# Fano $4$-folds with a small contraction

C. Casagrande  Università di Torino, Dipartimento di Matematica, via Carlo
Alberto 10, 10123 Torino - Italy<EMAIL_ADDRESS>
(Date: May 18, 2022)

###### Abstract.

Let $X$ be a smooth complex Fano 4-fold. We show that if $X$ has a small
elementary contraction, then $\rho_{X}\leq 12$, where $\rho_{X}$ is the Picard
number of $X$. This result is based on a careful study of the geometry of $X$,
on which we give a lot of information. We also show that in the boundary case
$\rho_{X}=12$ an open subset of $X$ has a smooth fibration with fiber
$\mathbb{P}^{1}$. Together with previous results, this implies that if $X$ is
a Fano 4-fold with $\rho_{X}\geq 13$, then every elementary contraction of $X$
is divisorial and sends a divisor to a surface. The proof is based on
birational geometry and the study of families of rational curves. More
precisely the main tools are: the study of families of lines in Fano 4-folds
and the construction of divisors covered by lines, a detailed study of fixed
prime divisors, the properties of the faces of the effective cone, and a
detailed study of rational contractions of fiber type.

###### 2020 Mathematics Subject Classification:

14J45,14J35,14E30

##  1\. Introduction

The classification of smooth, complex Fano varieties has been achieved up to
dimension $3$ and attracts a lot of attention also in higher dimensions. Let
us focus on dimension $4$, the first open case: the context of this paper is
the study of Fano $4$-folds with “large” Picard number (e.g. $\rho\geq 6$) by
means of birational geometry and families of rational curves, with the aim of
gaining a good understanding of the geometry and behaviour of these $4$-folds.
The main result of this paper is the following.

######  Theorem 1.1.

Let $X$ be a smooth Fano $4$-fold and $\rho_{X}$ its Picard number. If $X$ has
a small elementary contraction, then $\rho_{X}\leq 12$.

The proof of this result is based on a careful study of the geometry of $X$,
on which we give a lot of information. We also show an additional property in
the boundary case $\rho_{X}=12$, see Th. 9.1.

Apart from products of del Pezzo surfaces, to the author’s knowledge all the
known examples of Fano $4$-folds have $\rho\leq 9$; there is just one known
family with $\rho=9$, and it has small elementary contractions, see [CCF19].
Thus we do not know whether the bound of Th. 1.1 is sharp.

Let us notice that if $X$ has an elementary contraction of fiber type, then
$\rho_{X}\leq 11$ by [Cas08, Cor. 1.2(ii)-(iii)], and if $X$ has an elementary
divisorial contraction sending a divisor to a point or to a curve, then
$\rho_{X}\leq 5$ [Cas17, Rem. 2.17(1)]. Therefore we have the following.

######  Corollary 1.2.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 13$. Then every
elementary contraction of $X$ is divisorial and sends the exceptional divisor
to a surface.

This is the behaviour of products of del Pezzo surfaces, indeed we expect that
for $\rho_{X}$ large enough these should be the only Fano $4$-folds.

Let us now explain the content of the paper and strategy of the proof of Th.
1.1. In the sequel $X$ is a smooth, complex Fano $4$-fold.

Previous work. Our starting point is the results on the geometry of Fano
$4$-folds developed in [Cas13a, Cas17, Cas20], in particular: the Lefschetz
defect, the classification of fixed prime divisors, and the structure of
rational contractions of fiber type.

The Lefschetz defect. 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 for every closed subset $Z\subset X$, we denote by
$\mathcal{N}_{1}(Z,X)$ the linear subspace of $\mathcal{N}_{1}(X)$ spanned by
classes of curves contained in $Z$. If $X$ is Fano, the Lefschetz defect
${\delta_{X}}$ of $X$, introduced in [Cas12], is defined as follows:

(1.3)
$\delta_{X}:=\max\bigl{\\{}\operatorname{codim}\mathcal{N}_{1}(D,X)\,|\,D\text{
is a prime divisor in }X\bigr{\\}}.$

If $X$ is not a product of surfaces and $\delta_{X}\geq 2$, then $\rho_{X}\leq
12$ (see Th. 2.6 and 2.7), so that we can reduce to the case $\delta_{X}\leq
1$.

