilde{X}}+E+2(D_{1}+\cdots+D_{m})$, and let $R$ be an extremal ray of $\operatorname{NE}(\widetilde{X})$. If $K_{\widetilde{X}}\cdot R\geq 0$, then there is an exceptional line $\ell\subset\widetilde{X}$ with $[\ell]\in R$ (see Lemma 2.14), and we have $(D_{1}+\cdots+D_{m})\cdot\ell>0$, $-K_{\widetilde{X}}\cdot\ell=-1$, $E\cdot\ell=0$, thus $f^{*}(-K_{W})\cdot\ell>0$. Suppose instead that $R$ is $K$-negative. The extremal rays of $\operatorname{NE}(\widetilde{X})$ that are negative for some exceptional divisor of $f$ are precisely those in $\operatorname{NE}(f)$, therefore if $R\not\subset\operatorname{NE}(f)$ we have $-K_{\widetilde{X}}\cdot R>0$, $(E+2(D_{1}+\cdots+D_{m}))\cdot R\geq 0$, and again $f^{*}(-K_{W})\cdot R>0$. We conclude that $f^{*}(-K_{W})$ is nef and $f^{*}(-K_{W})^{\perp}\cap\operatorname{NE}(\widetilde{X})=\operatorname{NE}(f)$, which implies that $-K_{W}$ is ample.

On the other hand from $\widetilde{X}$ we can also consider the remaining $(D_{1}+\cdots+D_{m})$-negative flips, that will be given by the extremal rays generated by $[C_{L_{1}}],\dotsc,[C_{L_{m}}]$ (which are $E$-positive), and get a SQM $\xi_{2}\colon\widetilde{X}\dasharrow\widehat{X}$. Now we get a birational map $\sigma\colon\widehat{X}\to Z$ which is the blow-up of $m$ pairwise disjoint smooth irreducible curves $\Gamma^{\prime}_{1},\dotsc,\Gamma^{\prime}_{m}\subset Z$, and $Z$ is smooth and Fano (this can be seen with a similar argument as for $W$). In $Z$ we have $E\cdot\Gamma^{\prime}_{i}>0$ for every $i=1,\dotsc,m$. Moreover $[C_{E}]$ is extremal in $\operatorname{NE}(Z)$, and there is an associated elementary contraction of type $(3,2)$ $\alpha_{0}\colon Z\to W$.

[TABLE]

Set $g:=\alpha_{0}\circ\sigma\colon\widehat{X}\to W$. Then $g$ is another resolution of the birational map $X\dasharrow W$, and now we have $\operatorname{NE}(g)=\langle[C_{D_{1}}],\dotsc,[C_{D_{m}}],[\ell_{1}],\dotsc,[\ell_{m}]\rangle$ where $\ell_{i}\subset\widehat{X}$ is the exceptional line corresponding to the exceptional plane $L_{i}\subset\widetilde{X}$.999Or $\operatorname{NE}(g)=\langle[C_{D_{1}}],\dotsc,[C_{D_{m}}],[\ell_{1}],\dotsc,[\ell_{m-1}]\rangle$ if $E\cap D_{m}=\emptyset$ in $X$. Moreover $C_{E}\equiv C_{D_{i}}+\ell_{i}$ for every $i$.

Lemma 5.21**.**

In the setting of Rem. 5.20, let $i\in\{1,\dotsc,m\}$ be such that $E\cap D_{i}\neq\emptyset$, $A\subset W$ be the image of $E$, and $\Gamma_{i}\subset W$ the image of $D_{i}$, so that $A$ is an irreducible surface and $\Gamma_{i}$ a smooth irreducible curve contained in $W_{\text{\it reg}}$. Then $A$ and $\Gamma_{i}$ intersect transversally at finitely many points, contained in $A_{\text{\it reg}}$.

Proof.

Let $S$ be a connected component of $E\cap D_{i}$ in $X$. Then $S\cong\mathbb{F}_{1}$ and if $\ell\subset\mathbb{F}_{1}$ is the fiber of the $\mathbb{P}^{1}$-bundle and $e\subset\mathbb{F}_{1}$ is the $(-1)$-curve, we have $\ell\equiv C_{E}$, $e=S\cap L_{i}$, and $e\equiv C_{L_{i}}$ (notation as in Rem. 5.20).

The surface $S$ is contained in $\operatorname{dom}(\xi_{1})$, therefore it is mapped isomorphically to $\widetilde{X}$, where we still denote it by $S$. Set $A_{Y}:=\tilde{\alpha}(E)\subset\widetilde{Y}$ and $\Lambda:=\tilde{\alpha}(S)\subset A_{Y}$; then $\Lambda$ is a connected component of $A_{Y}\cap\tilde{\alpha}(D_{i})$.

We note that $S=\tilde{\alpha}^{-1}(\Lambda)$ and $\tilde{\alpha}_{|S}$ is a $\mathbb{P}^{1}$-bundle, so that $\tilde{\alpha}$ has one-dimensional fibers over $\Lambda$, hence both $A_{Y}$ and $\widetilde{Y}$ are smooth at every point of $\Lambda$ (Lemma 2.9).

Moreover $\Lambda\cong\mathbb{P}^{1}$ and $\Lambda\equiv C_{D_{i}}$. If $\tilde{\alpha}(D_{i})_{|A_{Y}}=m\Lambda+B$ where $B$ is a divisor in $A_{Y}$ supported on the other connected components of $A_{Y}\cap\tilde{\alpha}(D_{i})$ (note that $A_{Y}\cap\tilde{\alpha}(D_{i})\subset(A_{Y})_{\text{\it reg}}$), we get $-1=\tilde{\alpha}(D_{i})\cdot C_{D_{i}}=\tilde{\alpha}(D_{i})\cdot\Lambda=\tilde{\alpha}(D_{i})_{|A_{Y}}\cdot_{A_{Y}}\Lambda=m\Lambda^{2}$, so that $m=1$ and $\Lambda$ is a $(-1)$-curve in $A_{Y}$.

This shows that, in $W$, $w_{0}:=\sigma_{Y}(\Lambda)=f(S)$ is a point, and $A$ is smooth at $w_{0}$. Moreover $\sigma_{Y|A_{Y}}$ is the blow-up of the reduced point $w_{0}$, therefore the curve $\Gamma_{i}$ and $A$ intersect transversally at $w_{0}$. ∎

Lemma 5.22**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, $X\dasharrow\widetilde{X}$ a SQM, and $f\colon\widetilde{X}\to Y$ a special contraction with $\dim Y=3$ and $\rho_{X}-\rho_{Y}=2$. Then $\operatorname{NE}(f)=\langle[C_{E_{1}}],[C_{E_{2}}]\rangle$ where $E_{1},E_{2}\subset\widetilde{X}$ are fixed prime divisors of type $(3,2)$ such that $\mathcal{N}_{1}(E_{i},\widetilde{X})=\mathcal{N}_{1}(\widetilde{X})$ for $i=1,2$.

Proof.

The Fano $4$-folds as in the statement are classified in [CS24, Th. 5.1]. The description of $\operatorname{NE}(f)$ is in [ibid., Lemma 5.6], and we keep the same notation. In particular let $\tilde{\alpha}\colon\widetilde{X}\to\widetilde{W}$ the elementary contraction of type $(3,2)$ with exceptional divisor $E_{1}$, and $S:=\tilde{\alpha}(E_{1})\subset\widetilde{W}$. Then $\mathcal{N}_{1}(S,\widetilde{W})=\mathcal{N}_{1}(\widetilde{W})$ by [ibid., 5.67]; on the other hand $\ker\tilde{\alpha}_{*}\subset\mathcal{N}_{1}(E_{1},\widetilde{X})$ and $\tilde{\alpha}_{*}(\mathcal{N}_{1}(E_{1},\widetilde{X}))=\mathcal{N}_{1}(S,\widetilde{W})$, and we conclude that $\mathcal{N}_{1}(E_{1},\widetilde{X})=\mathcal{N}_{1}(\widetilde{X})$.

Let us consider now $E_{2}$. If we are in case $(b)$ of [ibid., Lemma 5.39], then the situation for $E_{1}$ and $E_{2}$ is symmetric [ibid., Lemma 5.46, 5.60], and as before we conclude that $\mathcal{N}_{1}(E_{2},\widetilde{X})=\mathcal{N}_{1}(\widetilde{X})$. If we are in case $(a)$ and the surface $S$ is singular, then $\tilde{\alpha}\colon\widetilde{X}\to\widetilde{W}$ has a $2$-dimensional fiber $F_{0}$ (Lemma 2.9). We have $E_{2}\cdot\operatorname{NE}(\tilde{\alpha})>0$, hence $\dim(F_{0}\cap E_{2})\geq 1$ and $\ker\tilde{\alpha}_{*}\subset\mathcal{N}_{1}(E_{2},\widetilde{X})$. On the other hand $T:=\tilde{\alpha}(E_{2})\supset S$, so that $\tilde{\alpha}_{*}(\mathcal{N}_{1}(E_{2},\widetilde{X}))=\mathcal{N}_{1}(T,\widetilde{W})\supset\mathcal{N}_{1}(S,\widetilde{W})=\mathcal{N}_{1}(\widetilde{W})$, and we conclude again that $\mathcal{N}_{1}(E_{2},\widetilde{X})=\mathcal{N}_{1}(\widetilde{X})$.

We are left with the case where we are in case $(a)$ and $S$ is smooth. Then by [ibid., Th. 5.1, Lemma 5.68, 5.70] there is a SQM $\widetilde{W}\dasharrow\widehat{W}:=\operatorname{Bl}_{q_{0},\dotsc,q_{r}}\mathbb{P}^{4}$ (see §11.1), with $r=\rho_{X}-3$, and $S\subset\widetilde{W}$ is the transform of a cubic scroll $A\subset\mathbb{P}^{4}$ containing the points $q_{0},\dotsc,q_{r}$. Moreover $Y\cong\operatorname{Bl}_{p_{1},\dotsc,p_{r}}\mathbb{P}^{3}$ and there is a $\mathbb{P}^{1}$-bundle $\pi\colon\widetilde{W}\to Y$, induced by the projection $\mathbb{P}^{4}\dasharrow\mathbb{P}^{3}$ from $q_{0}$, such that $f=\pi\circ\tilde{\alpha}$.

The points $q_{0},\dotsc,q_{r}\in\mathbb{P}^{4}$ are in general linear position [ibid., 5.63], therefore up to exchanging some points we can assume that $q_{0}$ does not belong to the $(-1)$-curve in $A\cong\mathbb{F}_{1}$. Then the projection of $A$ from $q_{0}$ is a smooth quadric surface $B_{0}\subset\mathbb{P}^{3}$, and $B:=\pi(S)=\operatorname{Bl}_{p_{1},\dotsc,p_{r}}B_{0}$ is a smooth surface with $\rho_{B}=2+r=\rho_{X}-1$. Finally $E_{2}\subset\widetilde{X}$ is the transform of $T=\pi^{-1}(B)\subset\widetilde{W}$, which is a $\mathbb{P}^{1}$-bundle over $B$, thus $T$ is smooth with $\rho_{T}=\rho_{X}=\rho_{\widetilde{W}}+1$. We have $E_{2}\cong T$ because both $T$ and $S$ are smooth, $S\subset T$, and $\tilde{\alpha}$ is the blow-up along $S$.

Let us consider the restriction $r\colon\mathcal{N}^{1}(\widetilde{W})\to\mathcal{N}^{1}(T)$. Since $\mathcal{N}_{1}(T,\widetilde{W})=\mathcal{N}_{1}(S,\widetilde{W})=\mathcal{N}_{1}(\widetilde{W})$, dually $r$ is injective.

We claim that the class $[S]\in\mathcal{N}^{1}(T)$ is not in the image $r(\mathcal{N}^{1}(\widetilde{W}))$. Let us first show that this implies the statement. Consider the commutative diagram:

[TABLE]

where the horizontal maps are the restrictions. Since $r$ is injective and $\varphi$ is an isomorphism, $\varphi(r(\mathcal{N}^{1}(\widetilde{W})))$ is a hyperplane contained in $\operatorname{Im}r^{\prime}$. Moreover $\varphi([S])$ is outside this hyperplane and $\varphi([S])=[E_{1|E_{2}}]\in\operatorname{Im}r^{\prime}$, and we conclude that $r^{\prime}$ is surjective, hence an isomorphism as $\rho_{E_{2}}=\rho_{X}$, and finally that $\mathcal{N}_{1}(E_{2},\widetilde{X})=\mathcal{N}_{1}(\widetilde{X})$.

To prove our claim, note that the canonical class $K_{A}$ is not the restriction of a class in $\mathcal{N}^{1}(\mathbb{P}^{4})$. Let $\sigma\colon\widehat{W}\to\mathbb{P}^{4}$ be the blow-up of $q_{0},\dotsc,q_{r}$, and $S^{\prime}\subset\widehat{W}$ the transform of $A$. Then $\sigma_{|S^{\prime}}$ is the blow-up of $A$ along $q_{0},\dotsc,q_{r}$, and $K_{S^{\prime}}=(\sigma_{|S^{\prime}})^{*}(K_{A})+\sum_{i=0}^{r}e_{i}$, where $e_{i}\subset S^{\prime}$ are the $(-1)$-curves, that are restrictions to $S^{\prime}$ of the exceptional divisors of $\sigma$. Hence we see that $K_{S^{\prime}}$ is not the restriction of a class in $\mathcal{N}^{1}(\widehat{W})$. Moreover $S\subset\widetilde{W}$ is contained in the open subset where the SQM $\widetilde{W}\dasharrow\widehat{W}$ is an isomorphism [CS24, Lemma 5.48], therefore $K_{S}$ is not the restriction of a class in $\mathcal{N}^{1}(\widetilde{W})$. We have $S\subset T$ and $K_{S}=(K_{T}+S)_{|S}$, hence $K_{T}+S\not\in r(\mathcal{N}^{1}(\widetilde{W}))$. On the other hand $K_{T}=(K_{\widetilde{W}}+T)_{|T}$, and we conclude that $[S]\not\in r(\mathcal{N}^{1}(\widetilde{W}))$. ∎

6. Quasi-elementary rational contractions onto surfaces

In this section we prove the following result (Th. 1.11 from the Introduction), that will be needed in the next section; it improves Theorems 4.1 and 4.5 in the case of a quasi-elementary rational contraction onto a surface.

Theorem 6.1**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and $f\colon X\dasharrow S$ a quasi-elementary rational contraction onto a surface. Then $\rho_{X}-\rho_{S}\leq 3$, and if $\rho_{X}-\rho_{S}>1$, then $f$ factors as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}Y\stackrel{{\scriptstyle h}}{{\dasharrow}}S$ where $\dim Y=3$, $g$ and $h$ are rational contractions, $g$ is special, and $h$ is elementary.

The condition $\rho_{X}\geq 7$ is necessary, see Ex. 4.6; see moreover §11.4 for an example of a Fano $4$-fold $X$ with $\rho_{X}=3$ and a quasi-elementary contraction $X\to\mathbb{P}^{2}$, with general fiber $\operatorname{Bl}_{2\mskip 1.0mu\text{\it pts}}\mathbb{P}^{2}$, that does not factor through a $3$-fold. We also note that the bound $\rho_{X}-\rho_{S}\leq 3$ is sharp, as the following examples show.

Example 6.2**.**

Let $X$ be one of the two families of Fano $4$-folds with $\rho_{X}=7$ described in §11.2. Then there are a quasi-elementary rational contraction $f\colon X\dasharrow S$ with $\dim S=2$ and $d_{f}=\rho_{X}-\rho_{S}=3$, and a rational contraction $f^{\prime}\colon X\dasharrow\mathbb{P}^{1}$ with $d_{f^{\prime}}=4$.

Example 6.3**.**

Let $X$ be the Fano model of the blow-up of $\mathbb{P}^{4}$ at $r$ general points, with $r\in\{6,7,8\}$, so that $\rho_{X}\in\{7,8,9\}$ (see §11.1). Then there is a a quasi-elementary rational contraction $f\colon X\dasharrow S$ with $\dim S=2$ and $d_{f}=\rho_{X}-\rho_{S}=2$.

Theorems 6.1 and 1.5 allow to classify complete the case where $\rho_{X}-\rho_{S}=3$, as follows.

Corollary 6.4**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and $f\colon X\dasharrow S$ a quasi-elementary rational contraction onto a surface with $\rho_{X}-\rho_{S}\geq 3$. Then $\rho_{X}-\rho_{S}=3$ and $X$ is as in Th. 1.5$(ii)$.

6.5****Overview of the proof of Th. 6.1.

We first show a preliminary property in Lemma 6.6, then we prove Th. 6.1. The proof is quite long and will take the whole section.

The bound $\rho_{X}-\rho_{S}\leq 3$ follows easily from Th. 4.5; the main work is to prove that $f$ factors through a $3$-fold, when it is not elementary. So suppose that $\rho_{X}-\rho_{S}\in\{2,3\}$ and let $\tilde{f}\colon\widetilde{X}\to S$ be a $K$-negative resolution of $f$. We analyse the possible factorizations of $f$ in elementary contractions, using Lemmas 3.16 and 3.18, and show that either the statement holds, or $\tilde{f}$ has two distinct factorizations $\widetilde{X}\stackrel{{\scriptstyle\sigma_{i}}}{{\to}}Y_{i}\stackrel{{\scriptstyle g_{i}}}{{\to}}S$ where $\sigma_{i}$ is elementary of type $(3,2)$ and $g_{i}$ is quasi-elementary and $K$-negative, for $i=1,2$. Set $E_{i}:=\operatorname{Exc}(\sigma_{i})\subset\widetilde{X}$.

The surface $S$ is smooth del Pezzo with $\rho_{S}\geq 4$; for a $(-1)$-curve $\Gamma\subset S$, $D:=\tilde{f}^{*}\Gamma$ is a fixed prime divisor in $\widetilde{X}$. Using results from Section 5 we show that $D$ is of type $(3,1)^{\text{\it sm}}$, and that up to exchanging $E_{1}$ and $E_{2}$ we have $E_{1}\cdot C_{D}=0$, $E_{2}\cdot C_{D}=E_{2}\cdot C_{E_{1}}=1$, and $E_{1}\cdot C_{E_{2}}>1$. Then by applying Lemma 6.6 we deduce that the general fiber of $g_{2}$ is $\mathbb{P}^{1}\times\mathbb{P}^{1}$. Thanks to this, if $\rho_{X}-\rho_{S}=3$, we factor again $g_{2}$ using Lemma 3.18, and we get the statement.

We finally prove that in this setting we cannot have $\rho_{X}-\rho_{S}=2$, proceeding by contradiction. We show that $Y_{1}$ is smooth and $g_{1}\colon Y_{1}\to S$ is a $\mathbb{P}^{2}$-bundle, while the general fiber of $\tilde{f}$ is $\operatorname{Bl}_{2\mskip 1.0mu\text{\it pts}}\mathbb{P}^{2}$. Then we consider a birational map $\alpha\colon S\to\mathbb{P}^{2}$, and show that $\alpha\circ g_{1}\colon Y_{1}\to\mathbb{P}^{2}$ also factors as $Y_{1}\stackrel{{\scriptstyle\text{SQM}}}{{\dasharrow}}\widetilde{Y}_{1}\stackrel{{\scriptstyle\beta}}{{\to}}\mathbb{P}^{2}\times\mathbb{P}^{2}\stackrel{{\scriptstyle h}}{{\to}}\mathbb{P}^{2}$, where $h$ is the first projection and $\beta$ is the blow-up of $\rho_{X}-3$ lines contained in distinct fibers of $h$ (compare with §11.3).

Let $B_{2}\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$ be the transform of $E_{2}\subset\widetilde{X}$; then $B_{2}$ is a prime divisor of bidegree $(2,1)$, containing the lines blown-up by $\beta$. We also consider $A\subset B_{2}$ the surface image of $E_{1}\subset\widetilde{X}$. A careful analysis of the possible configurations for $B_{2}$ and $A$ shows that this case gives a contradiction.

Lemma 6.6**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $X\dasharrow\widetilde{X}$ a SQM. Let $f\colon\widetilde{X}\to S$ be a $K$-negative contraction onto a surface that factors as $g\circ\sigma$, where $\sigma\colon\widetilde{X}\to Y$ is a divisorial elementary contraction of type $(3,2)$.

[TABLE]

Suppose also that there is a $(-1)$-curve $\Gamma\subset S_{\text{\it reg}}$ such that $D:=f^{*}(\Gamma)$ is a fixed prime divisor of type $(3,1)^{\text{\it sm}}$ with $E\cdot C_{D}>0$, where $E:=\operatorname{Exc}(\sigma)\subset\widetilde{X}$.

Then the general fiber of $g$ is $\mathbb{P}^{1}\times\mathbb{P}^{1}$.

Proof.

Since $f(C_{E})=\{\text{\it pt}\}$, we have $D\cdot C_{E}=0$, therefore $D$ and $E$ are adjacent by Lemma 5.9.

Let $\tau\colon S\to S_{1}$ be the contraction of the $(-1)$-curve $\Gamma$, and set $p:=\tau(\Gamma)\in S_{1}$; the composition $\tau\circ f\colon\widetilde{X}\to S_{1}$ contracts ${D}$ to the point $p$. We run an MMP for ${D}$, relative to $\tau\circ f$: this means that we consider $D$-negative small extremal rays in $\operatorname{NE}(\tau\circ f)$ and their flips, until we find a $D$-negative divisorial extremal ray in the relative cone. In this way we get a commutative diagram:

[TABLE]

where $\varphi$ is a sequence of $D$-negative flips, $\alpha$ is a divisorial elementary contraction with exceptional divisor the transform $\widehat{D}\subset\widehat{X}$ of $D$, and $h$ is a contraction of fiber type. Note that both $\widehat{D}$ and the indeterminacy locus of $\varphi^{-1}$ are contained in the fiber $(h\circ\alpha)^{-1}(p)$.

By Th.-Def. 5.1, $Z$ is smooth and is a SQM of a smooth Fano $4$-fold, and $\alpha$ is the blow-up of a smooth curve $C\subset Z$.

Since $[C_{E}]$ is extremal in $\operatorname{NE}(\widetilde{X})$, by Lemma 5.6 $E$ is disjoint from all exceptional lines in $\widetilde{X}$. Let us factor $\varphi$ as a sequence of $D$-negative flips:

[TABLE]

Let $\widehat{E}\subset\widehat{X}$ and $E_{i}\subset\widetilde{X}_{i}$ be the transforms of $E$. We show by induction on $i=1,\dotsc,r$ that $E_{i}$ is disjoint from all exceptional lines in $\widetilde{X}_{i}$. For $i=1$ we have already remarked this.

Suppose that the claim holds in $\widetilde{X}_{i}$, and consider the flip $\sigma_{i}\colon\widetilde{X}_{i}\dasharrow\widetilde{X}_{i+1}$. If $\sigma_{i}$ is not $K$-negative, then its indeterminacy locus is a disjoint union of exceptional lines (see Lemma 2.14), and $X_{i+1}$ will have less exceptional lines than $X_{i}$, still disjoint from $E_{i+1}$. If instead $\sigma_{i}$ is $K$-negative, then its indeterminacy locus is a finite disjoint union of exceptional planes $L_{j}\subset X_{i}$ (see Lemma 2.13) such that $D\cdot C_{L_{j}}<0$, because $\sigma_{i}$ is $D$-negative. Then $E_{i}\cap L_{j}=\emptyset$ by Lemma 5.13, thus in $X_{i+1}$ the divisor $E_{i+1}$ stays disjoint from the new exceptional lines in the indeterminacy locus of $\sigma_{i}^{-1}$.

We conclude that $\widehat{E}\subset\widehat{X}$ is disjoint from all exceptional lines in $\widehat{X}$, and by Lemma 5.6 the class $[C_{\widehat{E}}]$ generates an extremal ray of $\operatorname{NE}(\widehat{X})$, which is contracted by the map $h\circ\alpha\colon\widehat{X}\to S_{1}$. Let $\hat{\sigma}\colon\widehat{X}\to\widehat{Y}$ be the associated contraction, of type $(3,2)$ with exceptional divisor $\widehat{E}$ (see Th.-Def. 5.1).

By Lemma 5.14 we have a commutative diagram:

[TABLE]

where $\sigma^{\prime}$ is a divisorial elementary contraction of type $(3,2)$ with $\operatorname{Exc}(\sigma^{\prime})=\alpha(\widehat{E})$, $\alpha^{\prime}$ is the blow-up of a smooth point $q\in W$ with $\operatorname{Exc}(\alpha^{\prime})=\hat{\sigma}(\widehat{D})=:D^{\prime}\subset\widehat{Y}$, and $g_{1}$ is a contraction of fiber type.

Moreover $\varphi_{Y}$ is a SQM such that $\hat{\sigma}(\widehat{E})\subset\operatorname{dom}(\varphi_{Y}^{-1})$ (by Th.-Def. 5.1), therefore $\widehat{Y}\smallsetminus\operatorname{dom}(\varphi_{Y}^{-1})$ is contained in the fiber $(g_{1}\circ\alpha^{\prime})^{-1}(p)$, and the same for $D^{\prime}$. We conclude that $g\colon Y\to S$, $g_{1}\circ\alpha^{\prime}\colon\widehat{Y}\to S_{1}$, and $g_{1}\colon W\to S_{1}$, all have isomorphic general fibers.

We show that $F\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ where $F\subset W$ is the general fiber of $g_{1}$; this gives the statement.

By Lemma 2.9 both $\widehat{Y}$ and $W$ have at most isolated, locally factorial, and terminal singularities; in particular $F$ is smooth.

We show that $g_{1}$ is $K$-negative. Let $\ell\subset W$ be an irreducible curve with $-K_{W}\cdot\ell\leq 0$, and $\ell_{\widehat{Y}}\subset\widehat{Y}$ its transform. Note that $\ell_{\widehat{Y}}\not\subset D^{\prime}$ because $\alpha^{\prime}(D^{\prime})=\{q\}$, thus $D^{\prime}\cdot\ell_{\widehat{Y}}\geq 0$ and $0\geq-K_{W}\cdot\ell=(-K_{\widehat{Y}}+3D^{\prime})\cdot\ell_{\widehat{Y}}\geq-K_{\widehat{Y}}\cdot\ell_{\widehat{Y}}$. By Lemma 5.8 $\ell_{\widehat{Y}}$ is an exceptional line, disjoint from $\hat{\sigma}(\widehat{E})$, and $K_{\widehat{Y}}\cdot\ell_{\widehat{Y}}=1$ also gives $D^{\prime}\cdot\ell_{\widehat{Y}}=0$, thus $\ell_{\widehat{Y}}\cap D^{\prime}=\emptyset$.

Then the transform $\ell_{\widehat{X}}\subset\widehat{X}$ is again an exceptional line, disjoint from both $\widehat{E}$ and $\widehat{D}$. It is not difficult to see that $\ell_{\widehat{X}}$ must be contained in $\operatorname{dom}(\varphi^{-1})$, so that its transform $\ell_{\widetilde{X}}\subset\widetilde{X}$ is an exceptional line, disjoint from $D$. Since $f$ is $K$-negative, $f(\ell_{\widetilde{X}})\subset S$ is a curve. Moreover it cannot be the $(-1)$-curve $\Gamma$, otherwise $\ell_{\widetilde{X}}$ would be contained in $D$. We conclude that $\tau(f(\ell_{\widetilde{X}}))=g_{1}(\ell)\subset S_{1}$ is a curve, and $g_{1}$ is $K$-negative, therefore its general fiber $F$ is a smooth del Pezzo surface.

Suppose by contradiction that $F\not\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$. Then $F$ is covered by a family of curves of anticanonical degree $3$, thus $W$ is covered by a family of curves of anticanonical degree $3$, contracted by $g_{1}$. If $C\subset W$ is a curve of the family containing the point $q$ blown-up by $\alpha^{\prime}\colon\widehat{Y}\to W$, then every irreducible component $C_{0}$ of $C$ satisfies $-K_{W}\cdot C_{0}>0$, because $g_{1}$ is $K$-negative. Let $C_{0}$ be an irreducible component of $C$ containing $q$, and $\widehat{C}_{0}\subset\widehat{Y}$ its transform. If $-K_{W}\cdot C_{0}=1$ we get $-K_{\widehat{Y}}\cdot\widehat{C}_{0}\leq-2$, and if $-K_{W}\cdot C_{0}=3$ we get $-K_{\widehat{Y}}\cdot\widehat{C}_{0}=0$ or $-K_{\widehat{Y}}\cdot\widehat{C}_{0}\leq-3$, in any case contradicting Lemma 5.8. Therefore it must be $-K_{W}\cdot C_{0}=2$, $-K_{\widehat{Y}}\cdot\widehat{C}_{0}\leq-1$, and $\widehat{C}_{0}$ is an exceptional line by Lemma 5.8. However we must also have $C=C_{0}\cup C_{1}$ with $C_{1}$ irreducible, $-K_{W}\cdot C_{1}=1$, and $q\not\in C_{1}$. Then the transform of $C_{1}$ in $\widehat{Y}$ is an irreducible curve of anticanonical degree one intersecting an exceptional line, contradicting Lemma 5.8. Therefore $F\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$. ∎

Proof of Th. 6.1.

Let us assume that $f$ is not elementary, so that $\rho_{X}-\rho_{S}>1$.

Since $f$ is quasi-elementary, we have $d_{f}=\rho_{X}-\rho_{S}$, and $S$ is a smooth del Pezzo surface (Lemma 3.20). By Th. 4.5 we have $\rho_{X}-\rho_{S}\leq 4$, and if $\rho_{X}-\rho_{S}=4$, then it should be $S\cong\mathbb{P}^{2}$, which is impossible because $\rho_{X}\geq 7$. Therefore we have $2\leq\rho_{X}-\rho_{S}\leq 3$, in particular $\rho_{S}\geq\rho_{X}-3\geq 4$.

We are left to show that $f$ factors as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}Y\stackrel{{\scriptstyle h}}{{\dasharrow}}S$ where $\dim Y=3$, $g$ is special, and $h$ is elementary. Let us consider a $K$-negative resolution $\tilde{f}\colon\widetilde{X}\to S$ of $f$.

6.7**.**

Either we get the statement, or $\widetilde{X}$ contains two fixed prime divisors $E_{1},E_{2}$ of type $(3,2)$ such that $C_{E_{1}}$ and $C_{E_{2}}$ are contracted by $\tilde{f}$ and $(E_{1}\cdot C_{E_{2}})(E_{2}\cdot C_{E_{1}})>1$.

Proof.

We consider first the case $\rho_{X}-\rho_{S}=2$, and apply Lemma 3.18 to $f$. If we get $(i)$, we have the statement. If we get $(ii)$, then $f$ admits two factorizations $X\stackrel{{\scriptstyle\alpha_{i}}}{{\dasharrow}}W_{i}\stackrel{{\scriptstyle h_{i}}}{{\to}}S$ where $\alpha_{i}$ is a divisorial elementary rational contraction with exceptional locus dominating $S$ for $i=1,2$, so that $\alpha_{i}$ is of type $(3,2)$. Moreover $\alpha_{1}^{*}\mathcal{N}^{1}(W_{1})\neq\alpha_{2}^{*}\mathcal{N}^{1}(W_{2})$, hence $\operatorname{Exc}(\alpha_{1})\neq\operatorname{Exc}(\alpha_{2})$ by Th.-Def. 5.1.

Let $E_{i}\subset\widetilde{X}$ be the transform of $\operatorname{Exc}(\alpha_{i})$; then $E_{i}$ is a fixed prime divisor of type $(3,2)$, $E_{1}\neq E_{2}$, and $C_{E_{i}}$ is contracted by $\tilde{f}$. Moreover $\mathbb{R}_{\geq 0}[C_{E_{i}}]$ is an extremal ray of $\operatorname{NE}(\tilde{f})$ by Lemma 5.7, and finally $\operatorname{NE}(\tilde{f})=\langle[C_{E_{1}}],[C_{E_{2}}]\rangle$. Since $f$ is quasi-elementary, we have $\dim(\operatorname{NE}(\tilde{f})\cap\operatorname{mov}(X))=2$ (see [Cas13a, Prop. 2.22]), hence $(E_{1}\cdot C_{E_{2}})(E_{2}\cdot C_{E_{1}})>1$ by [Cas20, Lemma 4.6(a)].

Suppose now that $\rho_{X}-\rho_{S}=3$; we proceed similarly. By Lemma 3.16 $f$ factors as

[TABLE]

where $g$ is elementary, either divisorial or of fiber type, and $h$ is of fiber type (because $f$ is quasi-elementary). Moreover $h$ is quasi-elementary by Lemma 3.15 and $\rho_{X_{2}}-\rho_{S}=2$, therefore $\dim X_{2}\neq 3$, and we conclude that $\dim X_{2}=4$ and $g$ is birational divisorial with $\operatorname{Exc}(g)$ dominating $S$.

Now we apply Lemma 3.18 to $h$. If we get $(i)$, then $h$ factors as $X_{2}\stackrel{{\scriptstyle g_{2}}}{{\dasharrow}}Y\stackrel{{\scriptstyle h_{2}}}{{\to}}S$ where $\dim Y=3$ and both $g_{2}$ and $h_{2}$ are elementary of fiber type. Since $\operatorname{Exc}(g)$ dominates $S$, its image in $Y$ is a divisor, hence the composition $g_{2}\circ g\colon X\dasharrow Y$ is special, and we get the statement.

If instead we get $(ii)$, for $i=1,2$ we have a factorization of $h$ as

[TABLE]

where $\alpha_{i}$ is a divisorial elementary rational contraction with exceptional locus dominating $S$ for $i=1,2$, so that $\alpha_{i}$ is of type $(3,2)$. Let $E_{i}\subset\widetilde{X}$ be the transform of $\operatorname{Exc}(\alpha_{i})\subset X_{2}$. Similarly as in the previous case $\rho_{X}-\rho_{S}=2$ we see that $E_{i}$ is a fixed prime divisor of type $(3,2)$, $C_{E_{i}}$ is contracted by $\tilde{f}$, and $E_{1}\neq E_{2}$. Let moreover $G\subset\widetilde{X}$ be the transform of $\operatorname{Exc}(g)$, which is again a fixed prime divisor of type $(3,2)$.

We note that $G$ and $E_{i}$ are adjacent, because they are both contracted by the composite birational map $\widetilde{X}\stackrel{{\scriptstyle\text{SQM}}}{{\dasharrow}}X\stackrel{{\scriptstyle g}}{{\dasharrow}}X_{2}\stackrel{{\scriptstyle\alpha_{i}}}{{\dasharrow}}W_{i}$, where $W_{i}$ is $\mathbb{Q}$-factorial and $\rho_{X}-\rho_{W_{i}}=2$ (see e.g. [Cas22, proof of Lemma 4.2]). By Lemma 5.9 we conclude that $G\cdot C_{E_{i}}=E_{i}\cdot C_{G}=0$ for $i=1,2$.

Let us show that $({E}_{1}\cdot C_{{E}_{2}})({E}_{2}\cdot C_{{E}_{1}})>1$. Let $\beta\colon\widehat{X}\to W_{1}$ be a resolution of the above map $\widetilde{X}\dasharrow W_{1}$; with a slight abuse of notation, we still denote by $G$ and $E_{i}$ their transforms in $\widehat{X}$. Then $G,E_{1}$ are the exceptional divisors for $\beta$, and $\beta(E_{2})\subset W_{1}$ is a divisor that dominates $S$ under $h_{1}\colon W_{1}\to S$, therefore $\beta(E_{2})\cdot\operatorname{NE}(h_{1})>0$. Moreover $\beta^{*}(\beta(E_{2}))=E_{2}+(E_{2}\cdot C_{E_{1}})E_{1}+(E_{2}\cdot C_{G})G=E_{2}+(E_{2}\cdot C_{E_{1}})E_{1}$, and $[\beta_{*}(C_{E_{2}})]\in\operatorname{NE}(h_{1})$. Hence:

[TABLE]

and we conclude that $(E_{1}\cdot C_{E_{2}})(E_{2}\cdot C_{E_{1}})>1$. ∎

6.8**.**

Let $i\in\{1,2\}$. Since $\tilde{f}$ is $K$-negative and contracts $C_{E_{i}}$, by Lemma 5.7 $\mathbb{R}_{\geq 0}[C_{E_{i}}]$ is an extremal ray of $\operatorname{NE}(\tilde{f})$, and we have a factorization:

[TABLE]

where $\sigma_{i}$ is the contraction of $\mathbb{R}_{\geq 0}[C_{E_{i}}]$, $Y_{i}$ has at most isolated terminal and locally factorial singularities (Lemma 2.9), $\sigma_{i}(E_{i})$ dominates $S$, and $g_{i}$ is quasi-elementary (Lemma 3.15).

We also note that $g_{i}$ is $K$-negative: suppose otherwise; then by Lemma 5.8 $g_{i}$ must contract some exceptional line of $Y_{i}$, disjoint from $\sigma_{i}(E_{i})$, and this contradicts the $K$-negativity of $\tilde{f}$.

Let $F_{i}\subset Y_{i}$ be a general fiber of $g_{i}$, so that $F_{i}$ is a smooth del Pezzo surface.

6.9**.**

Let $\Gamma\subset S$ be a $(-1)$-curve (recall that $\rho_{S}\geq 4$), and set $D:=\tilde{f}^{-1}(\Gamma)$; then $D$ is a prime divisor because $\tilde{f}$ is quasi-elementary (Lemma 3.9), and it is fixed because $\Gamma$ is. Moreover $\tilde{f}^{*}(\Gamma)=mD$ for some $m\in\mathbb{Z}_{>0}$, thus $D\cdot\gamma=0$ for every $\gamma\in\operatorname{NE}(\tilde{f})$, and $D$ is the pullback of some divisor in $S$; this implies that $m=1$ and $D=\tilde{f}^{*}(\Gamma)$.

We show that $D$ is of type $(3,1)^{\text{\it sm}}$. We have $\tilde{f}(D)=\Gamma$, therefore $\dim\mathcal{N}_{1}(D,\widetilde{X})\leq 1+\rho_{X}-\rho_{S}\leq 4$ (Lemma 2.6). On the other hand we have $\delta_{X}\leq 1$ by Th. 1.9, and we deduce from Lemma 5.3 that $D$ cannot be of type $(3,2)$.

Moreover for $i=1,2$ we have $D\cdot C_{E_{i}}=0$, hence $D$ is adjacent to $E_{i}$ by Lemma 5.9; in $\widetilde{X}$ we have ${D}\cap{E}_{i}\neq\emptyset$ because ${E}_{i}$ dominates $S$, thus $\dim({D}\cap{E}_{i})=2$ and since the indeterminacy locus of the map $\widetilde{X}\dasharrow X$ is one-dimensional (Lemma 2.14$(a)$), we have $\widetilde{D}\cap\widetilde{E}_{i}\neq\emptyset$ in $X$, where $\widetilde{D},\widetilde{E}_{i}\subset X$ are the transforms of $D,E_{i}$ respectively.

Then $D$ cannot be of type $(3,0)^{\text{\it sm}}$ by Lemma 5.11. Moreover $D$ cannot be of type $(3,0)^{Q}$, otherwise by Lemma 5.12 we get a contradiction with $(E_{1}\cdot C_{E_{2}})(E_{2}\cdot C_{E_{1}})>1$. We conclude that $D$ is of type $(3,1)^{\text{\it sm}}$.

We apply Lemma 5.17. Since $(E_{1}\cdot C_{E_{2}})(E_{2}\cdot C_{E_{1}})>1$, up to exchanging $E_{1}$ and $E_{2}$ we have:

[TABLE]

Note that $E_{1}\cdot C_{E_{2}}$ and $E_{2}\cdot C_{E_{1}}$ do not depend on $D$, therefore (6.10) must hold for the pullback $D$ of every $(-1)$-curve $\Gamma\subset S$.

Since $E_{2}\cdot C_{D}>0$, we can apply Lemma 6.6 to $\tilde{f}=g_{2}\circ\sigma_{2}$ and $D$, and deduce that $F_{2}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$, where $F_{2}\subset Y_{2}$ is a general fiber of $g_{2}$.

6.11**.**

Suppose that $\rho_{X}-\rho_{S}=3$, so that $\rho_{Y_{2}}-\rho_{S}=2$, and apply Lemma 3.18 to $g_{2}\colon Y_{2}\to S$. We show that $(ii)$ cannot happen. Otherwise, $g_{2}$ factors as $Y_{2}\stackrel{{\scriptstyle\alpha}}{{\dasharrow}}W\stackrel{{\scriptstyle h}}{{\to}}S$ where $\alpha$ is elementary and divisorial with exceptional locus dominating $S$. Consider a resolution $\alpha^{\prime}\colon Y_{2}^{\prime}\to W$ of $\alpha$. Then there is a sequence of flips $\psi\colon Y_{2}\dasharrow Y_{2}^{\prime}$, relative to $g_{2}$, such that $\alpha=\alpha^{\prime}\circ\psi$.

[TABLE]

By Lemma 5.8 each flip in $\psi$ has locus contained in the smooth locus of the $4$-fold, and is either $K$-negative, or $K$-positive. Therefore the locus of each flip is a finite disjoint union of exceptional planes or lines, and such locus must be contracted to finitely many points by $g_{2}$. We conclude that $h\circ\alpha^{\prime}$ sends the indeterminacy locus $Y_{2}^{\prime}\smallsetminus\operatorname{dom}(\psi^{-1})$ to a finite set of points, therefore $\psi$ is an isomorphism on the general fiber $F_{2}$, and $h\circ\alpha^{\prime}$ has general fiber $F_{2}^{\prime}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$. However this is impossible, because $\alpha^{\prime}$ must restrict to a non-trivial birational contraction on $F_{2}^{\prime}$.

Therefore we must get $(i)$ from Lemma 3.18, and $g_{2}$ factors as $Y_{2}\stackrel{{\scriptstyle\beta}}{{\dasharrow}}Z\stackrel{{\scriptstyle k}}{{\to}}S$ with $\dim Z=3$ and both maps are elementary of fiber type. Since $E_{2}$ dominates $S$, its image in $Z$ is a surface, $\beta\circ\sigma_{2}\colon\widetilde{X}\dasharrow Z$ is special, and we get the statement.

6.12**.**

We are left with the case where $\rho_{X}-\rho_{S}=2$ and $g_{1}$ and $g_{2}$ are elementary. We show that this case does not happen, as it leads to a contradiction.

Note that $\operatorname{NE}(\tilde{f})=\langle[C_{E_{1}}],[C_{E_{2}}]\rangle$ (see 6.8).

For $i=1,2$ let $r_{i}$ be the degree of $(g_{i})_{|\sigma_{i}(E_{i})}\colon\sigma_{i}(E_{i})\to S$. Then $\sigma_{i}$ blows-up $F_{i}$ in $r_{i}$ points, so if $F\subset\widetilde{X}$ is a general fiber of $\tilde{f}$, then $F$ is a del Pezzo surface with

[TABLE]

We show that:

[TABLE]

Indeed write $(E_{i})_{|F}=\Gamma_{i1}+\cdots+\Gamma_{ir_{i}}$; then $\Gamma_{ij}\equiv C_{E_{i}}$ for every $j$, and

[TABLE]

Now (6.13), (6.14), (6.10), and $\rho_{F_{2}}=2$, give:

[TABLE]

and we get $r_{2}=1$, $F_{1}\cong\mathbb{P}^{2}$, $\rho_{F}=3$, and $r_{1}=E_{1}\cdot C_{E_{2}}=2$.

6.15**.**

We show that:

[TABLE]

Indeed $(-K_{\widetilde{X}}-2E_{1}-3E_{2})\cdot C_{E_{i}}=0$ for $i=1,2$, thus $-K_{\widetilde{X}}-2E_{1}-3E_{2}=\tilde{f}^{*}M$ for some divisor $M$ on $S$.

By Lemma 5.19, for every $(-1)$-curve $\Gamma\subset S$ there exists an exceptional line $\ell\subset\widetilde{X}$ such that $\tilde{f}(\ell)=\Gamma$. We have $E_{i}\cdot\ell=0$ for $i=1,2$ by Lemma 5.6, thus

[TABLE]

and this implies that $M\cdot\Gamma=-1=K_{S}\cdot\Gamma$ for every $(-1)$-curve $\Gamma\subset S$. Since $S$ is a del Pezzo surface with $\rho_{S}=\rho_{X}-2\geq 5$, $\mathcal{N}_{1}(S)$ is generated by classes of $(-1)$-curves, and we conclude that $M=K_{S}$.

6.17**.**

Set $B_{2}^{\prime}:=\sigma_{1}(E_{2})\subset Y_{1}$. Applying the pushforward $(\sigma_{1})_{*}$ to (6.16) we get:

[TABLE]

Restricting to $F_{1}\cong\mathbb{P}^{2}$ gives $\mathcal{O}_{Y_{1}}(B_{2}^{\prime})_{|F_{1}}\cong\mathcal{O}_{\mathbb{P}^{2}}(1)$, in particular $B_{2}^{\prime}$ is $g_{1}$-ample. If $A$ is an ample divisor on $S$, then for $r\gg 0$ the divisor $B_{2}^{\prime}+rg_{1}^{*}(A)$ is ample on $Y_{1}$, and we still have $\mathcal{O}_{Y_{1}}(B_{2}^{\prime}+rg_{1}^{*}(A))_{|F_{1}}\cong\mathcal{O}_{\mathbb{P}^{2}}(1)$. By [BS95, Prop. 3.2.1], this implies that $g_{1}$ is a $\mathbb{P}^{2}$-bundle and $Y_{1}$ is smooth, so that $Y_{1}$ is a SQM of a smooth Fano $4$-fold $Y^{\prime}_{1}$ with $\rho_{Y_{1}^{\prime}}\geq 6$ (see Th.-Def. 5.1).

6.19**.**

Let us choose a birational morphism $\alpha\colon S\to\mathbb{P}^{2}$, which blows-up the points $p_{1},\dotsc,p_{m}\in\mathbb{P}^{2}$ with $m=\rho_{S}-1=\rho_{X}-3\geq 4$. For $i\in\{1,\dotsc,m\}$ let $\Gamma_{i}\subset S$ be the $(-1)$-curve over $p_{i}$, and set $D_{i}:=\tilde{f}^{*}\Gamma_{i}\subset\widetilde{X}$. Then $D_{i}$ is a fixed prime divisor of type $(3,1)^{\text{\it sm}}$ with $D_{i}\cdot C_{E_{1}}=D_{i}\cdot C_{E_{2}}=E_{1}\cdot C_{D_{i}}=0$ and $E_{2}\cdot C_{D_{i}}=1$, for every $i=1,\dotsc,m$ (see 6.9). Moreover for $i\neq j$ we have $D_{i}\cap D_{j}=\emptyset$, therefore $D_{i}\cdot C_{D_{j}}=0$. By Rem. 5.20 this implies that $\langle[E_{1}],[D_{1}],\dotsc,[D_{m}]\rangle$ is a fixed face of $\operatorname{Eff}(\widetilde{X})\cong\operatorname{Eff}(X)$, of dimension $\rho_{X}-2$.

For $i\in\{1,\dotsc,m\}$ set $D_{i}^{\prime}:=\sigma_{1}(D_{i})\subset Y_{1}$; the divisors $D_{1}^{\prime},\dotsc,D_{m}^{\prime}$ are contracted to points by $\alpha\circ g_{1}$. We run a MMP in $Y_{1}$ for $D_{1}^{\prime}+\cdots+D_{m}^{\prime}$, relative to $\alpha\circ g_{1}$, and get a commutative diagram:

[TABLE]

where $\xi$ is a SQM, $\beta$ is birational with exceptional divisors the transforms $D_{1}^{\prime\prime},\dotsc,D_{m}^{\prime\prime}\subset\widetilde{Y}_{1}$ of $D_{1}^{\prime},\dotsc,D_{m}^{\prime}\subset Y_{1}$, and $h\colon W\to\mathbb{P}^{2}$ is an elementary contraction with general fiber $F_{W}\cong\mathbb{P}^{2}$. Moreover it follows from Rem. 5.20 that $\beta$ is the blow-up of $m$ pairwise disjoint smooth curves $C_{1},\dotsc,C_{m}\subset W$, and $W$ is smooth and a SQM of a smooth Fano $4$-fold.

Let $\widetilde{B}_{2}^{\prime}\subset\widetilde{Y}_{1}$ and $B_{2}\subset W$ be the transforms of $E_{2}$, and set $H=h^{*}\mathcal{O}_{\mathbb{P}^{2}}(1)\in\operatorname{Pic}(W)$. We have $-K_{S}=\alpha^{*}(-K_{\mathbb{P}^{2}})-\sum_{i=1}^{m}\Gamma_{i}$, and by (6.18) $-K_{Y_{1}}+g_{1}^{*}\alpha^{*}(-K_{\mathbb{P}^{2}})-\sum_{i=1}^{m}D_{i}^{\prime}=3B_{2}^{\prime}$, which in $\widetilde{Y}_{1}$ gives $-K_{\widetilde{Y}_{1}}+3\beta^{*}(H)-\sum_{i=1}^{m}D_{i}^{\prime\prime}=3\widetilde{B}_{2}^{\prime}$. Finally via the pushforward $\beta_{*}$ we get:

[TABLE]

This implies that $B_{2|F_{W}}\cong\mathcal{O}_{\mathbb{P}^{2}}(1)$, therefore using again [BS95, Prop. 3.2.1] as in 6.17 we get $W=\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{E})$, where $\mathcal{E}$ is a rank $3$ vector bundle on $\mathbb{P}^{2}$. Recall that $W$ is a SQM of a smooth Fano $4$-fold. Moreover, since $-K_{W}$ is divisible by $3$ in $\operatorname{Pic}(W)$, $W$ cannot contain exceptional lines nor exceptional planes, so the second elementary contraction of $W$ cannot be small (see Lemma 2.14 and Lemma 2.13), and $W$ is Fano of index $3$. This also implies that $c_{1}(\mathcal{E})\equiv 0\mod 3$, and by [SW90] we conclude that $W\cong\mathbb{P}^{2}\times\mathbb{P}^{2}$ . We can assume that $h\colon\mathbb{P}^{2}\times\mathbb{P}^{2}\to\mathbb{P}^{2}$ is the first projection.

Let us consider the curves $C_{1},\dotsc,C_{m}\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$ blown-up by $\beta$. We have that $h(C_{i})=p_{i}\in\mathbb{P}^{2}$ with $p_{i}\neq p_{j}$ for $i\neq j$, so that $C_{i}$ is a smooth curve in $h^{-1}(p_{i})=\mathbb{P}^{2}$. If $L\subset h^{-1}(p_{i})$ is a general line, we have $-K_{\mathbb{P}^{2}\times\mathbb{P}^{2}}\cdot L=3$ and $L$ must meet $C_{i}$, hence $D_{i}^{\prime\prime}\cdot\widetilde{L}>0$, where $\widetilde{L}\subset\widetilde{Y}_{1}$ is its transform. On the other hand $\widetilde{L}$ cannot be an exceptional line, because $\widetilde{Y}_{1}$ can contain at most finitely many of them (see Lemma 2.14), therefore $-K_{\widetilde{Y}_{1}}\cdot\widetilde{L}>0$, which implies that $D_{i}^{\prime\prime}\cdot\widetilde{L}=1$ and $C_{i}\cdot_{\mathbb{P}^{2}}L=1$, so that $C_{i}$ is itself a line in $h^{-1}(p_{i})=\mathbb{P}^{2}$.

We have therefore shown that $Y_{1}$ is a SQM of a blow-up of $\mathbb{P}^{2}\times\mathbb{P}^{2}$ along $m$ lines contained in different fibers of the first projection (see Ex. 11.3); in particular $\widetilde{Y}_{1}\smallsetminus\operatorname{dom}(\xi^{-1})$ is the disjoint union of $m$ exceptional planes, given by the transforms of $h^{-1}(p_{1}),\dotsc,h^{-1}(p_{m})$, and $Y_{1}\smallsetminus\operatorname{dom}\xi$ is the disjoint union of $m$ exceptional lines.

6.21**.**

Consider the surface $\sigma_{1}(E_{1})\subset B_{2}^{\prime}\subset Y_{1}$, and let $\widetilde{A}\subset\widetilde{B}_{2}^{\prime}\subset\widetilde{Y}_{1}$ and $A\subset B_{2}\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$ be its transforms.

Let $i\in\{1,\dotsc,m\}$. It follows from Lemma 5.21 that, whenever $A\cap C_{i}\neq\emptyset$, $A$ and $C_{i}$ intersect transversally at finitely many points, where $A$ is smooth.

From (6.20) we see that $B_{2}$ has bidegree $(2,1)$ in $\mathbb{P}^{2}\times\mathbb{P}^{2}$, hence $h_{|B_{2}}\colon B_{2}\to\mathbb{P}^{2}$ is generically a $\mathbb{P}^{1}$-bundle. Moreover $C_{i}\subset B_{2}$, because $E_{2}\cdot C_{D_{i}}>0$ (see 6.19); on the other hand $B_{2}$ cannot contain the whole fiber $h^{-1}(p_{i})$, otherwise $\widetilde{B}_{2}^{\prime}\subset\widetilde{Y}_{1}$ would contain an exceptional plane in the indeterminacy locus of $\xi^{-1}$, $\widetilde{B}_{2}\subset{Y}_{1}$ would intersect an exceptional line, and $E_{2}\subset\widetilde{X}$ would intersect an exceptional line too, a contradiction (see Lemmas 5.8 and 5.6). Therefore $C_{i}=(h_{|B_{2}})^{-1}(p_{i})$, $h_{|B_{2}}$ is a $\mathbb{P}^{1}$-bundle over $p_{i}$, and $C_{i}\subset(B_{2})_{\text{\it reg}}$.

Finally we note that since $E_{1}\cdot C_{E_{2}}=2$, we have $A\cdot_{B_{2}}C_{i}=2$, and we conclude that $A$ and $C_{i}$ intersect at exactly two points.

We set $C_{i}=\{p_{i}\}\times L_{i}$ with $L_{i}\subset\mathbb{P}^{2}$ a line.

6.22**.**

Let $i,j,k\in\{1,\dotsc,m\}$ with $i<j<k$; we show that $L_{i}\cap L_{j}\cap L_{k}=\emptyset$ (in particular this also shows that $L_{i}\neq L_{j}$). By contradiction, suppose otherwise, and let $q\in L_{i}\cap L_{j}\cap L_{k}$.

Let $h^{\prime}\colon\mathbb{P}^{2}\times\mathbb{P}^{2}\to\mathbb{P}^{2}$ be the second projection, and in $(h^{\prime})^{-1}(q)=\mathbb{P}^{2}$ consider a general conic $\Gamma$ through $(p_{i},q)$, $(p_{j},q)$, and $(p_{k},q)$ (note that $p_{i},p_{j},p_{k}$ are not collinear in $\mathbb{P}^{2}$, because $\operatorname{Bl}_{p_{i},p_{j},p_{k}}\mathbb{P}^{2}$ is a del Pezzo surface, see 6.19). Then $-K_{\mathbb{P}^{2}\times\mathbb{P}^{2}}\cdot\Gamma=6$ and if $\widetilde{\Gamma}\subset\widetilde{Y}_{1}$ is its transform, we have $D_{i}^{\prime\prime}\cdot\widetilde{\Gamma}>0$, $D_{j}^{\prime\prime}\cdot\widetilde{\Gamma}>0$, $D_{k}^{\prime\prime}\cdot\widetilde{\Gamma}>0$, hence $-K_{\widetilde{Y}_{1}}\cdot\widetilde{\Gamma}\leq 0$, so that $\widetilde{\Gamma}$ is an exceptional line by Lemma 2.14. However this is impossible, because $\widetilde{Y}_{1}$ contains at most finitely many exceptional lines (again by Lemma 2.14), while $\Gamma$ moves in a positive dimensional family.

6.23**.**

Set $\Lambda:=h^{\prime}(A)\subset\mathbb{P}^{2}$, and let $i\in\{1,\dotsc,m\}$. We have $h^{\prime}(C_{i}\cap A)\subset L_{i}\cap\Lambda$, in particular $L_{i}\cap\Lambda\neq\emptyset$ because $C_{i}\cap A\neq\emptyset$. Moreover $\Lambda$ cannot be a point, otherwise it should lie in $L_{1}\cap L_{2}\cap L_{3}$ which is empty by 6.22 (recall that $m\geq 4$).

Let $q\in L_{i}\cap\Lambda$; we show that $(p_{i},q)\in A$. Suppose otherwise, and let $L$ be a line in $(h^{\prime})^{-1}(q)$ containing $(p_{i},q)$ and meeting $A$. Then $-K_{\mathbb{P}^{2}\times\mathbb{P}^{2}}\cdot L=3$, and if $\widetilde{L}\subset\widetilde{Y}_{1}$ is its transform, we have $-K_{\widetilde{Y}_{1}}\cdot\widetilde{L}\leq 1$, $\widetilde{L}\cap\widetilde{A}\neq\emptyset$, and $\widetilde{L}\not\subset\widetilde{A}$, contradicting Lemma 5.8. Therefore $(p_{i},q)\in A$.

Since we also have $(p_{i},q)\in C_{i}$, this implies that $h^{\prime}(C_{i}\cap A)=L_{i}\cap\Lambda$, hence $L_{i}$ intersects $\Lambda$ set-theoretically in two points (see 6.21). We conclude that $\Lambda$ is an irreducible curve of degree at least $2$.

6.24**.**

Let $\tau\colon Z\to\mathbb{P}^{2}\times\mathbb{P}^{2}$ be the blow-up of $A$, with exceptional divisor $E_{1}^{\prime}\subset Z$; we note that $Z$ is a smooth Fano $4$-fold with $\rho_{Z}=3$ (see Rem. 5.20), and there is a birational contraction $\varphi\colon X\dasharrow Z$.

Let $q\in\Lambda$ be a general point, $\Gamma\subset\mathbb{P}^{2}\times\{q\}$ a general line, and $\widetilde{\Gamma}\subset Z$ its transform. Note that $A$ has at most isolated singularities (Lemma 2.9), therefore $(h^{\prime}_{|A})^{-1}(q)\subset\mathbb{P}^{2}\times\{q\}$ is a smooth curve; let $d$ be its degree. Then $E_{1}^{\prime}\cdot\widetilde{\Gamma}=d$, hence $-K_{Z}\cdot\widetilde{\Gamma}=3-d\geq 1$, and we get $d\in\{1,2\}$.

Let $G_{Z}\subset Z$ be the transform of $G:=\mathbb{P}^{2}\times\Lambda\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$. We have $G\cdot\Gamma=0$ and $\tau^{*}(G)=G_{Z}+mE_{1}^{\prime}$ with $m\geq 1$, therefore $G_{Z}\cdot\widetilde{\Gamma}=-md<0$.

If $d=2$, we see that $G_{Z}$ is a fixed prime divisor covered by rational curves of anticanonical degree one. Consider the birational contraction $\varphi\colon X\dasharrow Z$, and let $G_{X}\subset X$ be the transform of $G_{Z}$. Then the indeterminacy locus of $\varphi^{-1}\colon Z\dasharrow X$ must be disjoint from the transform of $\mathbb{P}^{2}\times\{q\}$ (see [CS24, Prop. 2.10]), so that $G_{X}$ is again a fixed prime divisor covered by rational curves $C$ with $-K_{X}\cdot C=1$. By [Cas17, Lemma 2.18] $G_{X}$ must be of type $(3,2)$, but this contradicts $G_{X}\cdot C=-2m\leq-2$.

6.25**.**

Therefore $d=1$, and $A$ is $\mathbb{P}^{1}$-bundle over $\Lambda$; since $A$ has isolated singularities, this implies that $\Lambda$ is smooth.

Recall that $B_{2}$ has bidegree $(2,1)$ in $\mathbb{P}^{2}\times\mathbb{P}^{2}$, hence $h^{\prime}_{|B_{2}}\colon B_{2}\to\mathbb{P}^{2}$ is generically a conic bundle.

We notice that $h^{\prime}_{|B_{2}}$ does not have $2$-dimensional fibers, otherwise the fibers of $h_{|B_{2}}$, as curves in $\mathbb{P}^{2}$, should all have a common point. However the curves $C_{i}$ are fibers of $h_{|B_{2}}$ (see 6.21), and by 6.22 there is no common point on the lines $L_{i}$’s. Therefore $h^{\prime}_{|B_{2}}\colon B_{2}\to\mathbb{P}^{2}$ is a conic bundle; let $\Delta\subset\mathbb{P}^{2}$ be its discriminant locus, parametrising singular conics.

Since the general fiber of $h^{\prime}_{|A}$ is a line, we have $\Lambda\subset\Delta$, and $(h^{\prime}_{|B_{2}})^{*}(\Lambda)=A+A^{\prime}$ where $A^{\prime}$ is a prime divisor that dominates $\Lambda$ (possibly $A^{\prime}=A$), such that the general fiber of $h^{\prime}_{|A^{\prime}}$ is a line too.

This implies that $B_{2}$ must be singular, otherwise $h^{\prime}_{|B_{2}}$ would be an elementary contraction of a smooth Fano $3$-fold, and $A\cdot C=0$ for the general fiber $C\subset B_{2}$, so that $A$ should be the pullback of a divisor in $\mathbb{P}^{2}$, but it is not. Since $B_{2}$ is singular, $h_{|B_{2}}\colon B_{2}\to\mathbb{P}^{2}$ must have a $2$-dimensional fiber $F=\{p_{0}\}\times\mathbb{P}^{2}\subset B_{2}$, and all the fibers of the conic bundle $h^{\prime}_{B_{2}}$ contain the point $p_{0}$.

6.26**.**

Recall that $\deg\Lambda\geq 2$ by 6.23. Suppose first that $\Lambda$ is a conic, and recall that $C_{1}\subset(B_{2})_{\text{\it reg}}$ and $A\cdot_{B_{2}}C_{1}=2$ (see 6.21). Then $2=\Lambda\cdot_{\mathbb{P}^{2}}L_{1}=(h^{\prime}_{|B_{2}})^{*}(\Lambda)\cdot_{B_{2}}C_{1}=(A+A^{\prime})\cdot_{B_{2}}C_{1}=2+A^{\prime}\cdot_{B_{2}}C_{1}$, therefore $A^{\prime}\cdot_{B_{2}}C_{1}=0$. We also have $C_{1}\not\subset A^{\prime}$, because $L_{1}\not\subset\Lambda$, and we conclude that $A^{\prime}\cap C_{1}=\emptyset$, namely $A^{\prime}\cap(h_{|B_{2}})^{-1}(p_{1})=\emptyset$, and $p_{1}\not\in h(A^{\prime})$.

Then $h(A^{\prime})\subsetneq\mathbb{P}^{2}$, so that $A^{\prime}=L_{0}\times\Lambda$ where $L_{0}$ is a fixed line. Note that $B_{2}$ is the total space of a net of conics; let $\Gamma\subset\mathbb{P}^{2}$ be the subvariety parametrising conics of the net that have $L_{0}$ as a component. It is not difficult to see that $\Gamma$ must be linear, and since it contains the conic $\Lambda$, we get $\Gamma=\mathbb{P}^{2}$, but this is impossible because $B_{2}$ is irreducible. Therefore we conclude that $\deg\Lambda>2$.

6.27**.**

If $\Delta\subsetneq\mathbb{P}^{2}$, then $\Delta$ is a cubic curve and $\Lambda\subset\Delta$, therefore $\Lambda=\Delta$ and $\Delta$ is smooth (see 6.25). A local computation shows that this implies that $B_{2}$ is smooth too (see e.g. [CS24, Rem. 3.9 and its proof]), but we have already excluded this (see 6.25).

Therefore $\Delta=\mathbb{P}^{2}$ and every fiber of $h^{\prime}_{|B_{2}}$ is singular. Then it is not difficult to see that either all conics of the net have a fixed component, or they are all singular at the same point $p_{0}$ (see for instance [Wal77, Table 1]), and since $B_{2}$ is irreducible, we must be in the second case. The fibers of $h^{\prime}_{|A}$ are components of fibers of $h^{\prime}_{|B_{2}}$, hence they all contain $p_{0}$, and $F\cap A$ is a section of $h^{\prime}_{|A}$, where $F=\{p_{0}\}\times\mathbb{P}^{2}$. Therefore $F\cap A\cong\Lambda$, and $F\cap A$ has degree $\geq 3$. Let $L\subset F$ be a general line, and $\widetilde{L}\subset Z$ its transform (see 6.24). Then $E_{1}^{\prime}\cdot\widetilde{L}\geq 3$ and $-K_{\mathbb{P}^{2}\times\mathbb{P}^{2}}\cdot L=3$, thus $-K_{Z}\cdot\widetilde{L}\leq 0$, a contradiction. This concludes the proof of Th. 6.1. ∎

7. Fixed divisors of type $(3,2)$

In this section we show the following result, which implies Th. 1.10 from the Introduction.

Theorem 7.1**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and $E\subset X$ a fixed prime divisor of type $(3,2)$ such that $\mathcal{N}_{1}(E,X)\subsetneq\mathcal{N}_{1}(X)$.

Then $\rho_{X}\leq 9$, $X$ has an elementary rational contraction onto a $3$-fold, and $E$ is of type $(3,2)^{\text{\it sm}}$. More precisely there is a commutative diagram

[TABLE]

where:

1. $\bullet$

$W^{\prime}$* is a smooth Fano $4$-fold and $\sigma^{\prime}$ is of type $(3,2)^{\text{\it sm}}$, with exceptional divisor $E$;* 2. $\bullet$

$\xi$* and $W^{\prime}\dasharrow W$ are SQM’s, and $E\subset\operatorname{dom}(\xi)$;* 3. $\bullet$

$Y$* is a smooth weak Fano101010Namely $-K_{Y}$ is nef and big. $3$-fold and $\tilde{f}$ is an elementary contraction and a conic bundle, finite on $\xi(E)$;* 4. $\bullet$

$Z$* is a smooth del Pezzo surface.*

The bound $\rho_{X}\leq 9$ improves the bound $\rho_{X}\leq 12$ shown in [Cas17, Prop. 5.32]; we do not know if it is sharp. Let us also note that, up to flops, there are only three possibilities for the $3$-fold $Y$, see [CS24, Cor. 6.4].

7.3****Overview of the proof of Th. 7.1.

We consider the cone $\operatorname{Mov}(X)^{\vee}\subset\mathcal{N}_{1}(X)$, dual of the cone of movable divisors, that has $\langle[C_{E}]\rangle$ as a face of dimension one (see Rem. 5.5). The main step is to show that there exists a two-dimensional face $\langle[C_{E}]\rangle+\alpha$ of $\operatorname{Mov}(X)^{\vee}$ with $\alpha\subset\operatorname{mov}(X)$ (in particular this implies that $X$ has an elementary rational contraction of fiber type, see again Rem. 5.5). To prove this, we work by contradiction, and show that otherwise we can produce a special rational contraction of fiber type $\psi\colon X\dasharrow T$, with $\rho_{X}-\rho_{T}=2$, and contracting $C_{E}$. Then we reach a contradiction by applying Lemma 5.22 when $\dim T=3$, and Th. 6.1 when $\dim T=2$.

Once we have the one-dimensional face $\alpha$ of $\operatorname{Mov}(X)^{\vee}$ as above, this gives an elementary rational contraction of fiber type $X\dasharrow Y$. If $X\to W^{\prime}$ is the elementary contraction of type $(3,2)$ with exceptional divisor $E$, since $\langle[C_{E}]\rangle+\alpha$ is a face of $\operatorname{Mov}(X)^{\vee}$, we see that there are elementary rational contractions $Y\dasharrow Z$ and $W^{\prime}\dasharrow Z$ onto the same target. Then by analysing the maps, and using that $\mathcal{N}_{1}(E,X)\subsetneq\mathcal{N}_{1}(X)$, we get the statement.

Proof of Th. 7.1.

Since $X$ is not a product of surfaces and $\rho_{X}\geq 7$, we have $\delta_{X}\leq 1$ by Th. 1.9. Therefore $\delta_{X}=1$ and $\mathcal{N}_{1}(E,X)$ is a hyperplane in $\mathcal{N}_{1}(X)$.

Consider the cone $\operatorname{Mov}(X)^{\vee}\subset\mathcal{N}_{1}(X)$, dual of the cone of movable divisors; we have $\operatorname{Mov}(X)^{\vee}=\langle[C_{D}]\rangle_{D\text{ fixed}}+\operatorname{mov}(X)$, and $\langle[C_{E}]\rangle$ is a one-dimensional face of $\operatorname{Mov}(X)^{\vee}$, see Rem. 5.5.

7.4**.**

There exists a one-dimensional face $\alpha$ of $\operatorname{Mov}(X)^{\vee}$, contained in $\operatorname{mov}(X)$, such that $\langle[C_{E}]\rangle+\alpha$ is a face of $\operatorname{Mov}(X)^{\vee}$.

Proof.

We proceed by contradiction, and assume that every $2$-dimensional face of $\operatorname{Mov}(X)^{\vee}$ containing $\langle[C_{E}]\rangle$ has the form $\langle[C_{E}],[C_{D}]\rangle$ for some fixed prime divisor $D\subset X$.

Then as in [Cas17, proof of Prop. 5.32] we show that there exists a $2$-dimensional face $\langle[C_{E}],[C_{E^{\prime}}]\rangle$ of $\operatorname{Mov}(X)^{\vee}$ where $E^{\prime}$ is a fixed prime divisor of type $(3,2)$ such that $[C_{E^{\prime}}]\not\in\mathcal{N}_{1}(E,X)$, $E\cap E^{\prime}\neq\emptyset$, $E\cdot C_{E^{\prime}}>0$, and $E^{\prime}\cdot C_{E}>0$. In particular $\langle[C_{E}],[C_{E^{\prime}}]\rangle\cap\operatorname{mov}(X)\neq\{0\}$ by [Cas20, Lemma 4.6(a)].

The $2$-dimensional face $\langle[C_{E}],[C_{E^{\prime}}]\rangle$ of $\operatorname{Mov}(X)^{\vee}$ corresponds by duality to the codimension $2$ face $\eta:=\operatorname{Mov}(X)\cap C_{E}^{\perp}\cap C_{E^{\prime}}^{\perp}$ of $\operatorname{Mov}(X)$. Note that $\eta\subset\partial\operatorname{Eff}(X)$, because $\langle[C_{E}],[C_{E^{\prime}}]\rangle\cap\operatorname{mov}(X)\neq\{0\}$. Let $\eta_{0}\in\operatorname{MCD}(X)$ be a cone of dimension $\rho_{X}-2$ with $\eta_{0}\subset\eta$, and $\psi\colon X\dasharrow T$ the rational contraction such that $\eta_{0}=\psi^{*}(\operatorname{Nef}(T))$; then $\rho_{T}=\rho_{X}-2\geq 5$, $\psi$ is of fiber type, and $\dim T\in\{2,3\}$.

We show that $\psi$ is special.

Let $\tau_{\psi}$ be the minimal face of $\operatorname{Eff}(X)$ containing $\eta$ (see Def.-Rem. 3.2). By Lemma 3.10, we need to show that $\tau_{\psi}\cap\operatorname{Mov}(X))=\eta$.

Note that $\tau_{\psi}\cap\operatorname{Mov}(X)$ is a proper face of $\operatorname{Mov}(X)$ and contains $\eta$, therefore if $\tau_{\psi}\cap\operatorname{Mov}(X))\supsetneq\eta$, then $\tau_{\psi}\cap\operatorname{Mov}(X)$ must be a facet of $\operatorname{Mov}(X)$ containing $\eta$, and contained in $\partial\operatorname{Eff}(X)$.

On the other hand $\eta$ is contained in exactly two facets of $\operatorname{Mov}(X)$, that are $\operatorname{Mov}(X)\cap C_{E}^{\perp}$ and $\operatorname{Mov}(X)\cap C_{E^{\prime}}^{\perp}$, and both of them are not contained in $\partial\operatorname{Eff}(X)$. Therefore $\tau_{\psi}\cap\operatorname{Mov}(X)$ cannot be a facet, it must be $\tau_{\psi}\cap\operatorname{Mov}(X)=\eta$, and $\psi$ is special.

Let

[TABLE]

be a $K$-negative resolution of $\psi$, and let $\widetilde{E},\widetilde{E}^{\prime}\subset\widetilde{X}$ be the transforms of $E,E^{\prime}$. Since $\eta_{0}\subset C_{E}^{\perp}\cap C_{E^{\prime}}^{\perp}$, $\tilde{\psi}$ contracts both $C_{\widetilde{E}}$ and $C_{\widetilde{E}^{\prime}}$, and by Lemma 5.7 we have $\operatorname{NE}(\tilde{\psi})=\langle[C_{\widetilde{E}}],[C_{\widetilde{E}^{\prime}}]\rangle$. In particular, since $\operatorname{NE}(\tilde{\psi})$ does not contain small extremal rays, there are no flips relative to $\tilde{\psi}$, and $\tilde{\psi}\colon\widetilde{X}\to T$ is the unique resolution of $\psi$.

We have $\dim\mathcal{N}_{1}(\widetilde{E},\widetilde{X})=\dim\mathcal{N}_{1}(E,X)=\rho_{X}-1$ by Lemma 5.3. If $\dim T=3$, then $\mathcal{N}_{1}(\widetilde{E},\widetilde{X})\subsetneq\mathcal{N}_{1}(\widetilde{X})$ contradicts Lemma 5.22, therefore $\dim T=2$.

Since $\psi$ is special, the images of $\widetilde{E},\widetilde{E}^{\prime}$ in $T$ are either a curve, or $T$ itself. If $\tilde{\psi}(\widetilde{E})$ is a curve in $T$, then $\dim\mathcal{N}_{1}(\widetilde{E},\widetilde{X})\leq 1+\rho_{X}-\rho_{T}=3$ (see Lemma 2.6), a contradiction, and similarly for $\widetilde{E}^{\prime}$. Therefore $\tilde{\psi}(\widetilde{E})=\tilde{\psi}(\widetilde{E}^{\prime})=T$ and $\tilde{\psi}$ is quasi-elementary, because the general fiber will contain curves $C_{\widetilde{E}}$, $C_{\widetilde{E}^{\prime}}$.

Then Th. 6.1 implies that $\psi$ factors as the composition of two rational elementary contractions $X\stackrel{{\scriptstyle f}}{{\dasharrow}}Y\stackrel{{\scriptstyle g}}{{\dasharrow}}T$ where $\dim Y=3$. Up to composing with a SQM of $Y$, we can assume that $g$ is regular. By taking a resolution $\hat{f}\colon\widehat{X}\to Y$ of $f$, we get also a resolution $g\circ\hat{f}\colon\widehat{X}\to T$ of $\psi$, hence it must be $\widehat{X}=\widetilde{X}$ and $\tilde{\psi}=g\circ\hat{f}$. This implies that $\operatorname{NE}(\hat{f})$ is an extremal ray of $\operatorname{NE}(\tilde{\psi})$, which gives a contradiction, because $\operatorname{NE}(\tilde{\psi})=\langle[C_{\widetilde{E}}],[C_{\widetilde{E}^{\prime}}]\rangle$. ∎

7.5**.**

Let $\langle[C_{E}]\rangle+\alpha$ be a $2$-dimensional face of $\operatorname{Mov}(X)^{\vee}$, with $\alpha$ one-dimensional face of $\operatorname{mov}(X)$ (see 7.4). Dually this means that both $C_{E}^{\perp}\cap\operatorname{Mov}(X)$ and $\alpha^{\perp}\cap\operatorname{Mov}(X)$ are facets of $\operatorname{Mov}(X)$, and they intersect in a face $\eta$ of $\operatorname{Mov}(X)$ of dimension $\rho_{X}-2$.

Let us choose two cones $\eta_{0},\tau\in\operatorname{MCD}(X)$ as follows: $\eta_{0}\subset\eta$ of dimension $\rho_{X}-2$, and $\tau\subset\alpha^{\perp}\cap\operatorname{Mov}(X)$ of dimension $\rho_{X}-1$ and containing $\eta_{0}$.

Let $f\colon X\dasharrow Y$ be the elementary rational contraction with $\tau=f^{*}(\operatorname{Nef}(Y))$; $f$ is of fiber type, because $\alpha\subset\partial\operatorname{mov}(X)$, hence $\tau\subset\partial\operatorname{Eff}(X)$. Since $\eta_{0}\subset\tau$, there is a contraction $g\colon Y\to Z$ such that $(g\circ f)^{*}(\operatorname{Nef}(Z))=\eta_{0}$ (see Rem. 2.2), and $g$ is elementary because $\dim\eta_{0}=\rho_{X}-2$. Note that since $\rho_{Z}=\rho_{X}-2\geq 5$, we have $\dim Z\in\{2,3\}$.

Let $X\stackrel{{\scriptstyle\xi}}{{\dasharrow}}\widetilde{X}\stackrel{{\scriptstyle\tilde{f}}}{{\to}}Y$ be a $K$-negative resolution of $f$ and $\varphi\colon\widehat{X}\to Z$ a $K$-negative resolution of $g\circ f$. We have $\eta_{0}\subset C_{E}^{\perp}$, thus $\varphi$ contracts $C_{E}$, moreover $\varphi$ is $K$-negative: by Lemma 5.7 it factors through an elementary contraction $\sigma\colon\widehat{X}\to W$, of type $(3,2)$, with exceptional divisor the transform $\widehat{E}\subset\widehat{X}$ of $E$. We have a diagram:

[TABLE]

We also note that $\dim\mathcal{N}_{1}(\widehat{E},\widehat{X})=\dim\mathcal{N}_{1}(E,X)=\rho_{X}-1$ by Lemma 5.3.

7.6**.**

Since $\sigma$ is of type $(3,2)$ and $h$ is elementary of fiber type, both $W$ and $Z$ are $\mathbb{Q}$-factorial. Moreover $\tilde{f}$ is elementary of fiber type, $Y$ is $\mathbb{Q}$-factorial too, and $g$ cannot be small.

We show that $g$ is of fiber type. By contradiction, if $g$ is divisorial, then there is a prime divisor $D\subset\widetilde{X}$ such that $\tilde{f}(D)=\operatorname{Exc}(g)$ and hence $\operatorname{codim}g(\tilde{f}(D))>1$. If $\widehat{D}\subset\widehat{X}$ is the transform of $D$, then $h(\sigma(\widehat{D}))=g(\tilde{f}(D))$, thus it must be $\operatorname{codim}\sigma(\widehat{D})>1$ and $\widehat{D}=\widehat{E}$. On the other hand $\dim\varphi(\widehat{D})\leq 1$, therefore $\dim\mathcal{N}_{1}(\widehat{D},\widehat{X})\leq 3$ (see Lemma 2.6), a contradiction.

We conclude that $g$ is of fiber type, $\dim Y=3$, $\dim Z=2$, and the composition $g\circ\tilde{f}$ is quasi-elementary. This implies that $\sigma(\widehat{E})$ dominates $Z$; moreover $Z$ is a smooth del Pezzo surface (Lemma 3.20).

In particular $f\colon X\dasharrow Y$ is a rational contraction onto a $3$-fold, and we get $\rho_{X}\leq 9$ by Th. 1.5.

7.7**.**

Let us consider the $2$-dimensional cone $\operatorname{NE}(\varphi)=\operatorname{NE}(\sigma)+R$, where $R$ is an extremal ray of $\operatorname{NE}(\widehat{X})$. We have $\mathcal{N}_{1}(\widehat{E},\widehat{X})\subsetneq\mathcal{N}_{1}(\widehat{X})$ while $\varphi_{*}(\mathcal{N}_{1}(\widehat{E},\widehat{X}))=\mathcal{N}_{1}(\varphi(\widehat{E}),Z)=\mathcal{N}_{1}(Z)$, therefore $\ker\varphi_{*}\not\subset\mathcal{N}_{1}(\widehat{E},\widehat{X})$. On the other hand $\operatorname{NE}(\sigma)\subset\mathcal{N}_{1}(\widehat{E},\widehat{X})$, therefore we must have $R\not\subset\mathcal{N}_{1}(\widehat{E},\widehat{X})$. We also have $\widehat{E}\cdot\operatorname{NE}(\sigma)<0$, and there are curves $C\subset\widehat{X}$ contracted by $\varphi$ such that $\widehat{E}\cdot C>0$, therefore $\widehat{E}\cdot R>0$.

Let $F\subset\widehat{X}$ be a non-trivial fiber of the contraction of $R$, so that $\mathcal{N}_{1}(F,\widehat{X})=\mathbb{R}R$. Then $\widehat{E}\cap F\neq\emptyset$ (because $\widehat{E}\cdot R>0$) and $\dim(\widehat{E}\cap F)=0$ (because $R\not\subset\mathcal{N}_{1}(\widehat{E},\widehat{X})$), hence $\dim F=1$. Since $\varphi$ is $K$-negative, this implies that $R$ cannot be small (Lemma 2.13). Then there are no small rays in $\operatorname{NE}(\varphi)$, therefore $\widetilde{X}=\widehat{X}$, $\varphi=g\circ\tilde{f}$, $R=\operatorname{NE}(\tilde{f})$, and we get a diagram as (7.2). Moreover $\tilde{f}$ is finite on $\widetilde{E}$ because $\operatorname{NE}(\tilde{f})\not\subset\mathcal{N}_{1}(\widetilde{E},\widetilde{X})$, hence every fiber of $\tilde{f}$ has dimension one, so that $\tilde{f}$ is a conic bundle and $Y$ is smooth (see [Wiś91a, Th. (1.2)]), and $Y$ is weak Fano by [CS24, Cor. 6.4].

We show that $\sigma$ is of type $(3,2)^{\text{\it sm}}$. By Lemma 2.9, we have to show that every fiber of $\sigma$ has dimension $\leq 1$. If $\sigma$ had some $2$-dimensional fiber $F_{0}$, then $\tilde{f}(F_{0})\subset Y$ would be a $2$-dimensional fiber of $g$, which is impossible. This also implies that $W$ is smooth.

By Th.-Def. 5.1, there is an elementary contraction $\sigma^{\prime}\colon X\to W^{\prime}$ of type $(3,2)$ with exceptional divisor $E$, and $W^{\prime}$ is Fano; moreover $\sigma^{\prime}$ and $\sigma$ are locally isomorphic, therefore $E\subset\operatorname{dom}\xi$ where $\xi\colon X\dasharrow\widetilde{X}$ is the SQM, $\sigma^{\prime}$ is of type $(3,2)^{\text{\it sm}}$ too, and $W^{\prime}$ is smooth. This concludes the proof of Th. 7.1. ∎

We conclude this section with some technical lemmas that follow from our previous results, and are needed in the sequel.

Lemma 7.8** (refined version of [Cas22], Lemma 7.2).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, and $D\subset X$ a fixed prime divisor of type $(3,0)^{Q}$. Suppose that there are two fixed prime divisors $E_{1},E_{2}$, of type $(3,2)$, both adjacent to $D$. Then $X$ has a rational contraction onto a $3$-fold.

Proof.

Note that $X$ is not isomorphic to a product of surfaces, because it has a fixed prime divisor of type $(3,0)^{Q}$, hence $\delta_{X}\leq 1$ by Th. 1.9. Then [Cas22, proof of Lemma 7.2] shows that either $\mathcal{N}_{1}(E_{i},X)\subsetneq\mathcal{N}_{1}(X)$ for some $i\in\{1,2\}$, or $X$ has a rational contraction onto a $3$-fold, thus the statement follows from Th. 7.1. ∎

Lemma 7.9** (refined version of [Cas22], Lemma 5.13).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and $f\colon X\dasharrow S$ a rational contraction onto a surface with $\rho_{S}=1$. Suppose that there is a unique prime divisor in $X$ contracted to a point by $f$. Then there is a fixed prime divisor $E\subset X$ of type $(3,2)$ such that $\mathcal{N}_{1}(E,X)\subsetneq\mathcal{N}_{1}(X)$.

Proof.

As in [Cas22, proof of Lemma 5.13] we see that there is a special rational contraction $g\colon X\dasharrow T$ with $\dim T=\rho_{T}=2$. We have $\rho_{X}-\rho_{T}\geq 5$ while $d_{g}\leq 4$ by Th. 4.5 (see Def.-Rem. 3.2), hence $g$ is not quasi-elementary, and the statement follows from [ibid., Lemma 5.12]. ∎

Proposition 7.10** (refined version of [Cas22], Prop. 7.1).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, $D$ a fixed prime divisor of type $(3,1)^{\text{\it sm}}$ or $(3,0)^{Q}$, and $X\dasharrow\widetilde{X}\stackrel{{\scriptstyle\sigma}}{{\to}}Y$ the associated contraction as in Th.-Def. 5.1. Suppose that $Y$ contains a nef prime divisor $H$ covered by a family of rational curves of anticanonical degree one, and such that $H\cap\sigma(\operatorname{Exc}\sigma)=\emptyset$. Then one of the following holds:

1. $(i)$

$X$* has a (regular) contraction onto a $3$-fold, that sends $D$ to a point;* 2. $(ii)$

there is a fixed prime divisor $E\subset X$ of type $(3,2)$ such that $\mathcal{N}_{1}(E,X)\subsetneq\mathcal{N}_{1}(X)$.

Proof.

We note that since $X$ contains a fixed prime divisor of type $(3,1)^{\text{\it sm}}$ or $(3,0)^{Q}$, $X$ is not a product of surfaces, and $\delta_{X}\leq 1$ by Th. 1.9.

We follow [Cas22, proof of Prop. 7.1], which shows the existence of a commutative diagram:

[TABLE]

where $g$ and $h$ are contractions, and $h(D)=g(\sigma(\operatorname{Exc}(\sigma)))=\{\text{\it pt}\}$.

We show that $g$ and $h$ are of fiber type. If $D$ is $(3,1)^{\text{\it sm}}$, then $Y$ is a smooth Fano $4$-fold with $\rho_{Y}\geq 6$, and $\delta_{Y}\leq 2$ by Lemma 5.18. Now if $[C]\in\operatorname{mov}(Y)$, then $g$ is of fiber type. Otherwise as in [ibid., proof of Prop. 7.1] we see that $[C]$ generates an extremal ray of type $(3,2)$ of $\operatorname{NE}(Y)$, with locus $E_{1}$, and there is another fixed prime divisor $E_{2}\subset Y$, of type $(3,2)$, such that $E_{2}\cdot C>0$ and $[C_{E_{2}}]\in\operatorname{NE}(g)$. Then $E_{1}\cdot C_{E_{2}}>0$ [Cas13b, Lemma 2.2(b)], hence $[C]+[C_{E_{2}}]\in\operatorname{mov}(Y)$ [Cas20, Lemma 4.6 and its proof], and again $g$ is of fiber type. The case where $D$ is $(3,0)^{Q}$ is treated in [Cas22, proof of Prop. 7.1].

Therefore $g$ and $h$ are of fiber type, and [ibid., proof of Prop. 7.1] shows that $\dim Z>1$. If $\dim Z=3$ we have $(i)$, while if $\dim Z=2$, then by [ibid., Lemma 5.14] we have $\rho_{Z}=1$ and $D$ is the unique prime divisor contracted to a point by $h$. Hence Lemma 7.9 gives $(ii)$. ∎

8. Fano $4$-folds with no small contractions and blow-ups of cubic $4$-folds

In this section we consider Fano $4$-folds with no small contraction, and show the following result (Th. 1.2 from the Introduction).

Theorem 8.1**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and without small elementary contractions.

Then $\rho_{X}\leq 9$ and there are a cubic $4$-fold $Z\subset\mathbb{P}^{5}$ with at most isolated ordinary double points, and $s=\rho_{X}-1$ distinct planes $A_{1},\dotsc,A_{s}\subset Z$ intersecting pairwise in a point, such that $X$ is obtained from $Z$ by blowing-up one plane $A_{i}$ and successively the transforms of the other ones.111111The resulting $4$-fold does not depend on the order of the blow-ups, see Lemma 2.10.

We have $\operatorname{Sing}(Z)\subset\cup_{i}A_{i}$;121212The blow-up of the surfaces $A_{i}$ resolves the ordinary double points, see Rem 2.8. for each $i<j$ the point $A_{i}\cap A_{j}$ is in $Z_{\text{\it reg}}$, and $A_{i}\cap A_{j}\cap A_{k}=\emptyset$ for $i<j<k$.

Let us note that products of del Pezzo surfaces do not have small elementary contractions. Moreover the condition $\rho_{X}\geq 7$ is necessary in Th. 8.1, as the following example shows.

Example 8.2**.**

Let $X$ be a Fano $4$-fold with $\rho_{X}=6$ and $\delta_{X}=3$ which is not a product of surfaces; there are 9 such families, see Ex. 4.6. Then $X$ has no small elementary contraction. Indeed there is a quasi-elementary contraction $f\colon X\to S$ where $S$ is a surface with $\rho_{S}=2$; the cone $\operatorname{NE}(f)$ is described in [CRS22, §6], and it does not contain small extremal rays. On the other hand, if $\operatorname{NE}(X)$ had a small extremal ray $R\not\subset\operatorname{NE}(f)$, then $f$ should be finite on $\operatorname{Locus}(R)$ which is a union of exceptional planes (see Lemma 2.13), but this is impossible because $\rho_{S}=2$.

As remarked in the Introduction, we do not know whether a Fano $4$-fold as in Th. 8.1 exists, see §11.7 and Question 11.6. See also §11.7 for the geometry of the blow-up of a smooth cubic $4$-fold along two planes intersecting in a point; in particular we show that it is Fano.

Remark 8.3**.**

Let $X$ be as in Th. 8.1, and consider the blow-up $f\colon X\to Y$ of the last surface $S\subset Y$. The proof of Th. 8.1 (in particular Cor. 8.21) will show that $Y$ is Fano with at most isolated ordinary double points lying in $S$, and $S$ is a smooth del Pezzo surface with $\rho_{S}=\rho_{Y}=\rho_{X}-1\in\{6,7,8\}$. Moreover $\operatorname{NE}(S)\cong\operatorname{NE}(Y)$ under the natural map given by the inclusion $S\hookrightarrow Y$, and $\operatorname{Eff}(Y)\cong\operatorname{NE}(S)$ under the restriction. Every conic bundle $S\to\mathbb{P}^{1}$ extends to a contraction $Y\to\mathbb{P}^{2}$ sending $S$ to a line, and every birational map $S\to\mathbb{P}^{2}$ extends to a blow-up map $Y\to Z^{\prime}$, where $Z^{\prime}$ is a cubic $4$-fold, sending $S$ to a plane.

8.4****Overview of the proof of Th. 8.1.

We show first of all that if $X\to T$ is a non-trivial contraction of fiber type, then $T\cong\mathbb{P}^{2}$ (Lemma 8.5). This relies on both Th. 4.5, Th. 1.9, and [CS24].

Then we consider birational contractions of $X$. For the case of an elementary contraction $f\colon X\to Y$, it is shown in [Cas24] that $f$ is the blow-up of a smooth del Pezzo surface $S\subset Y$ with $K_{S}=K_{Y|S}$, and $Y$ is Fano with at most isolated ordinary double points (Prop. 8.7).

We first generalize this description to every birational contraction $\sigma\colon X\to Z$ (Prop. 8.8): $\sigma$ is the blow-up of smooth del Pezzo surfaces $A_{1},\dotsc,A_{\rho_{X}-\rho_{Z}}\subset Z$ that intersect pairwise transversally in (at most) finitely many points, and $Z$ is Fano with at most isolated ordinary double points. Moreover for every $i=1,\dotsc,\rho_{X}-\rho_{Z}$ we have $K_{A_{i}}=K_{Z|A_{i}}$, and if $E_{i}\subset X$ is the exceptional divisor over $A_{i}$, it follows from Th. 7.1 that $\mathcal{N}_{1}(E_{i},X)=\mathcal{N}_{1}(X)$ and hence $\mathcal{N}_{1}(A_{i},Z)=\mathcal{N}_{1}(Z)$, so that $\rho_{A_{i}}\geq\rho_{Z}$. This also implies that $E_{i}\cap E_{j}\neq\emptyset$ and hence $A_{i}\cap A_{j}\neq\emptyset$ for every $i<j$.

Let us fix $i$ and factor $\sigma$ as $X\stackrel{{\scriptstyle\sigma_{i}}}{{\to}}Y_{i}\stackrel{{\scriptstyle h_{i}}}{{\to}}Z$, where $h_{i}$ blows-up all the surfaces $A_{j}$ for $j\neq i$, $\rho_{Y_{i}}=\rho_{X}-1$, and $\sigma_{i}$ blows-up the transform $S_{i}\subset Y_{i}$ of $A_{i}$. Then $h_{i|S_{i}}\colon S_{i}\to A_{i}$ blows-up all the points where $A_{i}$ intersects the other surfaces $A_{j}$, $j\neq i$. On the other hand, $S_{i}$ is del Pezzo, thus $\rho_{S_{i}}\leq 9$. This allows to bound both $\rho_{A_{i}}$ and the cardinality of $A_{i}\cap A_{j}$ (Rem. 8.19), and is used repeatedly.

Then we show that, if for some $i\in\{1,\dotsc,\rho_{X}-\rho_{Z}\}$ we have $\rho_{A_{i}}=\rho_{Z}\geq 3$, then $\operatorname{NE}(A_{i})\cong\operatorname{NE}(Z)$, and one can extend contractions from $A_{i}$ to $Z$ (Cor. 8.21).

We use all these preliminary results to prove that there exists a birational contraction $X\to Z$ with $\rho_{Z}=1$ and such that one of the blown-up surfaces, say $A_{1}\subset Z$, also has $\rho_{A_{1}}=1$ (Lemma 8.23). With this we can finally show Th. 8.1, let us give an outline.

We have $A_{1}\cong\mathbb{P}^{2}$, $K_{Z|A_{1}}=K_{A_{1}}$, and we show that the restriction $\operatorname{Pic}(Z)\to\operatorname{Pic}(A_{1})$ is an isomorphism. This implies that $Z$ is Fano with index $3$, namely a del Pezzo $4$-fold, and these $4$-folds are classified. Let $H\in\operatorname{Pic}(Z)$ be such that $-K_{Z}=3H$, and set $d:=H^{4}$; then $h^{0}(Z,-K_{Z})=15d+10$ (see 8.30).

Next we show that there are two other blown-up surfaces, say $A_{2}$ and $A_{3}$, that intersect $A_{1}$ only in one point, trasversally. If $Z^{\prime}\to Z$ is the blow-up of $A_{2}$ and $A_{3}$, and $A_{1}^{\prime}\subset Z^{\prime}$ is the transform of $A_{1}$, we have $A_{1}^{\prime}\cong\operatorname{Bl}_{2\mskip 1.0mu\text{\it pts}}\mathbb{P}^{2}$ and $\rho_{A_{1}^{\prime}}=\rho_{Z^{\prime}}=3$ (see 8.28), and hence $\operatorname{NE}(A_{1}^{\prime})\cong\operatorname{NE}(Z^{\prime})$ by the previous part of the proof; we also compute that $h^{0}(Z^{\prime},-K_{Z^{\prime}})\in\{15d-10,\dotsc,15d-6\}$. Then we extend the contraction $A_{1}^{\prime}\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ to an elementary birational contraction $Z^{\prime}\to W$ which is again the blow-up of a surface; $W$ is a Fano $4$-fold with $\rho_{W}=2$ and index $2$. By studying the contractions of $W$, we show that $W$ is a double cover of $\mathbb{P}^{2}\times\mathbb{P}^{2}$ with branch divisor of degree $(2,2)$ (see 8.34). This gives $h^{0}(W,-K_{W})=45$, and allows to compute that $h^{0}(Z^{\prime},-K_{Z^{\prime}})=36$ and that $d=3$, so that $Z$ is a cubic $4$-fold, each $A_{i}$ is a plane, and $A_{i}\cap A_{j}$ is just one point for every $i<j$. We also get $h^{0}(Z,-K_{Z})=55$ and we can compute $h^{0}(X,-K_{X})$ in terms of the number of blown-up planes, that are $\rho_{X}-1$. Finally, using that $h^{0}(X,-K_{X})\geq 0$, we get $\rho_{X}\leq 9$ (see 8.35).

Lemma 8.5**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and without small elementary contractions.

Let $f\colon X\dasharrow Y$ be a non-trivial rational contraction of fiber type. Then $f$ is regular and equidimensional, and $Y\cong\mathbb{P}^{2}$.

Proof.

Since $X$ has no small elementary contraction, $f$ is regular. Let $F\subset X$ be a general fiber; we have $\dim\mathcal{N}_{1}(F,X)=d_{f}\leq 4$ by Th. 4.5 (see Def.-Rem. 3.2). In particular $F$ cannot be a divisor because $\delta_{X}\leq 1$ (Th. 1.9) and $\rho_{X}\geq 7$, hence $\dim Y>1$.

Suppose by contradiction that $\dim Y=3$. Then by Prop. 3.12 $f$ factors through a special rational contraction $f^{\prime}\colon X\dasharrow Y^{\prime}$ with $\dim Y^{\prime}=3$. As above $f^{\prime}$ is regular, and $\rho_{Y^{\prime}}\leq 3$ by Lemma 2.16. On the other hand $\rho_{X}-\rho_{Y^{\prime}}\leq 2$ by [CS24, Lemma 3.12], and we have a contradiction.

We conclude that $\dim Y=2$. As above we can factor $f$ as $X\stackrel{{\scriptstyle f^{\prime}}}{{\to}}Y^{\prime}\to Y$ where $f^{\prime}$ is equidimensional and $\dim Y^{\prime}=2$; in particular $Y^{\prime}$ is a smooth rational surface (Lemma 3.19). If $\rho_{Y^{\prime}}>1$, then there is a morphism $Y^{\prime}\to\mathbb{P}^{1}$, and the composition $X\to\mathbb{P}^{1}$ contradicts the previous part of the proof. Therefore $\rho_{Y^{\prime}}=1$ and $Y\cong Y^{\prime}\cong\mathbb{P}^{2}$. ∎

Corollary 8.6**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and without small elementary contractions. Let $E\subset X$ be a fixed prime divisor. Then $E$ is of type $(3,2)$ and $\mathcal{N}_{1}(E,X)=\mathcal{N}_{1}(X)$.

Proof.

Since $X$ has no small elementary contraction, $E$ must be of type $(3,2)$ by Th. -Def.5.1$(c)$. Then $\mathcal{N}_{1}(E,X)=\mathcal{N}_{1}(X)$ by Th. 7.1 and Lemma 8.5. ∎

Proposition 8.7** (refinement of [Cas24], Th. 4.1).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and without small elementary contractions.

Let $f\colon X\to Y$ be an elementary contraction. Then $f$ is of type $(3,2)$, $Y$ is Fano with at most isolated ordinary double points, and $S:=f(\operatorname{Exc}(f))\subset Y$ is a smooth del Pezzo surface with $K_{S}=K_{Y|S}$.

Proof.

The statement follows from [Cas24, Th. 4.1]. Note that this reference has the assumption $\rho_{X}\geq 8$, but this is used only to show that a contraction $g\colon X\to Z$ with $\rho_{X}-\rho_{Z}\leq 3$ cannot be of fiber type. In our setting this follows from Lemma 8.5, and the same proof applies under the assumption $\rho_{X}\geq 7$. Finally $Y$ is Fano by Th.-Def. 5.1, and has at most isolated ordinary double points by Lemma 2.9. ∎

Proposition 8.8**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and without small elementary contractions.

Let $\sigma\colon X\to Z$ be a birational contraction; then $\operatorname{NE}(\sigma)=R_{1}+\cdots+R_{s}$ with $R_{i}$ extremal ray of type $(3,2)$ of $\operatorname{NE}(X)$ for every $i=1,\dotsc,s$, and $s=\rho_{X}-\rho_{Z}$. Set $E_{i}:=\operatorname{Locus}(R_{i})$. The following holds:

1. $(a)$

$Z$* is Fano with at most isolated ordinary double points, and $A_{i}:=\sigma(E_{i})$ is a smooth del Pezzo surface with $K_{A_{i}}=K_{Z|A_{i}}$ and $\mathcal{N}_{1}(A_{i},Z)=\mathcal{N}_{1}(Z)$ for every $i=1,\dotsc,s$;* 2. $(b)$

for every $i\neq j$ the surfaces $A_{i}$ and $A_{j}$ intersect transversally in finitely many points where $Z$ is regular, while $A_{i}\cap A_{j}\cap A_{k}=\emptyset$ if $i<j<k$. Each singular point of $Z$ is contained in some $A_{i}$. Moreover $\operatorname{Exc}(\sigma)=E_{1}\cup\cdots\cup E_{s}$ and $\sigma$ is obtained by blowing-up one surface $A_{i}$ and successively the transforms of the other ones, in any order; 3. $(c)$

every elementary contraction of $Z$ is either of type $(3,2)$, or $Z\to\mathbb{P}^{2}$.

Proof.

8.9**.**

By Prop. 8.7 each $R_{i}$ is of type $(3,2)$, and if $\sigma_{i}\colon X\to Y_{i}$ is the contraction of $R_{i}$, then $Y_{i}$ has at most isolated ordinary double points, and $S_{i}:=\sigma_{i}(E_{i})$ is a smooth del Pezzo surface with $K_{S_{i}}=K_{Y_{i}|S_{i}}$.

By [Cas24, Lemma 2.5] we also have $E_{i}\cdot R_{j}=0$ for every $i\neq j$. In particular $[C_{E_{1}}],\dotsc,[C_{E_{s}}]\in\mathcal{N}_{1}(X)$ are linearly independent, and $\operatorname{NE}(\sigma)$ is a simplicial cone of dimension $s=\rho_{X}-\rho_{Z}$.

8.10**.**

Clearly $E_{i}\subset\operatorname{Exc}(\sigma)$ for every $i=1,\dotsc,s$. Conversely if $C\subset X$ is an irreducible curve with $[C]\in\operatorname{NE}(\sigma)$, we have $C\equiv\sum_{j=1}^{s}\lambda_{j}C_{E_{j}}$ with $\lambda_{j}\in\mathbb{Q}_{\geq 0}$. Then for every $i=1,\dotsc,s$ we have $E_{i}\cdot C=-\lambda_{i}\leq 0$, and either $C\subset E_{i}$, or $\lambda_{i}=0$. Therefore either $[C]\in R_{i}$ for some $i\in\{1,\dotsc,s\}$, or $C\subset E_{i}\cap E_{j}$ for some $i\neq j$. In particular $\operatorname{Exc}(\sigma)=E_{1}\cup\cdots\cup E_{s}$.

Moreover, on the open subset $X\smallsetminus\cup_{j\neq i}E_{j}$, $\sigma$ coincides with $\sigma_{i}$, therefore $\dim A_{i}=2$, $A_{i}\smallsetminus\cup_{j\neq i}A_{j}$ is smooth, and $Z$ has at most isolated ordinary double points at $A_{i}\smallsetminus\cup_{j\neq i}A_{j}$. We also note that $\mathcal{N}_{1}(E_{i},X)=\mathcal{N}_{1}(X)$ by Cor. 8.6, hence $\mathcal{N}_{1}(A_{i},Z)=\sigma_{*}(\mathcal{N}_{1}(E_{i},X))=\mathcal{N}_{1}(Z)$, for every $i=1,\dotsc,s$.

8.11**.**

We show that $\sigma$ factors as the composition of $s$ elementary divisorial contractions of type $(3,2)$ with exceptional divisors $E_{1},\dotsc,E_{s}$, in arbitrary order, and $Z$ is $\mathbb{Q}$-factorial. We proceed by induction on $s$, the statement being clear for $s=1$. Consider a facet of $\operatorname{NE}(\sigma)$, up to renumbering we can assume that it is $R_{1}+\cdots+R_{s-1}$, and factor $\sigma$ as:

[TABLE]

where $\operatorname{NE}(\sigma^{\prime})=R_{1}+\cdots+R_{s-1}$. Then the statement holds for $\sigma^{\prime}$ by induction, $Z^{\prime}$ is $\mathbb{Q}$-factorial, and $f$ is an elementary birational contraction with exceptional locus $\sigma^{\prime}(E_{s})$, therefore $f$ is divisorial and $Z$ is $\mathbb{Q}$-factorial. Moreover $f$ is of type $(3,2)$ because $\dim A_{s}=2$. This gives the statement.

8.12**.**

If $g\colon Z\to W$ is an elementary contraction, we can consider the composition $g\circ\sigma\colon X\to W$. If $g$ is of fiber type, then $W\cong\mathbb{P}^{2}$ by Lemma 8.5. If instead $g$ is birational, then it is of type $(3,2)$ by 8.11. This gives $(c)$.

8.13**.**

Let $i,j\in\{1,\dotsc,s\}$ with $i<j$. We have $E_{i}\cap E_{j}\neq\emptyset$ by Cor. 8.6 and Rem. 2.7. Moreover every connected component of $E_{i}\cap E_{j}$ is isomorphic to $\mathbb{P}^{1}\times\mathbb{P}^{1}$ with the rulings being curves in $R_{i}$ and $R_{j}$ (see [Cas22, Lemma 4.15] and [Cas24, Prop. 3.4]). Therefore if $C\subset X$ is an irreducible curve contained in $E_{i}\cap E_{j}$, we have $[C]\in R_{i}+R_{j}$. Together with 8.10, this implies that for every irreducible curve $C\subset X$ contracted by $\sigma$ we have $[C]\in R_{i}+R_{j}$ for some $i<j$.

Let $\sigma_{ij}\colon X\to Y_{ij}$ be the contraction of $R_{i}+R_{j}$, and set $B_{i}:=\sigma_{ij}(E_{i})$, $B_{j}:=\sigma_{ij}(E_{j})$. The behaviour of $\sigma_{ij}$ is analyzed in [Cas24, Prop. 3.4 and its proof], where it is shown that:

1. (1)

$\dim(B_{i}\cap B_{j})=0$ and $E_{i}\cap E_{j}=\sigma_{ij}^{-1}(B_{i}\cap B_{j})$; 2. (2)

if $B_{i}\cap B_{j}=\{y_{1},\dotsc,y_{m}\}$, then $B_{i}$, $B_{j}$, and $Y_{ij}$ are smooth at each $y_{a}$, and the fiber $T_{a}:=\sigma_{ij}^{-1}(y_{a})$ is an isolated $2$-dimensional fiber of $\sigma_{ij}$ and a connected component of $E_{i}\cap E_{j}$; 3. (3)

if we factor $\sigma_{ij}$ as

[TABLE]

where $g$ is an elementary contraction of type $(3,2)$ with exceptional divisor $E_{j}^{\prime}:=\sigma_{i}(E_{j})$, then $(E_{j}^{\prime})_{|S_{i}}=C_{1}+\cdots+C_{m}$ (recall that $S_{i}=\sigma_{i}(E_{i})\subset Y_{i}$, see 8.9) where $C_{1},\dotsc,C_{m}$ are pairwise disjoint $(-1)$-curves in $S_{i}$, and fibers of $g$; moreover $C_{a}=\sigma_{i}(T_{a})$ and $C_{a}\subset(Y_{i})_{\text{\it reg}}$ for every $a=1,\dotsc,m$.

In particular, locally around $y_{1},\dotsc,y_{m}$, $g$ is just the blow-up of the smooth surface $B_{j}$ in the smooth $4$-fold $Y_{ij}$, thus the restriction $g_{|S_{i}}\colon S_{i}\to B_{i}$ is the blow-up of $B_{i}$ at the scheme intersection $B_{i}\cap B_{j}$. Since $S_{i}$ is smooth (see 8.9), this intersection must be reduced.

8.14**.**

Let $k\in\{1,\dotsc,s\}$, $k\neq i,j$, and let $T$ be a connected component of $E_{i}\cap E_{j}$; note that $\mathcal{N}_{1}(T,X)=\mathbb{R}R_{i}\oplus\mathbb{R}R_{j}$ is $2$-dimensional. If $T\cap E_{k}\neq\emptyset$, then $T\subset E_{k}$, because $E_{k}\cdot R_{i}=E_{k}\cdot R_{j}=0$. Then the contraction $\sigma_{k}$ of $R_{k}$ must be finite on $T$, and $\sigma_{k}(T)=S_{k}\subset Y_{k}$. This gives $\mathcal{N}_{1}(S_{k},Y_{k})=(\sigma_{k})_{*}(\mathcal{N}_{1}(T,X))$, $\dim\mathcal{N}_{1}(S_{k},Y_{k})\leq 2$, and $\dim\mathcal{N}_{1}(E_{k},X)\leq 1+\dim\mathcal{N}_{1}(S_{k},Y_{k})\leq 3$ (see Lemma 2.6), a contradiction. Hence $E_{i}\cap E_{j}\cap E_{k}=\emptyset$ and $A_{i}\cap A_{j}\cap A_{k}=\emptyset$.

On the open subset $X\smallsetminus\cup_{k\neq i,j}E_{k}$, $\sigma$ coincides with $\sigma_{ij}$, and $A_{i}\cap A_{j}\subset\sigma(X\smallsetminus\cup_{k\neq i,j}E_{k})$. We conclude from 8.13 that every $A_{i}$ is smooth, $A_{i}$ and $A_{j}$ intersect transversally at finitely many points where $Z$ is smooth, and $Z$ has at most isolated ordinary double points contained in some $A_{i}$; in particular $Z$ has locally factorial, terminal singularities (see Rem. 2.8). We have shown $(b)$.

8.15**.**

We have $-K_{X}+E_{1}+\cdots+E_{s}=\sigma^{*}(-K_{Z})$. If $R$ is an extremal ray of $\operatorname{NE}(X)$ different from $R_{1},\dotsc,R_{s}$, then $E_{i}\cdot R\geq 0$ for every $i=1,\dotsc,s$ by [Cas24, Lemma 2.3], thus $-K_{X}+E_{1}+\cdots+E_{s}$ is nef and $(-K_{X}+E_{1}+\cdots+E_{s})^{\perp}\cap\operatorname{NE}(X)=\operatorname{NE}(\sigma)$. This implies that $-K_{Z}$ is ample, namely $Z$ is Fano.

8.16**.**

Fix $i\in\{1,\dotsc,s\}$. We have a factorization:

[TABLE]

set $E^{\prime}_{j}:=\sigma_{i}(E_{j})\subset Y_{i}$ for every $j\neq i$. Then $h_{i|S_{i}}\colon S_{i}\to A_{i}$ is a blow-up of smooth points, with exceptional locus $\cup_{j\neq i}E^{\prime}_{j|S_{i}}$, which is a union of pairwise disjoint $(-1)$-curves $\Gamma_{a}$ (see 8.13(3)). We have $K_{S_{i}}=(h_{i|S_{i}})^{*}K_{A_{i}}+\sum_{a}\Gamma_{a}$ and $K_{Y_{i}}=h_{i}^{*}K_{Z}+\sum_{j\neq i}E_{j}^{\prime}$; then $K_{S_{i}}=K_{Y_{i}|S_{i}}$ (see 8.9) implies that $(h_{i|S_{i}})^{*}K_{A_{i}}=(h_{i|S_{i}})^{*}K_{Z|A_{i}}$ and finally that $K_{A_{i}}=K_{Z|A_{i}}$, so we have $(a)$.∎

Remark 8.18**.**

In the setting of Prop. 8.8, let $g\colon Z\to W$ be an elementary birational contraction. Then $B:=g(\operatorname{Exc}(g))\subset W$ is a smooth del Pezzo surface and

[TABLE]

Indeed by Prop. 8.8 both $Z$ and $W$ are Fano with locally factorial and terminal singularities, so that $h^{i}(Z,-K_{Z})=0$ for $i>0$ and $h^{0}(Z,-K_{Z})=\chi(Z,-K_{Z})$, and similarly for $W$. Moreover, still by Prop. 8.8 applied to $g\circ\sigma\colon X\to W$, we have that $B$ is a smooth del Pezzo surface with $K_{B}=K_{W|B}$, hence the formula follows from Lemma 2.11 and $\chi(B,-K_{B})=h^{0}(B,-K_{B})=11-\rho_{B}$.

Remark 8.19**.**

In the setting of Prop. 8.8, set $m_{ij}:=\#(A_{i}\cap A_{j})$ for every $i,j=1,\dotsc,\rho_{X}-\rho_{Z}$, $i\neq j$; note that $m_{ij}$ is also the number of connected components of $E_{i}\cap E_{j}$ in $X$.

Let $i\in\{1,\dotsc,\rho_{X}-\rho_{Z}\}$. The blow-up of $A_{i}$ at the points of intersection with $\cup_{j\neq i}A_{j}$ is the del Pezzo surface $S_{i}\subset Y_{i}$ as in (8.17), and $\rho_{S_{i}}\leq 9$, therefore:

[TABLE]

Corollary 8.21**.**

In the setting of Prop. 8.8, suppose that $\rho_{Z}\geq 3$ and $\rho_{A_{i}}=\rho_{Z}$ for some $i\in\{1,\dotsc,\rho_{X}-\rho_{Z}\}$. Let $\iota\colon A_{i}\hookrightarrow Z$ be the inclusion. Then $\iota_{*}\colon\mathcal{N}_{1}(A_{i})\to\mathcal{N}_{1}(Z)$ is an isomorphism such that $\iota_{*}(\operatorname{NE}(A_{i}))=\operatorname{NE}(Z)$. For every $(-1)$-curve $C\subset A_{i}$ there exists a (unique) extremal ray $R$ of $\operatorname{NE}(Z)$ such that $[C]\in R$; moreover $R$ is of type $(3,2)$ and $E_{R|A_{i}}=C$, where $E_{R}:=\operatorname{Locus}(R)$.

Proof.

We have $K_{A_{i}}=K_{Z|A_{i}}$ and $\mathcal{N}_{1}(A_{i},Z)=\mathcal{N}_{1}(Z)$ by Prop. 8.8$(a)$. In particular $\iota_{*}\colon\mathcal{N}_{1}(A_{i})\to\mathcal{N}_{1}(Z)$ is surjective, thus it is an isomorphism because $\rho_{A_{i}}=\rho_{Z}$.

We show that $\iota_{*}(\operatorname{NE}(A_{i}))=\operatorname{NE}(Z)$. The inclusion $\subseteq$ is clear. Conversely, let $R$ be an extremal ray of $\operatorname{NE}(Z)$. Then $R$ is of type $(3,2)$ by Prop. 8.8$(c)$ and because $\rho_{Z}\geq 3$. By the proof of Prop. 8.8 (more precisely 8.13 and 8.14) we have $E_{R}\cap A_{i}\neq\emptyset$ and $E_{R|A_{i}}=C_{1}+\cdots+C_{m}$ with $C_{1},\dotsc,C_{m}$ pairwise disjoint $(-1)$-curves with $[C_{j}]\in R$, so that $R\in\iota_{*}(\operatorname{NE}(A_{i}))$, and we conclude that $\iota_{*}(\operatorname{NE}(A_{i}))=\operatorname{NE}(Z)$. Moreover $-K_{Z}\cdot C_{j}=-K_{A_{i}}\cdot C_{j}=1$ for every $j$, which implies that $C_{j}\equiv C_{1}$ in $Z$. Since $\iota_{*}$ is injective, we get that $C_{j}\equiv C_{1}$ in $A_{i}$, therefore $C_{j}=C_{1}$ and $m=1$, because these are $(-1)$-curves. ∎

Lemma 8.22**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and without small elementary contractions. If there is a contraction $f\colon X\to\mathbb{P}^{2}$, then $d_{f}\leq 3$.

We recall that $d_{f}=\dim\mathcal{N}_{1}(F,X)$ for a general fiber $F\subset X$ of $f$, see Def.-Rem. 3.2.

Proof.

Let us factor $f$ as $X\stackrel{{\scriptstyle\sigma}}{{\to}}Z\stackrel{{\scriptstyle g}}{{\to}}\mathbb{P}^{2}$, with $\rho_{Z}=2$ and $g$ elementary; let $F\subset X$ be a general fiber of $f$ and $F_{Z}:=\sigma(F)\subset Z$ a general fiber of $g$.

Then $\sigma$ is birational by Lemma 8.5, $g$ is elementary of fiber type, and $\dim\mathcal{N}_{1}(F_{Z},Z)=1$. We apply Prop. 8.8 to $\sigma$; consider the surfaces $A_{1},\dotsc,A_{\rho_{X}-2}\subset Z$ blown-up by $\sigma$. Up to renumbering let $A_{1},\dotsc,A_{r}$ be those dominating $\mathbb{P}^{2}$ under $g$, with $r\in\{0,\dotsc,\rho_{X}-2\}$. Then

[TABLE]

Suppose by contradiction that $d_{f}\geq 4$. Then by Th. 4.5 we have $d_{f}=4$, $r=3$, and $F$ is a del Pezzo surface of degree one. Let us further factor $f$ as

[TABLE]

where $\sigma^{\prime}$ is the blow-up of $A_{1},A_{2},A_{3}\subset Z$, $\rho_{Z^{\prime}}=5$, and set $F^{\prime}:=\sigma^{\prime\prime}(F)\subset Z^{\prime}$. Then the surfaces blown-up by $\sigma^{\prime\prime}$ do not dominate $\mathbb{P}^{2}$, hence $F^{\prime}$ is contained in the open subset where $\sigma^{\prime\prime}$ is an isomorphism, so that $F^{\prime}\cong F$ and $F^{\prime}$ is a del Pezzo surface of degree one. By considering the anticanonical pencil in $F^{\prime}$, we see that $Z^{\prime}$ is covered by a proper family $V$ of irreducible curves of anticanonical degree one (recall that $Z^{\prime}$ is Gorenstein Fano by Prop. 8.8).

We have $\rho_{Z^{\prime}}<\rho_{X}$, hence $\sigma^{\prime\prime}$ is not an isomorphism; let $B\subset Z^{\prime}$ be a surface blown-up by $\sigma^{\prime\prime}$. By Prop. 8.8 we know that $B$ is a smooth del Pezzo surface with $K_{Z^{\prime}|B}=K_{B}$. Let $p\in B$ be a general point. There is an irreducible curve $\Gamma$ of the family $V$ such that $p\in\Gamma$. Since $-K_{Z^{\prime}}\cdot\Gamma=1$, we must have $\Gamma\subset B$ (see [CS24, Prop. 2.10]); moreover $-K_{B}\cdot\Gamma=-K_{Z^{\prime}}\cdot\Gamma=1$.

We conclude that $B$ is covered by curves of anticanonical degree one, hence $\rho_{B}=9$. On the other hand (8.20) applied to $\sigma^{\prime\prime}$ and $A_{i}=B$ gives $9\geq\rho_{B}+\rho_{X}-\rho_{Z^{\prime}}-1=\rho_{X}+3$, a contradiction. ∎

Lemma 8.23**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, not isomorphic to a product of surfaces, and without small elementary contractions.

Then there exists a birational contraction $\sigma\colon X\to Z$ with $\rho_{Z}=1$ and such that $\rho_{A_{i}}=1$ for some $i\in\{1,\dotsc,\rho_{X}-1\}$ (notation as in Prop. 8.8).

Proof.

8.24**.**

Let $\tau\colon X\to W$ be a contraction with $\rho_{W}=2$; it is birational by Lemma 8.5, and Prop. 8.8 applies; in particolar $W$ is a Fano $4$-fold with at most ordinary double points as singularities, and $\tau$ blows-up $\rho_{X}-2\geq 5$ surfaces $B_{1},\dotsc,B_{\rho_{X}-2}\subset W$. For every $i=1,\dotsc,\rho_{X}-2$, (8.20) gives $9\geq\rho_{B_{i}}+\rho_{X}-3\geq\rho_{B_{i}}+4$, therefore $\rho_{B_{i}}\leq 5$.

Let us consider the two elementary contractions $g_{j}\colon W\to Z_{j}$ of $W$, with $j\in\{1,2\}$, and set $f_{j}:=g_{j}\circ\tau\colon X\to Z_{j}$.

If $g_{j}$ is of fiber type, then by Lemmas 8.5 and 8.22 we have $Z_{j}\cong\mathbb{P}^{2}$, $f_{j}$ is equidimensional, and $d_{f_{j}}\leq 3$; moreover as in the proof of Lemma 8.22 we see that at most two surfaces $B_{i}$ can dominate $\mathbb{P}^{2}$ under $g_{j}$.

If $g_{j}$ is birational, then it must be of type $(3,2)$ by Prop. 8.8. Set $E_{j}:=\operatorname{Exc}(g_{j})\subset W$ and $A_{j}:=g_{j}(E_{j})\subset Z_{j}$. Suppose that $\rho_{A_{j}}>1$; we show that there are at most two indices $i\in\{1,\dotsc,\rho_{X}-2\}$ such that $E_{j|B_{i}}$ is reducible.

We apply Rem. 8.19 to $f_{j}=g_{j}\circ\tau\colon X\to Z_{j}$ and the surface $A_{j}$; the other surfaces blown-up by $f_{j}$ are $g_{j}(B_{1}),\dotsc,g_{j}(B_{\rho_{X}-2})$. Set $m_{i}:=\#(g_{j}(B_{i})\cap A_{j})$, and note that this is also the number of components of $E_{j|B_{i}}$ in $W$. Then (8.20) gives $9\geq\rho_{A_{j}}+\sum_{i=1}^{\rho_{X}-2}m_{i}\geq 2+\sum_{i}m_{i}$, therefore $\sum_{i=1}^{\rho_{X}-2}m_{i}\leq 7$ and the sum has $\rho_{X}-2\geq 5$ terms, so we see that there are at most two indices $i\in\{1,\dotsc,\rho_{X}-2\}$ such that $m_{i}>1$, namely such that $E_{j|B_{i}}$ is reducible.

8.25**.**

Suppose that both $g_{1}$ and $g_{2}$ are of fiber type.

Since $\rho_{X}-2\geq 5$, by 8.24 there is at least one surface, say $B_{1}\subset W$, such that $g_{j}(B_{1})\subsetneq\mathbb{P}^{2}$ for $j=1,2$. On the other hand $g_{j}(B_{1})=f_{j}(E_{1})$ cannot be a point because $f_{j}$ is equidimensional, therefore $g_{j}(B_{1})$ is a curve in $\mathbb{P}^{2}$. Since $B_{1}$ is a smooth del Pezzo surface, $g_{j|B_{1}}$ factors through a conic bundle $h_{j}\colon B_{1}\to\mathbb{P}^{1}$, such that $h_{1}$ and $h_{2}$ define a finite map $h\colon B_{1}\to\mathbb{P}^{1}\times\mathbb{P}^{1}$. Since $\rho_{B_{1}}\leq 5$, [Dru16, Prop. 7.2] implies that $B_{1}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$.

Similarly as in 8.24, using (8.20) and $\rho_{B_{1}}=2$, we see that there is at least one index $i\in\{2,\dotsc,\rho_{X}-2\}$ such that $\#(B_{1}\cap B_{i})=1$; up to renumbering we can assume $i=2$. Consider the factorization of $\tau$ given by:

[TABLE]

where $\tau^{\prime}$ is the blow-up of $B_{2}$, and let $B_{1}^{\prime}\subset W^{\prime}$ be the transform of $B_{1}$. Then $B_{1}^{\prime}\cong\operatorname{Bl}_{2\mskip 1.0mu\text{\it pts}}\mathbb{P}^{2}$, so that $\rho_{B_{1}^{\prime}}=\rho_{W^{\prime}}=3$, and we can apply Cor. 8.21. In particular, let $C_{1},C_{2}\subset B_{1}^{\prime}$ be the $(-1)$-curves contracted by $B_{1}^{\prime}\to\mathbb{P}^{2}$; then for $i=1,2$ the class $[C_{i}]\in\operatorname{NE}(W^{\prime})$ generates an extremal ray $R_{i}$ of type $(3,2)$, with locus $G_{i}\subset W^{\prime}$ such that $G_{i|B_{1}^{\prime}}=C_{i}$. Moreover $R_{1}+R_{2}$ is a face of $\operatorname{NE}(W^{\prime})$, because $\operatorname{NE}(W^{\prime})\cong\operatorname{NE}(B_{1}^{\prime})$.

Let $h\colon W^{\prime}\to T$ be the contraction with $\operatorname{NE}(h)=R_{1}+R_{2}$, so that $\rho_{T}=1$. We have $G_{1}\cdot C_{2}=G_{1|B_{1}^{\prime}}\cdot_{B_{1}^{\prime}}C_{2}=C_{1}\cdot_{B_{1}^{\prime}}C_{2}=0$, and similarly $G_{2}\cdot C_{1}=0$. Then it is easy to see that $\operatorname{Exc}(h)\subset G_{1}\cup G_{2}$, and $h$ is birational. Moreover $A_{T}:=h(B_{1}^{\prime})\subset T$ is one of the surfaces blown-up by the birational contraction $h\circ\tau^{\prime\prime}\colon X\to T$, and $\rho_{A_{T}}=1$ by construction, so we get the statement.

8.26**.**

Suppose now that $g_{1}$ and $g_{2}$ are both birational; recall that $E_{j}=\operatorname{Exc}(g_{j})\subset W$ and $A_{j}=g_{j}(E_{j})\subset Z_{j}$ for $j=1,2$. We show that $\rho_{A_{j}}=1$ for some $j\in\{1,2\}$, which gives the statement.

By contradiction, suppose that $\rho_{A_{j}}>1$ for $j=1,2$. Then, by 8.24, there are at most $4$ indices $i\in\{1,\dotsc,\rho_{X}-2\}$ such that $E_{j|B_{i}}$ is reducible for some $j\in\{1,2\}$, and since $\rho_{X}-2\geq 5$, there exists an index $i$, say $i=1$, such that $E_{j|B_{1}}$ is irreducible for $j=1,2$. Then $E_{j|B_{1}}=C_{j}$ is a $(-1)$-curve in $B_{1}$, with $[C_{j}]\in\operatorname{NE}(g_{j})$ (see 8.13(3)).

We have $\operatorname{NE}(W)=\operatorname{NE}(g_{1})+\operatorname{NE}(g_{2})$ and $E_{1}\cdot\operatorname{NE}(g_{1})<0$, hence $E_{1}\cdot\operatorname{NE}(g_{2})>0$. Then $C_{1}\cdot_{B_{1}}C_{2}=E_{1|B_{1}}\cdot_{B_{1}}C_{2}=E_{1}\cdot C_{2}>0$, and since $B_{1}$ is a smooth del Pezzo surface with $\rho_{B_{1}}\leq 5$ (see 8.24), it is easy to see that it must be $C_{1}\cdot_{B_{1}}C_{2}=1$. Conversely this implies that $E_{1}\cdot C_{2}=1$, and similarly $E_{2}\cdot C_{1}=1$. This gives $(E_{1}+E_{2})\cdot C_{1}=(E_{1}+E_{2})\cdot C_{2}=0$, which is impossible because $\rho_{W}=2$, $[C_{1}]$ and $[C_{2}]$ generate $\mathcal{N}_{1}(W)$, and we would get $E_{1}+E_{2}\equiv 0$.

8.27**.**

Finally suppose that $g_{1}\colon W\to Z_{1}$ is birational and $g_{2}\colon W\to\mathbb{P}^{2}$ is of fiber type. Set $E:=\operatorname{Exc}(g_{1})\subset W$ and $A:=g_{1}(E)\subset Z_{1}$. We have $\operatorname{NE}(W)=\operatorname{NE}(g_{1})+\operatorname{NE}(g_{2})$, and $E\cdot\operatorname{NE}(g_{1})<0$, hence $E\cdot\operatorname{NE}(g_{2})>0$.

If $\rho_{A}=1$ we get the statement, thus let us assume that $\rho_{A}>1$. Then by 8.24 there at most two indices $i\in\{1,\dotsc,\rho_{X}-2\}$ such that $E_{|B_{i}}$ is reducible, and at most two indices $i\in\{1,\dotsc,\rho_{X}-2\}$ such that $B_{i}$ dominates $\mathbb{P}^{2}$ under $g_{2}$. Since $\rho_{X}-2\geq 5$, there is at least one index $i$, say $i=1$, such that $g_{2}(B_{1})\subsetneq\mathbb{P}^{2}$ and $E_{|B_{1}}=C$ a $(-1)$-curve in $B_{1}$.

We show that $\rho_{B_{1}}=2$. Then the surface $g_{1}(B_{1})\subset Z_{1}$ has Picard number $1$, and this gives the statement.

Assume by contradiction that $\rho_{B_{1}}>2$. As in 8.25 we see that $g_{2}(B_{1})\subset\mathbb{P}^{2}$ is a curve and $g_{2|B_{1}}$ factors through a conic bundle $h\colon B_{1}\to\mathbb{P}^{1}$; since $\rho_{B_{1}}>2$, $h$ has at least one reducible fiber $\Gamma_{1}+\Gamma_{2}$, where $\Gamma_{i}$ are $(-1)$-curves with $\Gamma_{1}\cdot_{B_{1}}\Gamma_{2}=1$. Let $i\in\{1,2\}$. Note that in $W$ we have $[\Gamma_{i}]\in\operatorname{NE}(g_{2})$ and hence $E\cdot\Gamma_{i}>0$, therefore $C\cdot_{B_{1}}\Gamma_{i}=E_{|B_{1}}\cdot_{B_{1}}\Gamma_{i}=E\cdot\Gamma_{i}>0$.

However it is impossible to have such a configuration $C,\Gamma_{1},\Gamma_{2}$ of $(-1)$-curves in $B_{1}$, because $\rho_{B_{1}}\leq 5$ (see 8.24). Indeed, consider a birational map $\varphi\colon B_{1}\to\mathbb{P}^{2}$ that contracts $C$ to a point $p\in\mathbb{P}^{2}$. The $(-1)$-curves of $B_{1}$ are given by the exceptional curves for $\varphi$, and the transforms of the lines through two blown-up points. Since $C\cdot\Gamma_{i}>0$, $\varphi(\Gamma_{i})$ must be a line $\overline{pq_{i}}$ where $q_{i}$ is another blown-up point, but then the two lines meet only at $p$ and $\Gamma_{1}$ and $\Gamma_{2}$ should be disjoint in $B_{1}$. This concludes the proof of Lemma 8.23.∎

Proof of Th. 8.1.

By Lemma 8.23 there exists a birational contraction $\sigma\colon X\to Z$ with $\rho_{Z}=1$ and such that $\rho_{A_{i}}=1$ for some $i\in\{1,\dotsc,\rho_{X}-1\}$ (notation as in Prop. 8.8). Up to renumbering we can assume that $\rho_{A_{1}}=1$, so that $A_{1}\cong\mathbb{P}^{2}$. Similarly as before (see the proof of Lemma 8.23), using (8.20) we see that there are at least two indexes $j\in\{2,\dotsc,\rho_{X}-1\}$ with $\#(A_{1}\cap A_{j})=1$; up to renumbering we can assume that $m_{12}=m_{13}=1$.

8.28**.**

Consider the factorization of $\sigma$ given by:

[TABLE]

where $\rho_{Z^{\prime}}=3$ and $\sigma^{\prime}$ is the blow-up of $A_{2}$ and $A_{3}$. Let $A_{1}^{\prime}\subset Z^{\prime}$ be the transform of $A_{1}\subset Z$. Then $\sigma^{\prime}_{|A_{1}^{\prime}}\colon A_{1}^{\prime}\to A_{1}$ blows-up two points, so that $A_{1}^{\prime}\cong\operatorname{Bl}_{2\mskip 1.0mu\text{\it pts}}\mathbb{P}^{2}$ and $\rho_{A_{1}^{\prime}}=\rho_{Z^{\prime}}=3$. For $i=2,3$ we denote by $C_{i}\subset A_{1}^{\prime}$ the $(-1)$-curve over the point $A_{1}\cap A_{i}$, and $C_{0}\subset A_{1}^{\prime}$ the $(-1)$-curve not contracted by $\sigma^{\prime}$; recall that $\operatorname{NE}(A_{1}^{\prime})=\langle[C_{0}],[C_{2}],[C_{3}]\rangle$.

Let $\iota\colon A^{\prime}_{1}\hookrightarrow Z^{\prime}$ be the inclusion. By Cor. 8.21 $\iota_{*}\colon\mathcal{N}_{1}(A^{\prime}_{1})\to\mathcal{N}_{1}(Z^{\prime})$ is an isomorphism that induces an isomorphism between $\operatorname{NE}(A_{1}^{\prime})$ and $\operatorname{NE}(Z^{\prime})$, hence $\operatorname{NE}(Z^{\prime})$ is a simplicial $3$-dimensional cone generated by $[C_{0}],[C_{2}],[C_{3}]$.

For $i=0,2,3$ let $R_{i}$ be the extremal ray of $\operatorname{NE}(Z^{\prime})$ generated by $[C_{i}]$ (so that $\operatorname{NE}(\sigma^{\prime})=R_{2}+R_{3}$); by Cor. 8.21 each $R_{i}$ is of type $(3,2)$, with locus $E_{i}\subset Z^{\prime}$ such that $E_{i|A_{1}^{\prime}}=C_{i}$.

This implies that restriction $r^{\prime}\colon\operatorname{Pic}(Z^{\prime})\to\operatorname{Pic}(A^{\prime}_{1})$ is an isomorphism. Indeed $r^{\prime}$ is injective because $\iota_{*}$ is surjective, and it is surjective because $\operatorname{Pic}(A^{\prime}_{1})$ is generated by the classes of $C_{0},C_{2},C_{3}$, which are restrictions of divisors in $Z^{\prime}$.

8.29**.**

The restriction $\operatorname{Pic}(Z)\to\operatorname{Pic}(A_{1})$ is an isomorphism.

Proof.

We have a commutative diagram:

[TABLE]

where $r^{\prime}$ is an isomorphism and $(\sigma^{\prime})^{*}$ and $(\sigma^{\prime}_{|A_{1}^{\prime}})^{*}$ are injective, so that $r$ is injective too.

If $L\in\operatorname{Pic}(A_{1})$, consider $(\sigma^{\prime}_{|A_{1}^{\prime}})^{*}(L)\in\operatorname{Pic}(A_{1}^{\prime})$. Since $r^{\prime}$ is surjective, there exists $M\in\operatorname{Pic}(Z^{\prime})$ such that $M_{|A_{1}^{\prime}}=(\sigma^{\prime}_{|A_{1}^{\prime}})^{*}(L)$. We have $M\cdot C_{i}=M_{|A_{1}^{\prime}}\cdot C_{i}=0$ for $i=2,3$, so that $M=(\sigma^{\prime})^{*}(M_{2})$ for some $M_{2}\in\operatorname{Pic}(Z)$, and $(\sigma^{\prime}_{|A_{1}^{\prime}})^{*}(M_{2|A_{1}})=(\sigma^{\prime})^{*}(M_{2})_{|A_{1}^{\prime}}=M_{|A_{1}^{\prime}}=(\sigma^{\prime}_{|A_{1}^{\prime}})^{*}(L)$, hence $M_{2|A_{1}}=L$ and $r$ is surjective. ∎

8.30**.**

By Prop. 8.8 $Z$ is a Fano $4$-fold with at most locally factorial, isolated ordinary double points, $\rho_{Z}=1$, and $K_{A_{1}}=K_{Z|A_{1}}$. Since $A_{1}\cong\mathbb{P}^{2}$, 8.29 implies that $Z$ has index $3$, namely $Z$ is a del Pezzo variety, see [IP99, §3.2]. Let $d\in\mathbb{N}$ be its degree, i.e. $d=H^{4}$ where $H$ is the ample generator of $\operatorname{Pic}(Z)$ and $-K_{Z}=3H$. We have $\chi(Z,H)=h^{0}(Z,H)=d+3$ [ibid., Rem. 3.2.2(ii)], $\chi(Z,-H)=\chi(Z,-2H)=0$, and $\chi(Z,\mathcal{O}_{Z})=1$, which yields $\chi(Z,tH)=\frac{1}{24}(t+1)(t+2)(dt^{2}+3dt+12)$, and finally $h^{0}(Z,-K_{Z})=15d+10$.

8.31**.**

The birational map $\sigma^{\prime}\colon Z^{\prime}\to Z$ factors as two blow-ups $Z^{\prime}\to Z^{\prime\prime}\to Z$ with center first $A_{2}\subset Z$ and then the transform $A_{3}^{\prime\prime}\subset Z^{\prime\prime}$ of $A_{3}\subset Z$. By Prop. 8.8, $A_{2}$ is a smooth del Pezzo surface with $-K_{A_{2}}=-K_{Z|A_{2}}=3H_{|A_{2}}$, so that $A_{2}$ has again index $3$ and $A_{2}\cong\mathbb{P}^{2}$. Rem. 8.18 gives $h^{0}(Z^{\prime\prime},-K_{Z^{\prime\prime}})=h^{0}(Z,-K_{Z})-10=15d$.

By applying (8.20) to $X\to Z^{\prime\prime}$ and $A_{3}^{\prime\prime}\subset Z^{\prime\prime}$ we get $\rho_{A_{3}^{\prime\prime}}\leq 10-(\rho_{X}-\rho_{Z^{\prime\prime}})=12-\rho_{X}\leq 5$, thus again by Rem. 8.18 we get

[TABLE]

8.33**.**

Fix $j\in\{2,3\}$ and consider the contraction $g_{j}\colon Z^{\prime}\to Z_{j}$ with $\operatorname{NE}(g_{j})=R_{0}+R_{j}$ (see 8.28), so that $\rho_{Z_{j}}=1$. In $Z^{\prime}$ we have $E_{j}\cdot C_{0}=E_{j|A_{1}^{\prime}}\cdot_{A_{1}^{\prime}}C_{0}=C_{j}\cdot C_{0}=1$, and similarly $E_{0}\cdot C_{j}=1$. This implies that $g_{j}$ must be of fiber type. Indeed if $g_{j}$ is birational, we can apply Prop. 8.8 to $g_{j}\circ\sigma^{\prime\prime}\colon X\to Z_{j}$, and we get extremal rays $\widetilde{R}_{0}$, $\widetilde{R}_{j}$ in $\operatorname{NE}(g_{j}\circ\sigma^{\prime\prime})$, with $(\sigma^{\prime\prime})_{*}(\widetilde{R}_{0})=R_{0}$ and $(\sigma^{\prime\prime})_{*}(\widetilde{R}_{j})=R_{j}$, and loci $\widetilde{E}_{0}=(\sigma^{\prime\prime})^{*}(E_{0})$, $\widetilde{E}_{j}=(\sigma^{\prime\prime})^{*}(E_{j})$ in $X$. Then we get $\widetilde{E}_{j}\cdot\widetilde{R}_{0}>0$ by the projection formula, but it is shown in the proof of Prop. 8.8 that $\widetilde{E}_{j}\cdot\widetilde{R}_{0}=0$.

Therefore $g_{j}$ is of fiber type, and again by considering the composition $g_{j}\circ\sigma^{\prime\prime}\colon X\to Z_{j}$, we get $Z_{j}\cong\mathbb{P}^{2}$ by Lemma 8.5.

Let us also note that $(E_{0}+E_{j})\cdot R_{0}=(E_{0}+E_{j})\cdot R_{j}=0$, so that $\mathcal{O}_{Z^{\prime}}(E_{0}+E_{j})=g_{j}^{*}\mathcal{O}_{\mathbb{P}^{2}}(a_{j})$ for some $a_{j}\in\mathbb{N}$.

We have $(E_{0}+E_{j})_{|A_{1}^{\prime}}=C_{0}+C_{j}$ fiber of a conic bundle $h_{j}\colon A_{1}^{\prime}\to\mathbb{P}^{1}$, therefore $g_{j}(A_{1}^{\prime})\subset\mathbb{P}^{2}$ is a curve, and we have a factorization

[TABLE]

where $\nu_{j}$ is a finite map. Then $h_{j}^{*}\mathcal{O}_{\mathbb{P}^{1}}(1)=C_{0}+C_{j}=(E_{0}+E_{j})_{|A_{1}^{\prime}}=h_{j}^{*}\nu_{j}^{*}\mathcal{O}_{\mathbb{P}^{2}}(a_{j})_{|g_{j}(A_{1}^{\prime})}$, which gives $\mathcal{O}_{\mathbb{P}^{1}}(1)=\nu_{j}^{*}\mathcal{O}_{\mathbb{P}^{2}}(a_{j})_{|g_{j}(A_{1}^{\prime})}$. We conclude that $a_{j}=1$, $g_{j}(A_{1}^{\prime})$ is a line in $\mathbb{P}^{2}$, $\nu_{j}$ is an isomorphism, and $g_{j|A_{1}^{\prime}}=h_{j}$.

8.34**.**

Let now $f\colon Z^{\prime}\to W$ be the contraction of $R_{0}$; we have $\rho_{W}=2$, and $f$ is of type $(3,2)$ by Prop. 8.8$(c)$. Then Prop. 8.8 applies to the composition $f\circ\sigma^{\prime\prime}\colon X\to W$; in particular $W$ is a Fano $4$-fold with at most locally factorial, isolated ordinary double points. Set $A_{W}:=f(A_{1}^{\prime})\subset W$; then $A_{W}$ is one of the surfaces blown-up by $f\circ\sigma^{\prime\prime}$, hence it is a smooth del Pezzo surface with $K_{A_{W}}=K_{W|A_{W}}$.

Moreover $g_{j}\colon Z^{\prime}\to\mathbb{P}^{2}$ factors through $f$, so that $W$ has two elementary contractions $\pi_{j}\colon W\to\mathbb{P}^{2}$ for $j=1,2$, that induce a finite map $\pi\colon W\to\mathbb{P}^{2}\times\mathbb{P}^{2}$.

The restriction $f_{|A_{1}^{\prime}}$ contracts the “middle” $(-1)$-curve $C_{0}$ of $A_{1}^{\prime}\cong\operatorname{Bl}_{2\mskip 1.0mu\text{\it pts}}\mathbb{P}^{2}$, and $A_{W}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$. Moreover by 8.33 for $j=2,3$ the image $\pi_{j}(A_{W})\subset\mathbb{P}^{2}$ is a line, and $\pi_{j|A_{W}}\colon A_{W}\to\mathbb{P}^{1}$ are the two projections.

As in 8.28 we see that the restriction $\operatorname{Pic}(W)\to\operatorname{Pic}(A_{W})$ is an isomorphism, and $K_{W|A_{W}}=K_{A_{W}}$, therefore $W$ has index two. Moreover $K_{A_{W}}=\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(-2,-2)$ implies that $K_{W}=\pi_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(-2)+\pi_{2}^{*}\mathcal{O}_{\mathbb{P}^{2}}(-2)$.

Consider the general fiber $F_{1}\subset W$ of $\pi_{1}\colon W\to\mathbb{P}^{2}$. It is a smooth del Pezzo surface with $K_{F_{1}}=K_{W|F_{1}}$, thus $F_{1}$ has index two and $F_{1}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$. Notice that $\deg\pi=\deg(\pi_{2|F_{1}}\colon F_{1}\to\mathbb{P}^{2})$. We have

[TABLE]

hence $(\pi_{2|F_{1}})^{*}\mathcal{O}_{\mathbb{P}^{2}}(1)=\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}(1,1)$, and we conclude that $\deg\pi=2$. Finally $K_{W}=\pi^{*}\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{2}}(-2,-2)=\pi^{*}K_{\mathbb{P}^{2}\times\mathbb{P}^{2}}+R$ with $R\sim\pi^{*}\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{2}}(1,1)$, and $W$ is a double cover of $\mathbb{P}^{2}\times\mathbb{P}^{2}$ with branch locus a divisor of degree $(2,2)$. We have $\pi_{*}\mathcal{O}_{W}=\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{2}}(-1,-1)$, and

[TABLE]

therefore $h^{0}(W,-K_{W})=45$.

8.35**.**

The blow-up map $f\colon Z^{\prime}\to W$ has center a smooth del Pezzo surface of index two (by Prop. 8.8 and because $W$ has index two), therefore $h^{0}(Z^{\prime},-K_{Z^{\prime}})=h^{0}(W,-K_{W})-9=36$ by Rem. 8.18. Together with (8.32), this implies that $d=3$ and $Z\subset\mathbb{P}^{5}$ is a cubic $4$-fold, see [IP99, Th. 3.3.1]; moreover $A_{i}\subset Z$ is a plane for every $i=1,\dotsc,\rho_{X}-1$.

The intersection $A_{i}\cap A_{j}$ is finite, therefore $A_{i}$ and $A_{j}$ intersect only in one point, namely $m_{ij}=1$ for every $i\neq j$ (see Rem. 8.19). Moreover $A_{i}\cap A_{j}\cap A_{k}=\emptyset$ if $i<j<k$ by Prop. 8.8.

Let us write $\rho:=\rho_{X}$ for simplicity. The map $X\to Z$ factors a sequence of $\rho-1$ blow-ups:

[TABLE]

where, for $i=1,\dotsc,\rho-1$, $\sigma_{i}\colon X_{i+1}\to X_{i}$ blows-up a smooth del Pezzo surface $S_{i}\subset X_{i}$ with $K_{X_{i|S_{i}}}=K_{S_{i}}$, given by the transform of $A_{i}\subset Z$. The induced map $S_{i}\to A_{i}$ blows-up $\sum_{j=1}^{i-1}m_{ij}=i-1$ points, therefore $\rho_{S_{i}}=i$, and Rem. 8.18 gives

[TABLE]

and $h^{0}(X_{1},-K_{X_{1}})=55$. This gives $h^{0}(X,-K_{X})=\frac{1}{2}(\rho^{2}-23\rho+132)$; on the other hand $h^{0}(X,-K_{X})\geq 2$ (see Th. 2.12), which implies that $\rho_{X}\leq 9$. This concludes the proof of Th. 8.1.∎

9. Fano $4$-folds with a divisor of type $(3,0)^{\text{\it sm}}$

Let $X$ be a Fano $4$-fold with $\rho_{X}\geq 7$, that is not a product of surfaces; we consider now the case where $X$ has some small elementary contraction. The following remark shows that such $X$ must contain a fixed prime divisor of type $(3,0)^{\text{\it sm}}$, $(3,1)^{\text{\it sm}}$, or $(3,0)^{Q}$.

Remark 9.1**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, having a small elementary contraction $f\colon X\to Y$. We have $\operatorname{NE}(f)\not\subset\operatorname{mov}(X)=\operatorname{Eff}(X)^{\vee}$, therefore there exists a one-dimensional face $\tau$ of $\operatorname{Eff}(X)$ such that $\tau\cdot\operatorname{NE}(f)<0$. By Cor. 4.10 $\tau$ is a fixed face, hence $\tau=\mathbb{R}_{\geq 0}[D]$ for some fixed prime divisor $D\subset X$, and $D\cdot\operatorname{NE}(f)<0$. In particular $D$ contains $\operatorname{Locus}(f)$ which is a union of exceptional planes (Lemma 2.13), and $D$ cannot be of type $(3,2)$ (Lemma 5.3). We conclude that $D$ must be of type $(3,0)^{\text{\it sm}}$, $(3,1)^{\text{\it sm}}$, or $(3,0)^{Q}$ (see Th.-Def. 5.1).

In this section we consider the case where $X$ has a fixed prime divisor of type $(3,0)^{\text{\it sm}}$, and show the following result (Th. 1.3 from the Introduction).

Theorem 9.2**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ having a fixed prime divisor of type $(3,0)^{\text{\it sm}}$. Then one of the following holds:

1. $(i)$

$X$* has an elementary rational contraction onto a $3$-fold and $\rho_{X}\leq 9$;* 2. $(ii)$

$\rho_{X}=7$, $h^{0}(X,-K_{X})\in\{2,\dotsc,15\}$, and there is a SQM $X\dasharrow X^{\prime}$ such that $X^{\prime}=\operatorname{Bl}_{6\mskip 1.0mu\text{\it pts}}Y$, $Y$ a smooth Fano $4$-fold with $\rho_{Y}=1$ and $h^{0}(Y,-K_{Y})=h^{0}(X,-K_{X})+90\in\{92,\dotsc,105\}$. Moreover every fixed prime divisor of $X$ is of type $(3,0)^{\text{\it sm}}$.

9.3****Overview of the proof of Th. 9.2.

The proof follows the same strategy as in [Cas17], where the bound $\rho_{X}\leq 12$ is shown. Let us note that, in case $(i)$ where $X$ has an elementary rational contraction onto a $3$-fold, the bound $\rho_{X}\leq 9$ follows from Th. 1.5.

We treat separately the two cases where every fixed prime divisor is of type $(3,0)^{\text{\it sm}}$ (Prop. 9.4), and where there are also fixed prime divisors of other types (Prop. 9.20).

In the first case, there are a smooth Fano $4$-fold $Y$ and a SQM $X\dasharrow X^{\prime}$ such that $X^{\prime}=\operatorname{Bl}_{\text{\it pts}}Y$, and $Y$ does not have divisorial elementary rational contractions. We compute that $h^{0}(Y,-K_{Y})=h^{0}(X,-K_{X})+15(\rho_{X}-\rho_{Y})$.

If $Y$ has an elementary rational contraction onto a $3$-fold, then we lift this contraction to $X$, and get $(i)$. If $\rho_{Y}=1$, then either $Y\cong\mathbb{P}^{4}$ and we get again $(i)$ (see §11.1), or we get $h^{0}(Y,-K_{Y})\leq 105$ from [Cas17], which implies $(ii)$ (see 9.6).

Otherwise, we show that we can reduce to the situation where $\rho_{Y}=2$ and $Y$ has two distinct elementary rational contractions of fiber type $Y\dasharrow B_{i}$ where $B_{i}$ is $\mathbb{P}^{1}$ or $\mathbb{P}^{2}$, for $i=1,2$; moreover $\rho_{X}-\rho_{Y}\geq 5$ gives $h^{0}(Y,-K_{Y})\geq 75$. We show that this case does not happen, by studying explicitly the geometry of $Y$. This is the longest part of the proof (see 9.7 - 9.17).

Consider now the case where $X$ has a fixed prime divisor of type $(3,0)^{\text{\it sm}}$ and also some fixed prime divisor of another type. We know that $\operatorname{Eff}(X)$ is generated by classes of fixed prime divisors, and that every $2$-dimensional face is fixed (Cor. 4.10). Then there must a fixed prime divisor $D$ of type $(3,0)^{\text{\it sm}}$ and a fixed prime divisor $B$, not of type $(3,0)^{\text{\it sm}}$, that are adjacent. We consider the contraction $X\dasharrow Y$ of $D$, so that $Y$ is a smooth Fano $4$-fold, and the transform $B_{Y}\subset Y$ is still a fixed prime divisor. By analysing the possibilities for $B_{Y}$, we show that in each case there is a fixed prime divisor $E\subset X$, of type $(3,2)$, such that $D\cap E=\emptyset$. Then we get $(i)$ by Th. 7.1.

Proposition 9.4** (refinement of [Cas17], Th. 4.3).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, and suppose that every fixed prime divisor of $X$ is of type $(3,0)^{\text{\it sm}}$. Then the statement of Th. 9.2 holds.

Proof.

Let us first note that, if $X$ has an elementary rational contraction onto a $3$-fold, then $\rho_{X}\leq 9$ by Th. 1.5, and we get $(i)$.

We follow [Cas17, proof of Lemma 4.22], where it is shown that there is a diagram

[TABLE]

where $\varphi$ is a sequence of flips and divisorial elementary contractions, and $f$ is an elementary contraction of fiber type.

Up to increasing the number of flips and divisorial contractions in $\varphi$, we can also assume that $Y$ has no divisorial elementary rational contractions, hence $Y$ contains no fixed prime divisors (see Rem. 2.3).

By [ibid., Lemma 4.19], up to SQM we can assume that $Y$ is a smooth Fano $4$-fold, and there is a resolution $\varphi^{\prime}\colon X^{\prime}\to Y$ of $\varphi$ such that $\varphi^{\prime}$ is the blow-up of $Y$ in $r:=\rho_{X}-\rho_{Y}$ distinct points. Then we have:

[TABLE]

by Th. 2.12 and [ibid., Prop. 3.3].

If $\dim Z=3$, then by [ibid., Lemma 4.21] we have $(i)$.

9.6**.**

Suppose that $\rho_{Y}=1$ and $Z=\{\text{\it pt}\}$. If $Y\cong\mathbb{P}^{4}$, then $X$ is the Fano model of $\operatorname{Bl}_{\text{\it pts}}\mathbb{P}^{4}$ and has an elementary rational contraction onto $\operatorname{Bl}_{\text{\it pts}}\mathbb{P}^{3}$ (see §11.1), thus we have $(i)$. If instead $Y\not\cong\mathbb{P}^{4}$, as in [ibid., proof of Th. 3.2, p. 380] we see that $h^{0}(Y,-K_{Y})\leq 105$, which together with (9.5) implies $(ii)$.

9.7**.**

Suppose that $\dim Z=2$, and note that $Z$ is smooth (Lemma 3.19). We notice that $Z$ cannot have divisorial elementary contractions, otherwise the pullback under $f$ of the exceptional divisor would be a fixed prime divisor in $Y$, against our reductions. Therefore $Z$ is a minimal smooth rational surface with no divisorial elementary contraction, and we conclude that either $Z\cong\mathbb{P}^{2}$ and $\rho_{Y}=2$, or $Z\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ and $\rho_{Y}=3$. We also note that if $\dim Z=1$, then $Z\cong\mathbb{P}^{1}$ and $\rho_{Y}=2$.

9.8**.**

Suppose that $Z\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$, and consider the blow-up of the first point $W:=\operatorname{Bl}_{p_{1}}Y\to Y$ in $\varphi^{\prime}\colon X^{\prime}\to Y$. The composition $W\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ is not equidimensional, and by Prop. 3.12 it factors as

[TABLE]

where $W\dasharrow S$ is an elementary rational contraction, $S$ is a smooth surface (see Lemma 3.19), and $S\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ is the blow-up of a smooth point, thus $S\cong\operatorname{Bl}_{2\mskip 1.0mu\text{\it pts}}\mathbb{P}^{2}$.

Consider a $(-1)$-curve of $S$ contracted by $S\to\mathbb{P}^{2}$, and its pullback $G$ in $W$. This is a fixed prime divisor, and up to to replacing $W$ with a SQM, we can assume that $G=\operatorname{Exc}(g)$ for some divisorial elementary contraction $g\colon W\to Y^{\prime}$ (see Rem. 2.3). Then $g$ must be the blow-up of a smooth point (by [Cas17, Lemma 4.19(2)] applied to the composition $X\dasharrow Y^{\prime}$), and there is an elementary rational contraction $Y^{\prime}\dasharrow\mathbb{F}_{1}$.

[TABLE]

Now we consider the blow-up $\mathbb{F}_{1}\to\mathbb{P}^{2}$ and the composition $Y^{\prime}\dasharrow\mathbb{P}^{2}$; it factors again as an elementary divisorial rational contraction $Y^{\prime}\dasharrow Y^{\prime\prime}$ and an elementary contraction $Y^{\prime\prime}\to\mathbb{P}^{2}$. Finally, by replacing $Y$ with $Y^{\prime\prime}$, and we can assume that $\rho_{Y}=2$ and $Z\cong\mathbb{P}^{2}$.

9.9**.**

We are left with the possibility that $\rho_{Y}=2$ and $Y$ has two distinct elementary rational contractions of fiber type $Y\dasharrow B_{i}$ where $B_{i}$ is $\mathbb{P}^{1}$ or $\mathbb{P}^{2}$, for $i=1,2$, so that $\operatorname{Mov}(Y)=\operatorname{Eff}(Y)$ and the two one-dimensional faces of this cone are the pullbacks of $\operatorname{Nef}(B_{i})$, $i=1,2$. We are going to show that this contradicts $\rho_{X}\geq 7$.

9.10**.**

Suppose that $Y$ has index $>1$. Then $Y$ cannot have small elementary contractions (see Lemma 2.13), therefore both maps $Y\to B_{i}$ must be regular. Since each map must be finite on the fibers of the other one, such fibers are at most $2$-dimensional, and we conclude that $Y$ has two elementary contractions onto $\mathbb{P}^{2}$.

Fano $4$-folds with index $>1$ and Picard number $>1$ are classified, see [IP99, Cor. 3.1.15, Th. 3.3.1, Th. 7.2.15]. An inspection of the classification shows that the only cases where $\rho_{Y}=2$ and $Y$ has two elementary contractions onto $\mathbb{P}^{2}$ are $\mathbb{P}^{2}\times\mathbb{P}^{2}$ and a double cover of $\mathbb{P}^{2}\times\mathbb{P}^{2}$ ramified over a divisor of degree $(2,2)$.

On the other hand $Y$ cannot be covered by a family of curves of anticanonical degree $3$ by [Cas17, Lemma 3.12], and this excludes $\mathbb{P}^{2}\times\mathbb{P}^{2}$. In the other case we have $h^{0}(Y,-K_{Y})=45$ (see 8.34), and by (9.5) this implies that $r\leq 2$ and $\rho_{X}\leq 4$, a contradiction.

9.11**.**

We suppose from now on that $Y$ has index one.

For $i=1,2$ let us consider a $K$-negative resolution $f_{i}\colon Y_{i}\to B_{i}$ of $Y\dasharrow B_{i}$; we have a diagram:

[TABLE]

where $\psi_{i}$ is a SQM. Fix $i\in\{1,2\}$ and let $D_{i}\subset Y_{i}$ be a general fiber of $f_{i}$ if $B_{i}=\mathbb{P}^{1}$, the pullback of a general line if $B_{i}=\mathbb{P}^{2}$. Let $\widetilde{D}_{i}\subset Y$ be the transform of $D_{i}$.

9.12**.**

We notice that the general fiber $F_{i}$ of $f_{i}$ cannot have a covering family of rational curves of anticanonical degree $3$. Otherwise, $Y_{i}$ would have a covering family $V$ of rational curves, of anticanonical degree $3$, contracted to points by $f_{i}$. However this is excluded as in [Cas17, 4.24].

This implies that $D_{i}$ contains an irreducible curve $\Gamma_{i}$, contracted by $f_{i}$, with $c_{i}:=-K_{D_{i}}\cdot\Gamma_{i}\in\{1,2,4\}$ [Kol96, Cor. IV.1.15], and $c_{i}\in\{1,2\}$ if $B_{i}=\mathbb{P}^{2}$. Notice that $D_{i}\cdot\Gamma_{i}=0$ and $-K_{Y_{i}}\cdot\Gamma_{i}=c_{i}$.

Since $F_{i}$ is Fano, it is rationally connected. Consider $f_{i}\colon Y_{i}\to B_{i}$ if $B_{i}=\mathbb{P}^{1}$, or $f_{i|D_{i}}\colon D_{i}\to\mathbb{P}^{1}$ if $B_{i}=\mathbb{P}^{2}$. These are families of rationally connected varieties over a curve, and by [GHS03] they have a section, namely there exists $C_{i}\subset Y_{i}$ such that $D_{i}\cdot C_{i}=1$.

9.13**.**

We have $\operatorname{Eff}(Y)=\langle[\widetilde{D}_{1}],[\widetilde{D}_{2}]\rangle$, so there exist positive integers $n,a_{1},a_{2}$ such that

[TABLE]

and we can assume that $\gcd(n,a_{1},a_{2})=1$. Now let $\widehat{D}_{2}\subset Y_{1}$ be the transform of $D_{2}$. In $Y_{1}$ we have

[TABLE]

and intersecting with $C_{1}$ we get

[TABLE]

This implies that $\gcd(a_{2},n)$ divides $a_{1}$, and hence that $\gcd(a_{2},n)=1$. Similarly we get that $\gcd(a_{1},n)=1$.

Set $d:=\gcd(a_{1},a_{2})$ and $a_{i}:=da_{i}^{\prime}$ for $i=1,2$. We have

[TABLE]

and $\gcd(n,d)=1$. This implies that $d=1$, because $Y$ has index $1$. Now intersecting with $\Gamma_{1}$ in $Y_{1}$ we get from (9.14):

[TABLE]

which implies that $a_{2}$ divides $c_{1}$ and hence that $a_{2}\in\{1,2,4\}$, and $a_{2}\in\{1,2\}$ if $B_{1}=\mathbb{P}^{2}$. Similarly we deduce that $a_{1}\in\{1,2,4\}$.

Since $\gcd(a_{1},a_{2})=1$, up to exchanging $Y_{1}$ and $Y_{2}$ we can assume that $a_{1}=1$, so that (9.14) becomes:

[TABLE]

9.16**.**

Suppose that $B_{1}\cong\mathbb{P}^{1}$. Then $D_{1}$ is a smooth Fano $3$-fold and $h^{0}(D_{1},-K_{Y_{1}|D_{1}})=h^{0}(D_{1},-K_{D_{1}})=\frac{1}{2}(-K_{D_{1}})^{3}+3\leq 35$ [IP99, Cor. 7.1.2]. Moreover by (9.15):

[TABLE]

and we know that $h^{0}(Y_{1},-K_{Y_{1}})=h^{0}(Y,-K_{Y})>0$ by Th. 2.12; thus we get

[TABLE]

Now the exact sequence

[TABLE]

yields

[TABLE]

that together with (9.5) implies that $r\leq 3$ and $\rho_{X}\leq 5$, a contradiction.

9.17**.**

Suppose now that $B_{1}\cong\mathbb{P}^{2}$, and consider the blow-up of the first point $\operatorname{Bl}_{p_{1}}Y\to Y$ in $\varphi^{\prime}\colon X^{\prime}\to Y$. By [Cas17, Lemma 4.19(1)], applied to the map $X\dasharrow\operatorname{Bl}_{p_{1}}Y$, we see that there is a smooth Fano $4$-fold $W$, with $\rho_{W}=3$, and a SQM $\operatorname{Bl}_{p_{1}}Y\dasharrow W$. Moreover there is a SQM of $X$ which is obtained by blowing-up $W$ at $\rho_{X}-3$ distinct points, so that using again [ibid., Prop. 3.3] we have:

[TABLE]

Since $p_{1}$ cannot belong to an exceptional plane [ibid., Lemma 4.4(3)], we see that $p_{1}\in\operatorname{dom}(\psi_{i})$ for $i=1,2$ (see 9.11); we still denote by $p_{1}$ its image in $Y_{i}$. Let

[TABLE]

be the blow-up of $p_{1}$, and $E_{i}\subset W_{i}$ the exceptional divisor. Then $W_{i}$ is another SQM of $\operatorname{Bl}_{p_{1}}Y$ and $W$.

Consider the composition $f_{1}\circ\sigma_{1}\colon W_{1}\to\mathbb{P}^{2}$. As in 9.8, we see that this map must factor through an elementary rational contraction $W_{1}\dasharrow\mathbb{F}_{1}$, where $h\colon\mathbb{F}_{1}\to\mathbb{P}^{2}$ is the blow-up of the point $f_{1}(p_{1})$.

[TABLE]

Recall that $D_{1}=f_{1}^{*}(l)$ where $l\subset\mathbb{P}^{2}$ is a general line (see 9.11). Using (9.15) we get:

[TABLE]

Let $\alpha\colon W_{1}^{\prime}\to\mathbb{F}_{1}$ be a $K$-negative resolution of $W_{1}\dasharrow\mathbb{F}_{1}$, $\widehat{D}_{2}^{\prime}\subset W_{1}^{\prime}$ the transform of $\sigma_{1}^{*}(\widehat{D}_{2})$, and $E_{1}^{\prime}\subset W_{1}^{\prime}$ the transform of $E_{1}=\operatorname{Exc}(\sigma_{1})$. We have $E_{1}^{\prime}=\alpha^{*}(e)$ where $e\subset\mathbb{F}_{1}$ is the $(-1)$-curve. Let also $\pi\colon\mathbb{F}_{1}\to\mathbb{P}^{1}$ be the $\mathbb{P}^{1}$-bundle; we have $h^{*}(l)=f+e$ where $f\subset\mathbb{F}_{1}$ is a general fiber of $\pi$. Finally let us consider $F=\alpha^{*}(f)$ a general fiber of $\pi\circ\alpha\colon W_{1}^{\prime}\to\mathbb{P}^{1}$.

[TABLE]

We note that $F$ is a smooth Fano $3$-fold. Indeed there is a $K$-negative resolution $W_{1}^{\prime\prime}\to\mathbb{P}^{1}$ of $\pi\circ\alpha$, and the indeterminacy locus of the SQM $W_{1}^{\prime}\dasharrow W_{1}^{\prime\prime}$ is contracted to points by $\alpha$, therefore it does not meet $F$. Hence $F$ is isomorphic to a general fiber of $W_{1}^{\prime\prime}\to\mathbb{P}^{1}$, which is Fano. We also note that $\alpha(F)=f$, so that $\rho_{F}\geq 2$.

In $W_{1}^{\prime}$ we get, from (9.19):

[TABLE]

Recall that $\mathcal{O}_{Y_{2}}(D_{2})=f_{2}^{*}\mathcal{O}_{B_{2}}(1)$ (see 9.11), and that $a_{2}\in\{1,2\}$ because $B_{1}\cong\mathbb{P}^{2}$ (see 9.13). Therefore we have:

[TABLE]

Moreover $-K_{W_{1}^{\prime}|F}=-K_{F}$ and $h^{0}(F,-K_{F})=\frac{1}{2}(-K_{F})^{3}+3\leq 34$ by classification [IP99, Ch. 12], and finally we get $h^{0}(W,-K_{W})=h^{0}(W_{1}^{\prime},-K_{W_{1}^{\prime}})\leq h^{0}(W_{1}^{\prime},-K_{W_{1}^{\prime}}-F)+h^{0}(F,-K_{F})\leq 40$. Together with (9.18) this implies that $\rho_{X}\leq 5$, again a contradiction. This concludes the proof of Prop. 9.4.∎

Proposition 9.20** (refinement of [Cas17], Th. 5.40).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ having both a fixed prime divisor of type $(3,0)^{\text{\it sm}}$ and a fixed prime divisor not of type $(3,0)^{\text{\it sm}}$.

Then $X$ contains a fixed prime divisor $D$ of type $(3,0)^{\text{\it sm}}$ and a fixed prime divisor $E$ of type $(3,2)$ such that $D\cap E=\emptyset$; in particular $\mathcal{N}_{1}(E,X)\subsetneq\mathcal{N}_{1}(X)$.

Proof.

We proceed as in [Cas17, proof of Th. 5.40]. Since $X$ has a fixed prime divisor of type $(3,0)^{\text{\it sm}}$, $X$ is not isomorphic to a product of surfaces, hence $\delta_{X}\leq 1$ (Th. 1.9) and every face of $\operatorname{Eff}(X)$ of dimension $\leq 2$ is fixed (Cor. 4.10).

Since every one-dimensional face of $\operatorname{Eff}(X)$ is generated by a fixed prime divisor, there must a $2$-dimensional face $\langle[D],[B]\rangle$ such that $D$ is of type $(3,0)^{\text{\it sm}}$ and $B$ is not. This face is fixed, hence $D$ and $B$ are adjacent.

Let $X\dasharrow\widetilde{X}\stackrel{{\scriptstyle\sigma}}{{\to}}Y$ be the contraction of $D$ as in Th.-Def. 5.1$(a)$, so that $X\dasharrow\widetilde{X}$ is a SQM, $Y$ is a smooth Fano $4$-fold with $\rho_{Y}\geq 6$, and $\sigma$ is the blow-up of a point $p\in Y$ with exceptional divisor the transform of $D$; we also have $\delta_{Y}\leq 2$ by Lemma 5.18, so that Th.-Def. 5.1 applies to $Y$ too.

We note that since every $2$-dimensional face of $\operatorname{Eff}(X)$ is fixed, the same holds for every one-dimensional face of $\operatorname{Eff}(Y)$, and the transform $B_{Y}\subset Y$ of $B$ is a fixed prime divisor of $Y$ (see [Cas17, Lemma 2.21]). Moreover $p\not\in B_{Y}$ by [ibid., Lemma 5.41], and $B_{Y}$ is not of type $(3,0)^{\text{\it sm}}$ by [ibid., Lemma 4.14].

Let $C_{1},\dotsc,C_{s}\subset Y$ be the images of the exceptional lines of $\widetilde{X}$; then $C_{i}$ cannot meet any irreducible curve of anticanonical degree one in $Y$, for every $i=1,\dotsc,s$ [ibid., Lemma 3.11(1)].

If $B_{Y}$ is of type $(3,2)$, then $B_{Y}$ is covered by irreducible curves of anticanonical degree one, therefore $B_{Y}\cap C_{i}=\emptyset$ for every $i=1,\dotsc,s$. Then $B_{Y}$ is contained in the domain of the birational map $Y\dasharrow X$, therefore $B\cong B_{Y}$ and $B$ is of type $(3,2)$ too. Moreover $B\cap D=\emptyset$, and we have the statement.

Suppose that $B_{Y}$ is of type $(3,0)^{Q}$. Note that $Y$ cannot have a covering family of curves of anticanonical degree $3$ (see [ibid., Lemma 3.12]), hence by [ibid., Prop. 5.18] $Y$ must contain a prime divisor $E$ covered by curves of anticanonical degree one. As before, we have $C_{i}\cap E=\emptyset$ for every $i=1,\dotsc,s$. Moreover, as in [ibid., 5.44] we see that $[C_{i}]\in\operatorname{mov}(Y)$ for every $i=1,\dotsc,s$. Thus $\tau:=E^{\perp}\cap\operatorname{mov}(Y)$ is a face of $\operatorname{mov}(Y)$ containing $[C_{1}],\dotsc,[C_{s}]$.

By [ibid., Lemma 5.7] we have

[TABLE]

Therefore $\tau$ is a facet of $\operatorname{mov}(Y)$, and dually $\mathbb{R}_{\geq 0}[E]$ is a one-dimensional face of $\operatorname{Eff}(Y)=\operatorname{mov}(Y)^{\vee}$. Hence $E$ is a fixed divisor, it must be of type $(3,2)$ [ibid., Lemma 2.18], and as in the previous part of the proof we see that the transform of $E$ in $X$ is a fixed prime divisor of type $(3,2)$ disjoint from $D$.

Finally, if $B_{Y}$ is of type $(3,1)^{\text{\it sm}}$, there is a SQM $\varphi\colon Y\dasharrow\widetilde{Y}$ and a smooth Fano $4$-fold $Z$ such that $\widetilde{Y}$ is the blow-up of $Z$ along a smooth irreducible curve $C\subset Z$, with exceptional divisor the transform $B_{\widetilde{Y}}\subset\widetilde{Y}$ of $B_{Y}$. Let $\alpha$ be a one-dimensional face of $\operatorname{Eff}(Z)$ with $\alpha\cdot C>0$. Then $\alpha\not\subset\operatorname{Mov}(Z)$ by [ibid., Lemma 5.17], so that $\alpha=\mathbb{R}_{\geq 0}[G]$ where $G\subset Z$ is a fixed prime divisor; let $G_{\widetilde{Y}}\subset\widetilde{Y}$, $G_{Y}\subset Y$, and $G_{X}\subset X$ be its transforms. By [Cas22, Lemmas 4.2 and 4.4], $\langle[D],[B],[G_{X}]\rangle$ is a fixed face of $\operatorname{Eff}(X)$, such that $G_{X}$ is adjacent to both $D$ and $B$ . Since $G\cdot C>0$, we have $G_{\widetilde{Y}}\cap B_{\widetilde{Y}}\neq\emptyset$, thus $G_{Y}\cap B_{Y}$ is non-empty and is not a union of exceptional planes (recall that $\dim(\widetilde{Y}\smallsetminus\operatorname{dom}\varphi)=1$ by Lemma 2.14). This implies that also $G_{X}\cap B$ is non-empty and is not a union of exceptional planes. Then $G_{X}$ must be of type $(3,2)$ by [Cas22, Lemma 4.13(b)], hence $G_{X}\cap D=\emptyset$ by Lemma 5.11, and we get the statement. ∎

Proof of Th. 9.2.

If every fixed prime divisor of $X$ is of type $(3,0)^{\text{\it sm}}$, we apply Prop. 9.4. Otherwise by Prop. 9.20 there are a fixed prime divisor $E$ of type $(3,2)$, and another fixed prime divisor $D$, such that $D\cap E=\emptyset$. Then we have $\mathcal{N}_{1}(E,X)\subsetneq\mathcal{N}_{1}(X)$ (see Rem. 2.7), and we get $(i)$ by Th. 7.1. ∎

10. Fano $4$-folds with a divisor of type $(3,1)^{\text{\it sm}}$ or $(3,0)^{Q}$

In this section we conclude our study of Fano $4$-folds with $\rho\geq 7$ having a small elementary contraction, by considering the case where $X$ has a fixed prime divisor of type $(3,1)^{\text{\it sm}}$ or $(3,0)^{Q}$ (Th. 10.1). Then we prove Cor. 10.5 characterizing Fano $4$-folds with $\rho\geq 9$ that are not products of surfaces; this result implies Theorems 1.1 and 1.4 from the Introduction. Finally we also give an application to Fano $4$-folds with $\rho=8$ (Cor. 10.6).

Theorem 10.1**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 9$ having a fixed prime divisor of type $(3,1)^{\text{\it sm}}$ or $(3,0)^{Q}$. Then $X$ has a rational contraction onto a $3$-fold and $\rho_{X}=9$.

Note that in the setting of the theorem, it is enough to show that $X$ has a rational contraction onto a $3$-fold, because then $\rho_{X}=9$ follows from Th. 1.5.

To prove Th. 10.1, we follow [Cas22, proofs of Th. 8.1 and Th. 9.1], where in the same setting the bound $\rho_{X}\leq 12$ is shown. Thanks to the results in the previous sections, in particular Cor. 4.10, Th. 6.1, and Th. 7.1, and also to Th. 1.9, we can improve the bound to $9$ and simplify the proofs. We first treat the case where $X$ has a fixed prime divisor of type $(3,1)^{\text{\it sm}}$, and then the case where $X$ has a fixed prime divisor of type $(3,0)^{Q}$ (and no fixed prime divisors of type $(3,1)^{\text{\it sm}}$). Since the proofs from [Cas22] are quite technical, instead of reporting them in full we will point out which modifications are needed to get our statement; for the case of a divisor of type $(3,1)^{\text{\it sm}}$ we will also give a highlight of the main points of the proof.

Proof of Th. 10.1, case $(3,1)^{\text{\it sm}}$.

We follow [Cas22, proof of Th. 8.1].

Let us first assume that: $X$ is a smooth Fano $4$-fold with $\rho_{X}\geq 8$, having a fixed prime divisor $D$ of type $(3,1)^{\text{\it sm}}$, and with no rational contraction onto a $3$-fold.

Since $X$ has a fixed prime divisor of type $(3,1)^{\text{\it sm}}$, $X$ is not isomorphic to a product of surfaces. Then we have the following properties:

1. $\bullet$

$\delta_{X}\leq 1$ (Th. 1.9); 2. $\bullet$

$\mathcal{N}_{1}(E,X)=\mathcal{N}_{1}(X)$ for every fixed prime divisor $E\subset X$ of type $(3,2)$ (Th. 7.1); 3. $\bullet$

every face of $\operatorname{Eff}(X)$ of dimension $\leq 3$ is fixed (Cor. 4.10); 4. $\bullet$

$X$ has no fixed prime divisor of type $(3,0)^{\text{\it sm}}$ (Prop. 9.20).

We consider the contraction $X\dasharrow\widetilde{X}\stackrel{{\scriptstyle\sigma}}{{\to}}Y$ of $D$ as in Th.-Def. 5.1, so that $Y$ is a smooth Fano $4$-fold with $\rho_{Y}\geq 7$ and $\sigma$ is the blow-up of a smooth irreducible curve $C\subset Y$. Then we have:

1. $(a)$

$Y$ has no rational contraction onto a $3$-fold; 2. $(b)$

$\mathcal{N}_{1}(E,Y)=\mathcal{N}_{1}(Y)$ for every fixed prime divisor $E\subset Y$ of type $(3,2)$; 3. $(c)$

$Y$ has no fixed prime divisor of type $(3,0)^{\text{\it sm}}$; 4. $(d)$

$Y$ is not isomorphic to a product of surfaces; 5. $(e)$

$\delta_{Y}\leq 1$.

Here $(a)$, $(b)$, and $(c)$ follow from the same properties of $X$ as shown in [Cas22, 8.2], and $(d)$ and $(e)$ are proved in [ibid., 8.3]. We also have:

1. $(f)$

every face of $\operatorname{Eff}(Y)$ of dimension $\leq 2$ is fixed (Cor. 4.10).

As in [ibid., 8.5], we introduce the following notation: if $C$ belongs to a family of rational curves of anticanonical degree one with locus a divisor, we denote this divisor by $E_{0}$. Then we have:

1. $(g)$

if $Y$ contains a nef prime divisor $H$ covered by a family $V$ of rational curves of anticanonical degree one, then $H=E_{0}$ and $[C]\equiv[V]$.

Indeed we must have $H\cap C\neq\emptyset$ by Prop. 7.10, $(a)$, and $(b)$. Then $C$ is a member of the family $V$ by [ibid., Lemma 4.21$(a)$], so that $H=E_{0}$.

As in [ibid., 8.6 - 8.7], we denote by $E_{1},\dotsc,E_{r}$ the set of fixed prime divisors of $Y$ of type $(3,2)$ and different from $E_{0}$. It is shown in [ibid., 8.12 - 8.16] that $r\geq\rho_{X}-5$, and that for every $i=1,\dotsc,r$ the transform $\widetilde{E}_{i}\subset X$ of $E_{i}$ is a fixed prime divisor of type $(3,2)$ having intersection zero with $C_{D}$. Note that [ibid., 8.9 and 8.10] are not needed because of $(f)$, and [ibid., 8.11] is replaced by $(g)$. Moreover [ibid., 8.18] shows that every fixed prime divisor of $Y$ is of type $(3,1)^{\text{\it sm}}$ or $(3,2)$ (and both types do occur), hence by [ibid., Lemma 4.22] every fixed prime divisor of $X$, adjacent to $D$, is still of type $(3,1)^{\text{\it sm}}$ or $(3,2)$. We get the following intermediate result.

Proposition 10.2** (refinement of [Cas22], Prop. 8.17).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 8$ and $D\subset X$ a fixed prime divisor of type $(3,1)^{\text{\it sm}}$. Then one of the following holds:

1. $(i)$

$X$* has a rational contraction onto a $3$-fold;* 2. $(ii)$

there are at least $\rho_{X}-5$ fixed prime divisors of type $(3,2)$ adjacent to $D$ and having intersection zero with $C_{D}$; moreover every fixed prime divisor adjacent to $D$ is of type $(3,1)^{\text{\it sm}}$ or $(3,2)$.

We assume from now on that $\rho_{X}\geq 9$, namely that $X$ is a smooth Fano $4$-fold with $\rho_{X}\geq 9$, having a fixed prime divisor $D$ of type $(3,1)^{\text{\it sm}}$, and with no rational contraction onto a $3$-fold. We show that this gives a contradiction.

Notice that $\rho_{Y}\geq 8$, in particular every face of $\operatorname{Eff}(Y)$ of dimension $\leq 3$ is fixed, by Cor. 4.10 and $(d)$; this replaces [ibid., 8.24].

Let $B\subset Y$ be a fixed prime divisor of type $(3,1)^{\text{\it sm}}$; by $(a)$ and Prop. 10.2 applied to $Y$ and $B$, we see that $B$ is adjacent to some divisor among $E_{1},\dotsc,E_{r}$.

Using this, [ibid., 8.26 - 8.27] shows that there exist a fixed prime divisor $B_{1}\subset Y$ of type $(3,1)^{\text{\it sm}}$, and $i\in\{1,\dotsc,r\}$, say $i=1$, such that $B_{1}+E_{1}$ is movable and not big, so that the linear system $m|B_{1}+E_{1}|$ for $m\gg 0$ defines a rational contraction of fiber type $f\colon Y\dasharrow Z$. By [ibid., Lemma 5.11], $(a)$, and $(b)$, we have $Z\cong\mathbb{P}^{2}$ and $f$ is special.

We note that $C$ cannot be contracted to a point by $f$, otherwise the composition $X\dasharrow\mathbb{P}^{2}$ has a unique prime divisor (precisely $D$) contracted to a point, and this contradicts Lemma 7.9. Thus $C$ is not contracted to a point, and as in [ibid., proof of 8.9], this implies that the general fiber of $f$ is $\mathbb{P}^{2}$, hence the minimal face $\tau_{f}$ of $\operatorname{Eff}(Y)$ containing $f^{*}\operatorname{Nef}(\mathbb{P}^{2})$ is a facet (see Def.-Rem. 3.2).

Then [ibid., 8.29] shows that there are irreducible curves $\Gamma_{1},\dotsc,\Gamma_{\rho_{Y}-2}\subset\mathbb{P}^{2}$, and fixed prime divisors $B_{2},\dotsc,B_{\rho_{Y}-2}\subset Y$ of type $(3,1)^{\text{\it sm}}$, such that (up to renumbering) $f^{*}(\Gamma_{i})=B_{i}+E_{i}$ for $i=1,\dotsc,\rho_{Y}-2$, and

[TABLE]

Moreover $\langle[B_{1}],\dotsc,[B_{\rho_{Y}-4}],[E_{\rho_{Y}-3}],[E_{\rho_{Y}-2}]\rangle$ is a face of $\tau_{f}$ of dimension $\rho_{Y}-2$, hence there exists a facet $\eta$ of $\operatorname{Eff}(Y)$ such that

[TABLE]

(see [ibid., 8.32]).

If $P\subset Y$ is a fixed prime divisor such that $[P]\in\eta\smallsetminus\tau_{f}$, [ibid., 8.33 - 8.35] shows that, up to renumbering, $P+B_{i}$ is movable for $i=1,\dotsc,\rho_{Y}-6$. Let $\eta_{1}$ be the minimal face of $\eta$ containing $[(\rho_{Y}-6)P+B_{1}+\cdots+B_{\rho_{Y}-6}]$, so that $\eta_{1}$ contains $[P],[B_{1}],\dotsc,[B_{\rho_{Y}-6}]$, and $[P+B_{i}]\in\eta_{1}\cap\operatorname{Mov}(Y)$ for every $i=1,\dotsc,\rho_{Y}-6$; in particular $\eta_{1}\cap\operatorname{Mov}(Y)$ is a face of $\operatorname{Mov}(Y)$ of dimension $\geq\rho_{Y}-6$. As in Rem. 3.11, we take a cone $\alpha\in\operatorname{MCD}(Y)$ such that $\alpha\subset\eta_{1}\cap\operatorname{Mov}(Y)$ and $\dim\alpha=\dim(\eta_{1}\cap\operatorname{Mov}(Y))$, and consider the rational contraction $h\colon Y\dasharrow S$ such that $\alpha=h^{*}(\operatorname{Nef}(S))$ (see [ibid., 8.36]).

Then $h$ is of fiber type and special, by Lemma 3.10; moreover $\rho_{S}=\dim(\eta_{1}\cap\operatorname{Mov}(Y))\geq\rho_{Y}-6\geq 2$, hence $S\not\cong\mathbb{P}^{1}$, and by $(a)$ we have $\dim S=2$. By [ibid., Lemma 5.12] and $(b)$, $h$ must be quasi-elementary.

Finally Th. 6.1, $(a)$, and $(d)$ imply that $h$ is elementary, therefore $\rho_{S}=\rho_{Y}-1$, $\eta_{1}=\eta$, and $h^{*}(\mathcal{N}^{1}(S))=\mathbb{R}\eta$. On the other hand $[E_{\rho_{Y}-2}]\in\eta$, and by [ibid., Lemma 5.11] and $(b)$ this gives a contradiction. This concludes the first part of the proof of Th. 10.1. ∎

Lemma 10.3** (refinement of [Cas22], Lemma 9.2).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and let $D,E$ be two adjacent fixed prime divisors of type $(3,0)^{Q}$ in $X$. Let $L\subset E$ be an exceptional plane such that $L\cap D=\emptyset$. Then one of the following holds:

1. $(i)$

$X$* has a rational contraction onto a $3$-fold;* 2. $(ii)$

there exists an exceptional plane $M\subset D$ such that $C_{M}+C_{E}\equiv C_{D}+C_{L}$ and $D\cdot C_{M}=-1$.

Proof.

Note that $X$ is not a product of surfaces, because it contains fixed prime divisors of type $(3,0)^{Q}$, hence $\delta_{X}\leq 1$ by Th. 1.9. The same proof as the one of [Cas22, Lemma 9.2] applies, just using Th. 7.1 instead of [ibid., Th. 4.8], and Prop. 7.10 and Th. 7.1 instead of [ibid., Prop. 7.1]. ∎

Lemma 10.4** (refinement of [Cas22], Lemma 9.3).**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and let $D,E$ be two adjacent fixed prime divisors of type $(3,0)^{Q}$ in $X$. Suppose that one of the following holds:

1. $(a)$

every fixed prime divisor is of type $(3,0)^{Q}$; 2. $(b)$

there is a fixed prime divisor $F$, of type $(3,2)$, such that $\langle[D],[E],[F]\rangle$ is a fixed face of $\operatorname{Eff}(X)$.

Then one of the following holds:

1. $(i)$

$X$* has a rational contraction onto a $3$-fold;* 2. $(ii)$

there exist $L_{1},\dotsc,L_{\rho_{X}-2}\subset E$ exceptional planes, disjoint from $D$, such that $[C_{E}],[C_{L_{1}}],\dotsc,[C_{L_{\rho_{X}-2}}]$ is a basis of $D^{\perp}$. In case $(b)$, we can moreover assume that $C_{E}\equiv C_{F}+C_{L_{1}}$ and $[C_{E}],[C_{F}],[C_{L_{2}}],\dotsc,[C_{L_{\rho_{X}-2}}]$ is a basis of $D^{\perp}$.

Proof.

Note that $X$ is not a product of surfaces, because it contains fixed prime divisors of type $(3,0)^{Q}$, hence $\delta_{X}\leq 1$ by Th. 1.9. The same proof as the one of [Cas22, Lemma 9.3] applies, just using Lemma 7.8 instead of [ibid., Lemma 7.2], and Th. 7.1 instead of [ibid., Th. 4.8]. ∎

Proof of Th. 10.1, case $(3,0)^{Q}$.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 9$ having a fixed prime divisor of type $(3,0)^{Q}$. Note that $X$ is not a product of surfaces, therefore $\delta_{X}\leq 1$ by Th. 1.9.

If $X$ has a rational contraction onto a $3$-fold, then $\rho_{X}=9$ by Th. 1.5. We assume that $X$ has no rational contraction onto a $3$-fold, and reach a contradiction. By Th. 9.2 and by the first part of the proof, every fixed prime divisor of $X$ is of type $(3,0)^{Q}$ or $(3,2)$.

The same proof as [Cas22, proof of Th. 9.1] works, with the following modifications.

1. $\bullet$

By Cor. 4.10, every face of $\operatorname{Eff}(X)$ of dimension $\leq 4$ is fixed; this simplifies [ibid., 9.4 and 9.6]. 2. $\bullet$

In [ibid., 9.4 and 9.5] we apply Lemma 7.8 instead of [ibid., Lemma 7.2]. 3. $\bullet$

In [ibid., 9.7] we apply Th. 7.1 instead of [ibid., Th. 4.8]. 4. $\bullet$

In [ibid., 9.12 and 9.13] we apply Lemma 10.4 instead of [ibid., Lemma 9.3]. 5. $\bullet$

In [ibid., 9.14 and 9.24] we apply Lemma 10.3 instead of [ibid., Lemma 9.2]. 6. $\bullet$

In [ibid., 9.20] we get a contradiction, because we are assuming $\rho_{X}\geq 9$. ∎

Corollary 10.5**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 9$, not isomorphic to a product of surfaces. Then $\rho_{X}=9$ and one of the following holds:

1. $(i)$

$X$* has an elementary rational contraction onto a $3$-fold;* 2. $(ii)$

$X$* is the blow-up of $W$ along a normal surface $S$, where $W$ is the Fano model of $\operatorname{Bl}_{7\mskip 1.0mu\text{\it pts}}\mathbb{P}^{4}$, and $S\subset W$ is the transform of a cubic scroll or a cone over a twisted cubic in $\mathbb{P}^{4}$, containing the blown-up points;* 3. $(iii)$

$X$* is a blow-up of a cubic $4$-fold as in Th. 1.2.*

Note that this result implies Theorems 1.1 and 1.4.

Proof.

If $X$ has no small elementary contractions, then by Th. 1.2 we have $\rho_{X}=9$ and $(iii)$.

If instead $X$ has a small elementary contraction, then by Rem. 9.1 $X$ has a fixed prime divisor of type $(3,0)^{\text{\it sm}}$, $(3,1)^{\text{\it sm}}$, or $(3,0)^{Q}$. If $X$ has a fixed prime divisor of type $(3,0)^{\text{\it sm}}$, then we apply Th. 9.2 and get $\rho_{X}=9$ and $(i)$.

Finally, if $X$ has a fixed prime divisor of type $(3,1)^{\text{\it sm}}$ or $(3,0)^{Q}$, by Th. 10.1 we get that $\rho_{X}=9$ and $X$ has a rational contraction onto a $3$-fold. Then we apply Th. 1.5, and get $(i)$ or $(ii)$. ∎

Corollary 10.6**.**

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}=8$. Then one of the following holds:

1. $(i)$

$X$* has a rational contraction onto a $3$-fold;* 2. $(ii)$

every fixed prime divisor of $X$ is of type $(3,1)^{\text{\it sm}}$ or $(3,2)$; 3. $(iii)$

every fixed prime divisor of $X$ is of type $(3,0)^{Q}$ or $(3,2)$, and two divisors of type $(3,2)$ cannot be adjacent.

Proof.

Let us assume that $(i)$ does not hold, namely $X$ has no rational contraction onto a $3$-fold; in particular $X$ is not isomorphic to a product of surfaces. By Th. 9.2, $X$ has no fixed prime divisors of type $(3,0)^{\text{\it sm}}$, hence every fixed prime divisor of $X$ is of type $(3,2)$, $(3,1)^{\text{\it sm}}$, or $(3,0)^{Q}$ (see Th.-Def. 5.1). By Prop. 10.2, in $X$ a divisor of type $(3,1)^{\text{\it sm}}$ and one of type $(3,0)^{Q}$ cannot be adjacent. Moreover by Lemma 7.8 a divisor of type $(3,0)^{Q}$ can be adjacent to at most one divisor of type $(3,2)$. Let us also note that every face of $\operatorname{Eff}(X)$ of dimension $\leq 3$ is fixed, by Cor. 4.10, and hence simplicial (see [Cas22, Lemma 4.2]).

Let $E$ be a fixed prime divisor of type $(3,2)$, and suppose that $E$ is adjacent to a fixed prime divisor $D$ of type $(3,0)^{Q}$. Then every $3$-dimensional face of $\operatorname{Eff}(X)$ containing $\langle[E],[D]\rangle$ must be $\langle[E],[D],[D^{\prime}]\rangle$ with $D^{\prime}$ of type $(3,0)^{Q}$. This implies that every fixed prime divisor adjacent to $E$ must be of type $(3,0)^{Q}$.

Consider now a sequence of fixed prime divisors $D_{1},\dotsc,D_{r}$ such that $D_{i}$ is adjacent to $D_{i+1}$ for every $i=1,\dotsc,r-1$, with $D_{1}$ of type $(3,0)^{Q}$. Then $D_{2}$ can be either $(3,0)^{Q}$ or $(3,2)$, and in this last case, $D_{3}$ must be $(3,0)^{Q}$. Proceeding in this way, we see that no $D_{i}$ can be of type $(3,1)^{sm}$. Since $\operatorname{Eff}(X)$ is connected, we must have $(ii)$ or $(iii)$. ∎

11. Examples

11.1. Fano models of the blow-ups of $\mathbb{P}^{4}$ in points

Let $X^{\prime}$ be the blow-up of $\mathbb{P}^{4}$ at $r$ general points, with $r\leq 8$. Then there is a SQM $X^{\prime}\dasharrow X$ such that $X$ is smooth and Fano, see [CS24, Ex. 7.2 and references therein]; we have $\rho_{X}=1+r\leq 9$.

If $r\geq 1$, there is an elementary rational contraction $f\colon X\dasharrow Y:=\operatorname{Bl}_{r-1\mskip 2.0mu\text{\it pts}}\mathbb{P}^{3}$.

Moreover (for $r\geq 2$) there is also an elementary rational contraction $\varphi\colon Y\dasharrow S:=\operatorname{Bl}_{r-2\mskip 2.0mu\text{\it pts}}\mathbb{P}^{2}$, and the composition $g:=\varphi\circ f\colon X\dasharrow S$ is a quasi-elementary rational contraction with $d_{g}=\rho_{X}-\rho_{S}=2$.

Finally, by composing $g$ with a conic bundle $S\to\mathbb{P}^{1}$ (for $r\geq 3$), we get a rational contraction $h\colon X\dasharrow\mathbb{P}^{1}$ with $d_{h}=3$.

We also note that every fixed prime divisor of $X$ is of type $(3,0)^{\text{\it sm}}$.

11.2. Two examples with $\rho=7$ from [CS24]

Let $W$ be the Fano model of $\operatorname{Bl}_{q_{1},\dotsc,q_{5}}\mathbb{P}^{4}$ (see §11.1), and for $i=1,2$ let $S_{i}\subset W$ be the transform of the following surface $A_{i}\subset\mathbb{P}^{4}$:

1. $\bullet$

$A_{1}$ is a general cubic rational normal scroll containing $q_{1},\dotsc,q_{5}$; 2. $\bullet$

$A_{2}$ is a sextic K3 surface with $\operatorname{Sing}(A_{2})=\{q_{1},\dotsc,q_{5}\}$ and having ordinary double points at each $q_{j}$; more precisely $A_{2}=Q\cap M$ where $Q$ is a general quadric hypersurface containing $q_{1},\dotsc,q_{5}$, and $M$ is a general cubic hypersurface with ordinary double points at $q_{1},\dotsc,q_{5}$.

Let $\sigma_{i}\colon X_{i}\to W$ be the blow-up along $S_{i}$; then $X_{i}$ is a smooth Fano $4$-fold with $\rho_{X_{i}}=7$ (see [CS24, §7.2 and §7.3]).

There is an elementary rational contraction $\psi\colon W\dasharrow Y:=\operatorname{Bl}_{4\mskip 1.0mu\text{\it pts}}\mathbb{P}^{3}$, and the composition $f_{i}:=\psi\circ\sigma_{i}\colon X_{i}\dasharrow Y$ is a special rational contraction with $\rho_{X_{i}}-\rho_{Y}=2$.

Moreover there is an elementary rational contraction $\varphi\colon Y\dasharrow S:=\operatorname{Bl}_{3\mskip 1.0mu\text{\it pts}}\mathbb{P}^{2}$, and the composition $g_{i}:=\varphi\circ f_{i}\colon X_{i}\dasharrow S$ is a quasi-elementary rational contraction with $d_{g_{i}}=\rho_{X_{i}}-\rho_{S}=3$.

Finally, by composing $g_{i}$ with a conic bundle $S\to\mathbb{P}^{1}$, we get a rational contraction $h_{i}\colon X_{i}\dasharrow\mathbb{P}^{1}$ with $d_{h_{i}}=4$.

We also note that $X_{i}$ contains (at least) six fixed prime divisors of type $(3,2)$: one is $\operatorname{Exc}(\sigma_{i})$, and the others are the transforms in $X_{i}$ of the cones in $\mathbb{P}^{4}$ over $A_{i}$ with vertex $q_{j}$, for $j=1,\dotsc,5$.

Moreover, for $j=1,\dotsc,5$, let $D_{j}\subset X_{i}$ be the transform of the exceptional divisor over $q_{j}$ in $\operatorname{Bl}_{q_{1},\dotsc,q_{5}}\mathbb{P}^{4}\to\mathbb{P}^{4}$. Then $D_{j}$ is a fixed prime divisor of type $(3,1)^{\text{\it sm}}$ in $X_{1}$, of type $(3,0)^{Q}$ in $X_{2}$ (see [ibid., Section 5, in particular Lemmas 5.39, 5.46, 5.68, 5.74]).

11.3. A family with an elementary rational contraction onto a surface

Let $p_{1},\dotsc,p_{m}\in\mathbb{P}^{2}$ be general points, $C_{1}^{\prime},\dotsc,C_{m}^{\prime}\subset\mathbb{P}^{2}$ general lines, $C_{i}:=\{p_{i}\}\times C_{i}^{\prime}\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$, and $h\colon\mathbb{P}^{2}\times\mathbb{P}^{2}\to\mathbb{P}^{2}$ the first projection.

Let $\beta\colon\widehat{X}\to\mathbb{P}^{2}\times\mathbb{P}^{2}$ be the blow-up of $C_{1},\dotsc,C_{m}$, $\widehat{D}_{i}\subset\widehat{X}$ the exceptional divisor over $C_{i}$, and $L_{i}\subset\widehat{X}$ the transform of the fiber $h^{-1}(p_{i})$, containing $C_{i}$.

Since $\mathcal{N}_{C_{i}/\mathbb{P}^{2}\times\mathbb{P}^{2}}\cong\mathcal{O}_{\mathbb{P}^{1}}(1)\oplus\mathcal{O}_{\mathbb{P}^{1}}^{\oplus 2}$, we have $\widehat{D}_{i}\cong\mathbb{P}_{\mathbb{P}^{1}}(\mathcal{O}\oplus\mathcal{O}(1)^{\oplus 2})$, and $\widehat{D}_{i}\cap L_{i}$ is the flopping curve in $\widehat{D}_{i}$. Moreover $L_{i}\cong\mathbb{P}^{2}$ and $\mathcal{N}_{L_{i}/\widehat{X}}\cong\mathcal{O}_{\mathbb{P}^{2}}(-1)^{\oplus 2}$, namely $L_{i}$ is an exceptional plane in $\widehat{X}$.

Let us consider the composition $h\circ\beta\colon\widehat{X}\to\mathbb{P}^{2}$. The class of every curve in $\widehat{D}_{i}\cup L_{i}$ is in $\langle[C_{L_{i}}]\rangle+\operatorname{NE}(\beta)$, so that

[TABLE]

and $h\circ\beta$ is $K$-negative. One can also check that $-K_{\widehat{X}}-\sum_{i=1}^{m}\widehat{D}_{i}$ has positive intersection with every non-zero class in $\operatorname{NE}(\beta)$, while it has zero intersection with every $C_{L_{i}}$, so that $\langle[C_{L_{1}}],\dotsc,[C_{L_{m}}]\rangle$ is an $m$-dimensional face of $\operatorname{NE}(h\circ\beta)$, whose contraction is small. There is a SQM $\xi\colon\widehat{X}\dasharrow\widetilde{X}$, relative to $\beta\circ h$:

[TABLE]

where $\widetilde{X}$ is smooth, the indeterminacy locus of $\xi$ is $L_{1}\cup\cdots\cup L_{m}$, and that of $\xi^{-1}$ is the disjoint union of $m$ exceptional lines $\ell_{1},\dotsc,\ell_{m}$.

Let $D_{i}\subset\widetilde{X}$ be the transform of $\widehat{D}_{i}$; we have $\ell_{i}\subset D_{i}$ and $D_{i}=(\tilde{h})^{-1}(p_{i})$. The restriction $(\xi)_{|\widehat{D}_{i}}\colon\widehat{D}_{i}\dasharrow D_{i}$ is a flop, we still have $D_{i}\cong\mathbb{P}_{\mathbb{P}^{1}}(\mathcal{O}\oplus\mathcal{O}(1)^{\oplus 2})$, but now the flopping curve in $D_{i}$ is the exceptional line $\ell_{i}$ (see Fig. 11.1). It is not difficult to check that the transform $\Gamma\subset\widetilde{X}$ of a line in a general fiber of $h\colon\mathbb{P}^{2}\times\mathbb{P}^{2}\to\mathbb{P}^{2}$ is numerically equivalent to a line in a fiber of the $\mathbb{P}^{2}$-bundle $D_{i}\to\mathbb{P}^{1}$, and that

[TABLE]

Let $\alpha\colon S\to\mathbb{P}^{2}$ be the blow-up of $p_{1},\dotsc,p_{m}$. Then $\tilde{h}$ factors as $\alpha\circ g$ where $g\colon\widetilde{X}\to S$ is a $\mathbb{P}^{2}$-bundle, and $g(D_{i})\subset S$ is the exceptional curve over $p_{i}$.

In conclusion, the composition $g\circ\xi\colon\widehat{X}\dasharrow S$ is an elementary rational contraction onto a surface. We have $\rho_{\widehat{X}}=m+2$.

**The case $m\leq 3$. ** If $m\leq 3$, then all these varieties are toric, and they can be described explicitly using Batyrev’s language of primitive collections and primitive relations [Bat91], and Sato’s description of how primitive relations change after smooth toric blow-ups and blow-downs [Sat00, §4]. Let us consider in detail each case.

For $m=1$, $X:=\widehat{X}$ is Fano with $\rho_{X}=3$; it is $G_{5}$ in Batyrev’s classification of toric Fano $4$-folds [Bat99].

For $m=2$, $\widehat{X}$ is not Fano: it contains one exceptional line $\ell$ whose image in $\mathbb{P}^{2}$ is the line $\overline{p_{1}p_{2}}$. More precisely, $\ell$ is the transform of the unique curve $\overline{p_{1}p_{2}}\times\{q\}\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$ that intersects both $C_{1}$ and $C_{2}$ (namely $q=C_{1}^{\prime}\cap C_{2}^{\prime}$). The class $[\ell]$ generates a small extremal ray of $\operatorname{NE}(\widehat{X})$; the associated flip $\widehat{X}\dasharrow X$ gives a smooth toric Fano $4$-fold $X$ with $\rho_{X}=4$, which is $Z_{2}$ in [Bat99].

[TABLE]

Finally let us consider the case $m=3$. Now $\widehat{X}$ contains three exceptional lines $\ell_{ij}$, whose images in $\mathbb{P}^{2}$ are the lines $\overline{p_{i}p_{j}}$, for $1\leq i<j\leq 3$. As before, $\ell_{ij}$ is the transform of the curve $\overline{p_{i}p_{j}}\times\{q_{ij}\}\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$ with $q_{ij}=C_{i}^{\prime}\cap C_{j}^{\prime}$. Again one can consider the flips of these three exceptional lines $\widehat{X}\dasharrow X$, and get a smooth toric Fano $4$-fold $X$ with $\rho_{X}=5$; this is Sato’s example [Sat00, Ex. 4.7].

Therefore for $m\in\{1,2,3\}$ this construction gives a Fano $4$-fold $X$ with $\rho_{X}=m+2\in\{3,4,5\}$ and an elementary rational contraction $X\dasharrow S$ onto a surface.

**The case $m\geq 4$. **

Lemma 11.1**.**

If there exists a SQM $\widehat{X}\dasharrow X$ such that $X$ is a smooth Fano $4$-fold, then $m\leq 6$.

Proof.

Assume that $X$ exists. We have $h^{0}(X,-K_{X})=\chi(X,-K_{X})$ because $X$ is Fano, and $\chi(X,-K_{X})=\chi(\widehat{X},-K_{\widehat{X}})$ by [Cas17, Cor. 3.10]. Since $C_{i}\cong\mathbb{P}^{1}$ and $-K_{\mathbb{P}^{2}\times\mathbb{P}^{2}}\cdot C_{i}=3$ for every $i=1,\dotsc,m$, by [ibid., Lemma 3.8] we get

[TABLE]

which implies that $m\leq 6$. ∎

Question 11.2**.**

For $m\in\{4,5,6\}$, does there exist a SQM $\widehat{X}\dasharrow X$ such that $X$ is a smooth Fano $4$-fold?

This would give a Fano $4$-fold $X$ with $\rho_{X}=m+2\in\{6,7,8\}$ and an elementary rational contraction $X\dasharrow S$ onto a surface. This construction shows up in the proof of Th. 6.1, see 6.19.

11.4. A family with $\rho=3$ and a quasi-elementary contraction onto $\mathbb{P}^{2}$

Let $A\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$ be a surface obtained as a complete intersection of two general divisors, $B$ of degree $(2,1)$ and $D$ of degree $(0,2)$; then $A$ is a del Pezzo surface with $K_{A}^{2}=2$. Let $\sigma\colon X\to\mathbb{P}^{2}\times\mathbb{P}^{2}$ be the blow-up along $A$; we show that $X$ is Fano. This example is inspired by the proof of Th. 6.1, see 6.24.

We have

[TABLE]

(see [CS24, Lemma 2.4]). One can compute that $(K_{\mathbb{P}^{2}\times\mathbb{P}^{2}})^{4}=486$, $(K_{{\mathbb{P}^{2}\times\mathbb{P}^{2}}|A})^{2}=90$, $K_{A}\cdot K_{{\mathbb{P}^{2}\times\mathbb{P}^{2}}|A}=18$, and $c_{2}(\mathcal{N}_{A/\mathbb{P}^{2}\times\mathbb{P}^{2}})=16$, thus $K_{X}^{4}=194$.

Let $E\subset X$ be the exceptional divisor, and $\widetilde{B}\subset X$ the transform of $B$; note that $\sigma_{|\widetilde{B}}\colon\widetilde{B}\to B$ is an isomorphism. By adjunction we have

[TABLE]

and $-K_{\mathbb{P}^{2}\times\mathbb{P}^{2}}-D$ is ample, thus $-K_{X|\widetilde{B}}$ is ample.

Let $C\subset X$ be an irreducible curve; we show that $-K_{X}\cdot C>0$. This is clear if $C\subset\widetilde{B}$ or if $\sigma(C)=\{\text{\it pt}\}$. Otherwise, let $H\in|\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{2}}(1,2)|$, so that $\widetilde{B}+\sigma^{*}H\in|-K_{X}|$. Then $\widetilde{B}\cdot C\geq 0$ and $\sigma^{*}(H)\cdot C>0$, hence $-K_{X}\cdot C>0$.

We conclude that $-K_{X}$ is strictly nef and big, hence ample by the base-point-free theorem. Here are the invariants of $X$:

[TABLE]

Let $\pi_{i}\colon\mathbb{P}^{2}\times\mathbb{P}^{2}\to\mathbb{P}^{2}$ be the two projections, and $f_{i}:=\pi_{i}\circ\sigma\colon X\to\mathbb{P}^{2}$, $i=1,2$.

Since $\pi_{1}(A)=\mathbb{P}^{2}$ and $(\pi_{1})_{|A}$ has degree $2$, $f_{1}$ is a quasi-elementary contraction with general fiber $F\cong\operatorname{Bl}_{2\mskip 1.0mu\text{\it pts}}\mathbb{P}^{2}$, and $d_{f_{1}}=\dim\mathcal{N}_{1}(F,X)=2$.

We also have $\widetilde{B}\cong B$ and $f_{1|\widetilde{B}}$ is a $\mathbb{P}^{1}$-bundle. If $C_{\widetilde{B}}\subset\widetilde{B}$ is a fiber of such $\mathbb{P}^{1}$-bundle, we have $-K_{X}\cdot C_{\widetilde{B}}=1$ and $\widetilde{B}\cdot C_{\widetilde{B}}=-1$, and $\widetilde{B}$ is the exceptional divisor of a divisorial elementary contraction $\alpha\colon X\to Y$ of type $(3,2)^{\text{\it sm}}$, such that $\operatorname{NE}(f_{1})=\operatorname{NE}(\sigma)+\operatorname{NE}(\alpha)$.

[TABLE]

Here $Y$ is a smooth Fano $4$-fold with $\rho_{Y}=2$, and the map $Y\to\mathbb{P}^{2}$ is a quadric bundle.

On the other hand, since $\pi_{2}(A)\subset\mathbb{P}^{2}$ is a conic, $f_{2}\colon X\to\mathbb{P}^{2}$ is equidimensional but not quasi-elementary; the general fiber is $\mathbb{P}^{2}$. Moreover $f_{2}^{*}(\pi_{2}(A))=\widetilde{D}+E$ where $\widetilde{D}\subset X$ is the transform of $D$, and $\widetilde{D}\cong\mathbb{P}^{2}\times\mathbb{P}^{1}$ with normal bundle $\mathcal{N}_{\widetilde{D}/X}\cong\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{1}}(-2,2)$. In fact $\widetilde{D}$ is the exceptional divisor of a divisorial elementary contraction $\beta\colon X\to Z$ of type $(3,1)$, where $Z$ is singular along the curve $\beta(\widetilde{D})$.

Finally we note that since $\widetilde{B}\cap\widetilde{D}=\emptyset$, one can check that $\operatorname{NE}(\alpha)+\operatorname{NE}(\beta)$ is a face of $\operatorname{NE}(X)$, with a birational contraction $\gamma\colon X\to W$ where $\rho_{W}=1$. If $H_{i}=f_{i}^{*}\mathcal{O}_{\mathbb{P}^{2}}(1)$, we have $\operatorname{Nef}(X)=\operatorname{Mov}(X)=\langle H_{1},H_{2},\gamma^{*}\operatorname{Nef}W\rangle$ and $\operatorname{Eff}(X)=\langle H_{1},\widetilde{B},\widetilde{D},E\rangle$, see Fig. 11.2.

11.5. Blow-ups of $\mathbb{P}^{4}$ along planes

Let $\sigma\colon X_{s}\to\mathbb{P}^{4}$ be the blow-up of $s$ general planes $A_{1},\dotsc,A_{s}\subset\mathbb{P}^{4}$, in the following sense: we blow-up one plane $A_{i}$ and successively the transforms of the other ones. Note that each pair of planes intersect pairwise in a point, and $\sigma$ does not depend on the order of the blow-ups, see Lemma 2.10. We will refer to such a map simply as the blow-up of $A_{1},\dotsc,A_{s}$.

We have $\rho_{X_{s}}=1+s$, and for $s=1,2$ the $4$-fold $X_{s}$ is toric and Fano: $X_{1}\cong\mathbb{P}_{\mathbb{P}^{1}}(\mathcal{O}^{\oplus 3}\oplus\mathcal{O}(1))$, and $X_{2}$ is $D_{17}$ in [Bat99].

Let $\pi\colon X_{1}\to\mathbb{P}^{1}$ be the $\mathbb{P}^{3}$-bundle, and $\sigma^{\prime}\colon X_{s}\to X_{1}$ the remaining blow-ups. Let also $\widetilde{A}_{i}\subset X_{1}$ be the transform of $A_{i}$ for $i=2,\dotsc,s$. Then $\widetilde{A}_{i}\cong\mathbb{F}_{1}$, $\pi_{|\widetilde{A}_{i}}$ is the $\mathbb{P}^{1}$-bundle, and $\widetilde{A}_{i}$ intersects the general fiber of $\pi$ in a line. The composition $\pi\circ\sigma^{\prime}\colon X_{s}\to\mathbb{P}^{1}$ is a quasi-elementary contraction with general fiber the blow-up $F_{s}$ of $\mathbb{P}^{3}$ along $s-1$ general lines. If $X_{s}$ is Fano, then $F_{s}$ must be Fano too; by classification $F_{s}$ is Fano if and only if $s\leq 4$ (see [IP99, Ch. 12]), hence we get the following.

Lemma 11.3**.**

If $X_{s}$ is Fano, then $s\leq 4$ and $\rho_{X_{s}}\leq 5$.

Question 11.4**.**

Is $X_{s}$ Fano for $s\in\{3,4\}$?

11.6. Blow-ups of quadrics along planes

Let $Q\subset\mathbb{P}^{5}$ be a smooth quadric. Recall that $Q$ contains two families of planes, and distinct planes in the same family intersect pairwise in a point. Let $A_{1},\dotsc,A_{s}\subset Q$ be general planes in the same family, and $\sigma\colon X_{s}\to Q$ the blow-up of $A_{1},\dotsc,A_{s}$ (see Lemma 2.10), so that $\rho_{X_{s}}=1+s$.

It is shown in [Man24] that $X_{5}$ is Fano; this implies that $X_{s}$ is Fano for every $s\leq 5$. We have $X_{1}\cong\mathbb{P}_{\mathbb{P}^{2}}(T_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(2))$ (see [SW90, Theorem on first page]), and we also note that $X_{2}$ is [FTT24, Fano 3-2] and $X_{3}$ is [FTT24, Fano 4-2].

For $s\geq 9$ we have $\rho_{X_{s}}\geq 10$, hence $X_{s}$ cannot be Fano by Th. 1.1.

Question 11.5**.**

Is $X_{s}$ Fano for $s\in\{6,7,8\}$?

For $i,j\in\{1,\dotsc,s\}$, $i<j$, let $H_{ij}\subset\mathbb{P}^{5}$ be the hyperplane spanned by $A_{i}$ and $A_{j}$; then $H_{ij|Q}$ is a cone over a smooth quadric surface. Moreover let $E_{i}\subset X_{s}$ be the exceptional divisor over $A_{i}\subset Q$, for $i=1,\dotsc,s$, and $H\subset X_{s}$ the pullback of a general hyperplane section.

**The case $s=2$. ** Let $\sigma_{1}\colon X_{1}\to Q$ be the blow-up of $A_{1}$, and $\sigma_{2}\colon X_{2}\to X_{1}$ the blow-up of the transform $\widetilde{A}_{2}\subset X_{1}$ of $A_{2}$. Moreover let $D_{12}^{\prime}\subset X_{1}$ and $D_{12}\subset X_{2}$ be the transforms of $H_{12|Q}$.

Under the $\mathbb{P}^{2}$-bundle $\pi_{1}\colon X_{1}\to\mathbb{P}^{2}$, the image of $\widetilde{A}_{2}\subset X_{1}$ is a line $\Gamma\subset\mathbb{P}^{2}$, and $\pi_{1}^{-1}(\Gamma)=D^{\prime}_{12}$. The composition $f_{1}:=\pi_{1}\circ\sigma_{2}\colon X_{2}\to\mathbb{P}^{2}$ is an equidimensional contraction with $f_{1}^{-1}(\Gamma)=E_{2}\cup D_{12}$, both fixed prime divisors, and $\tau_{f_{1}}=\langle[E_{2}],[D_{12}]\rangle$ (see Def.-Rem. 3.2 and Rem. 3.4).

Symmetrically, we can also factor $\sigma\colon X_{2}\to Q$ as $X_{2}\stackrel{{\scriptstyle\alpha}}{{\to}}X_{1}^{\prime}\to Q$ where $X_{1}^{\prime}\to Q$ is the blow-up of $A_{2}$, and $\alpha\colon X_{2}\to X_{1}^{\prime}$ the blow-up of the transform of $A_{1}$. We have a $\mathbb{P}^{2}$-bundle $\pi_{2}\colon X_{1}^{\prime}\to\mathbb{P}^{2}$, and the composition $f_{2}:=\pi_{2}\circ\alpha\colon X_{2}\to\mathbb{P}^{2}$ is an equidimensional contraction with $\tau_{f_{2}}=\langle[E_{1}],[D_{12}]\rangle$.

See Fig. 11.3 for a section of $\operatorname{Eff}(X_{2})$; we have $H\equiv D_{12}+H_{1}+H_{2}$, $f_{i}^{*}\mathcal{O}_{\mathbb{P}^{2}(1)}\equiv H-E_{i}$ for $i=1,2$, and $-K_{X_{2}}=4H-E_{1}-E_{2}$.

[TABLE]

The restriction $\sigma_{1|D^{\prime}_{12}}\colon D^{\prime}_{12}\to H_{12|Q}$ is a small resolution of the quadric cone, and $D^{\prime}_{12}\cong\mathbb{P}_{\mathbb{P}^{1}}(\mathcal{O}\oplus\mathcal{O}(1)^{\oplus 2})$. Then $D^{\prime}_{12}$ is smooth and contains $\widetilde{A}_{2}$, so that $D_{12}\cong D^{\prime}_{12}$, and if $C_{D_{12}}\subset D_{12}$ is a line in a fiber of the $\mathbb{P}^{2}$-bundle, we have $D_{12}\cdot C_{D_{12}}=-1$. In fact $D_{12}$ is the exceptional divisor of a blow-up $\tau\colon X_{2}\to\mathbb{P}^{2}\times\mathbb{P}^{2}$ along a smooth rational curve $C$, which is a complete intersection of divisors of degrees $(1,0),(0,1),(1,1)$.

This can be seen via the embedding $X_{2}\subset P_{2}$ where $P_{2}\to\mathbb{P}^{5}$ is the blow-up of $A_{1}$ and then of $\widetilde{A}_{2}$; $P_{2}$ is a toric variety, and it can be described explicitly using Batyrev’s language of primitive collections and primitive relations [Bat91], and Sato’s description of how primitive relations change after smooth blow-ups and blow-downs [Sat00, §4]. There is a blow-up $\tau_{P}\colon P_{2}\to Y:=\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{O}\oplus\mathcal{O}(1)^{\oplus 3})$ along a surface $S\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$, with exceptional divisor the transform in $P_{2}$ of the hyperplane $H_{12}\subset\mathbb{P}^{5}$ spanned by $A_{1}$ and $A_{2}$. We have $X_{2}\cdot\operatorname{NE}(\tau_{P})=0$, $\tau_{P|X_{2}}=\tau$, $\tau_{P}(X_{2})\in|\mathcal{O}_{Y}(1)|$, and $\tau_{P}(X_{2})\cong\mathbb{P}^{2}\times\mathbb{P}^{2}$. Moreover the curve $C:=S\cap\tau_{P}(X_{2})\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$ is the complete intersection of the divisors $\tau(E_{1})\in\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{2}}(1,0)$, $\tau(E_{2})\in\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{2}}(0,1)$, and a third divisor in $|\mathcal{O}_{\mathbb{P}^{2}\times\mathbb{P}^{2}}(1,1)|$.

**The case $s=3$. ** For $i,j\in\{1,2,3\}$, $i<j$, let $D_{ij}\subset X_{3}$ be the transform of $H_{ij|Q}$. Moreover let $\Lambda\subset\mathbb{P}^{5}$ be the plane spanned by the three points $A_{1}\cap A_{2}$, $A_{1}\cap A_{3}$, and $A_{2}\cap A_{3}$; we note that $\Lambda$ is contained in $Q$ and also in every hyperplane $H_{ij}$. Let $L\subset X_{3}$ be the transform of $\Lambda$; then $L$ is an exceptional plane contained in $D_{ij}$ for every $i,j$, and $[C_{L}]$ generates a small extremal ray of $\operatorname{NE}(X_{3})$ with locus $L$.

Let us consider the flip $X_{3}\dasharrow Y$ of $\mathbb{R}_{\geq 0}[C_{L}]$. We claim that $Y$ is the blow-up of $\mathbb{P}^{4}$ at three general lines, with exceptional divisors (the transforms of) $D_{12}$, $D_{13}$, $D_{23}$ (see [PPS21] for a description of $\operatorname{NE}(Y)$ and $\operatorname{Eff}(Y)$). The exceptional line $\ell\subset Y$ given by the flip is the transform of the unique line in $\mathbb{P}^{4}$ intersecting the three blown-up lines. Moreover the exceptional divisors $E_{1},E_{2},E_{3}$ of $\sigma\colon X_{3}\to Q$ are the transforms of the hyperplanes in $\mathbb{P}^{4}$ spanned by two blown-up lines.

This can be seen again by consider the toric variety $P_{3}$ obtained by blowing-up $P_{2}$ along the transform $\widetilde{A}_{3}$ of the third plane $A_{3}\subset\mathbb{P}^{5}$; we have $X_{3}\subset P_{3}$, and $[C_{L}]$ still generates a small extremal ray of $\operatorname{NE}(P_{3})$, with locus $L$, $K_{P_{3}}\cdot C_{L}=0$, and $X_{3}\cdot C_{L}=-1$. The flop of this small extremal ray yields a variety isomorphic to $P_{3}$:

[TABLE]

Then $\psi\colon P_{3}\to\mathbb{P}^{5}$ is again a sequence of three blow-up of planes, with exceptional divisors the transforms of first $H_{12}$, then $H_{13}$, and finally $H_{23}$. The image of $X_{3}$ is a hyperplane in $\mathbb{P}^{5}$, and $\psi$ blows it up in three disjoint lines.

As in the case $r=2$, for $i\in\{1,2,3\}$ there is an equidimensional contraction $f_{i}\colon X_{3}\to\mathbb{P}^{2}$ given by the $\mathbb{P}^{2}$-bundle obtained blowing-up $Q$ along $A_{i}$. We have $E_{2}+D_{12}\equiv E_{3}+D_{13}\equiv f_{1}^{*}\mathcal{O}_{\mathbb{P}^{2}}(1)$ and $\tau_{f_{1}}=\langle[E_{2}],[E_{3}],[D_{12}],[D_{13}]\rangle$, and similarly for $f_{2}$ and $f_{3}$. See Fig. 11.4 for a section of $\operatorname{Eff}(X_{3})$.

For $s\geq 3$, we still get $\binom{s}{2}$ fixed prime divisors $D_{ij}\subset X_{s}$ of type $(3,1)^{\text{\it sm}}$, given by the transforms of $H_{ij|Q}$, for $i,j\in\{1,\dotsc,s\}$, $i<j$. Moreover for $i,j,h\in\{1,\dotsc,s\}$, $i<j<h$, the plane $\Lambda_{ijh}\subset\mathbb{P}^{5}$ spanned by the three points $A_{i}\cap A_{j}$, $A_{i}\cap A_{h}$, and $A_{j}\cap A_{h}$, is contained in $Q$ and in $H_{ij},H_{ih},H_{jh}$; its transform $L_{ijh}\subset D_{ij}\cap D_{ih}\cap D_{jh}$ is an exceptional plane.

11.7. Blow-ups of cubic $4$-folds along planes

Let $Z\subset\mathbb{P}^{5}$ be a smooth cubic $4$-fold containing $s$ planes $A_{1},\dotsc,A_{s}\subset Z$ that intersect pairwise in a point, and $A_{i}\cap A_{j}\cap A_{k}=\emptyset$ for $i<j<k$.

Let $X_{s}\to Z$ the blow-up of $A_{1},\dotsc,A_{s}$ (see Lemma 2.10); then $X_{s}$ is a smooth projective $4$-fold with $\rho_{X_{s}}=1+s$, compare Th. 8.1 and Rem. 8.3.

For $s=1$, $X_{1}$ has two elementary contractions, the blow-up $X_{1}\to Z$, and a quadric bundle $X_{1}\to\mathbb{P}^{2}$, so that $X_{1}$ is Fano. For $s=2$, we show below that $X_{2}$ is still Fano. On the other hand for $s\geq 9$ we have $\rho_{X_{s}}\geq 10$, hence $X_{s}$ cannot be Fano by Th. 1.1.

Question 11.6**.**

For which values of $s\in\{3,\dotsc,8\}$ is $X_{s}$ Fano?

**The case $s=2$. ** We keep the same notation as in §11.6. We still have $X_{2}\subset P_{2}$, where $P_{2}$ is the toric $5$-fold obtained by blowing-up $\mathbb{P}^{5}$ along first $A_{1}$ and then $\widetilde{A}_{2}$. One can check that $-K_{P_{2}}-X_{2}$ is nef and big on $P_{2}$, and $(-K_{P_{2}}-X_{2})^{\perp}\cap\operatorname{NE}(P_{2})=R$ where $R$ is a small extremal ray with $K_{P_{2}}\cdot R=X_{2}\cdot R=0$. The locus of $R$ is a surface $T\cong\mathbb{P}^{2}$, thus either $T\subset X_{2}$, or $T\cap X_{2}=\emptyset$.

Let us consider the blow-up $P_{1}\to\mathbb{P}^{5}$ of $A_{1}$. The surface $T\subset P_{2}$ is the transform of the fiber $F\subset P_{1}$ of the blow-up over the point $A_{1}\cap A_{2}\in\mathbb{P}^{5}$. Then in $P_{1}$ we have $X_{1}\cap F=\widetilde{A}_{2}\cap F$ and this is a line in $F\cong\mathbb{P}^{2}$; therefore we see that $X_{2}$ can intersect $T$ at most in a curve, and we conclude that $X_{2}\cap T=\emptyset$. This shows that $-K_{X_{2}}=(-K_{P_{2}}-X_{2})_{|X_{2}}$ is ample, and $X_{2}$ is Fano.

Let us also consider the blow-up $\tau_{P}\colon P_{2}\to Y=\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{O}\oplus\mathcal{O}(1)^{\oplus 3})$ as in §11.6, and its restriction $\tau:=\tau_{P|X_{2}}\colon X_{2}\to W:=\tau_{P}(X_{2})$. If $C\subset P_{2}$ is a line in a general non-trivial fiber $F_{\tau}\cong\mathbb{P}^{2}$ of $\tau_{P}$, we have $X_{2}\cdot C=1$, therefore $X_{2}$ intersects $F_{\tau}$ in a line, and $W$ contains the surface $S\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ blown-up by $\tau_{P}$. Thus we see that $\tau$ is an elementary divisorial contraction of type $(3,2)$ with exceptional divisor $D_{12}$, the transform of $H_{12|Z}$.

The tautological class $\mathcal{O}_{Y}(1)$ is nef and big, and it is zero on a small extremal ray $R_{Y}$ of $\operatorname{NE}(Y)$ with locus $T_{Y}:=\tau_{P}(T)$, which is the negative section of the $\mathbb{P}^{3}$-bundle $Y\to\mathbb{P}^{2}$. In $P_{2}$ we have $T\cap X_{2}=\emptyset$ and $T\cap\operatorname{Exc}(\tau_{P})=\emptyset$, thus $W\cap T_{Y}=\emptyset$ in $Y$. Moreover $W\in|\mathcal{O}_{Y}(2)|$, $-K_{Y}\sim\mathcal{O}_{Y}(4)$, and again we see that $-K_{W}=\mathcal{O}_{Y}(2)_{|W}$ is ample, so that $W$ is Fano of index $2$. It is not difficult to see that $W$ is a double cover of $\mathbb{P}^{2}\times\mathbb{P}^{2}$ with branch divisor of degree $(2,2)$ (compare 8.34), and it has two elementary contractions $W\to\mathbb{P}^{2}$ that are quadric bundles. By composing with $\tau$, we get two equidimensional contractions $f_{i}\colon X_{2}\to\mathbb{P}^{2}$, $i=1,2$. Fig. 11.3 still gives a section of $\operatorname{Eff}(X_{2})$; here $H$ is the pullback of a general hyperplane section of $Z$, and $-K_{Z}=3H-E_{1}-E_{2}$.