Fixed prime divisors. A fixed prime divisor is a prime divisor $D$ which is
the stable base locus of the linear system $|D|$. Thanks to the bounds on the
Lefschetz defect, in [Cas13a] it is shown that when $\rho_{X}\geq 7$ there are
four possible types of fixed prime divisors of $X$, called $(3,2)$,
$(3,1)^{sm}$, $(3,0)^{sm}$, and $(3,0)^{Q}$, in relation to the associated
elementary divisorial contractions; see §4.2 for more details. If $X$ has a
fixed prime divisor of type $(3,0)^{sm}$, then $\rho_{X}\leq 12$ by [Cas17]
(see Th. 4.7), so that we can exclude this case and focus on the remaining
ones.

Rational contractions of fiber type. A rational contraction of fiber type is a
rational map $f\colon X\dasharrow Y$ that factors as sequence of flips
$X{\dasharrow}X^{\prime}$ followed by a contraction of fiber type
$f^{\prime}\colon X^{\prime}\to Y$ (i.e. a surjective map with connected
fibers, with $Y$ normal and projective, and $\dim Y\leq 3$, see Section 2).
Rational contractions of fiber type of Fano $4$-folds are studied in detail in
[Cas20], where in particular it is shown that if $X$ has a rational
contraction onto a $3$-fold, and $X$ is not a product of surfaces, then
$\rho_{X}\leq 12$ (see Th. 5.8).

New results. The new tools and results used for the proof of Th. 1.1 include:
the construction of families of lines and of divisors covered by lines, the
properties of the faces of the effective cone, a detailed study of fixed prime
divisors of type $(3,1)^{sm}$ and $(3,0)^{Q}$, and a detailed study of
rational contractions of fiber type onto surfaces and onto $\mathbb{P}^{1}$.

Families of lines. We define a line in $X$ as a rational curve $C$ such that
$-K_{X}\cdot C=1$, and a family of lines as a “maximal” irreducible subvariety
$V$ of $\operatorname{Chow}(X)$ whose general member is a line, see Section 3;
we study such families. By standard deformation theory one has $\dim V\geq 2$,
and we show that if $\rho_{X}\geq 6$, then $\dim V=2$ (Th. 3.7), because a
family of lines of larger dimension yields a prime divisor $D$ with small
$\dim\mathcal{N}_{1}(D,X)$, contradicting the results on the Lefschetz defect.
Moreover the curves of the family $V$ cover a surface or a prime divisor.
Divisors covered by lines play a crucial role in the paper; when $\rho_{X}\geq
7$ such a divisor is either nef, or fixed of type $(3,2)$; moreover the
covering family of lines is unique (Lemmas 3.11 and 3.12).

Movable and fixed faces of the effective cone. As usual we denote by
$\mathcal{N}^{1}(X)$ the real vector space of numerical equivalence classes of
divisors with real coefficients in $X$; the cones $\operatorname{Eff}(X)$ and
$\operatorname{Mov}(X)$, of classes of effective and movable divisors
respectively, are convex polyhedral cones in $\mathcal{N}^{1}(X)$.

We say that a (proper) face $\tau$ of $\operatorname{Eff}(X)$ is a movable
face if the relative interior of $\tau$ intersects the movable cone
$\operatorname{Mov}(X)$; there exists a movable face if and only if $X$ has
some non-zero, non-big movable divisor. Given a movable face $\tau$ of
$\operatorname{Eff}(X)$, we construct a rational contraction of fiber type
$f\colon X\dasharrow Y$ such that
$\rho_{Y}=\dim(\tau\cap\operatorname{Mov}(X))$ and
$\rho_{X}\leq\dim\tau+\rho_{F}$, where $F$ is a general fiber of
$f^{\prime}\colon X^{\prime}\to Y$ (where as above $f$ factors as a sequence
of flips followed by the regular contraction $f^{\prime}$) and it is smooth,
Fano, with $\dim F\leq 3$, so that $\rho_{F}\leq 10$ by classification (see
§5.2).

On the other hand, we say that a face $\tau$ of $\operatorname{Eff}(X)$ is a
fixed face if $\tau\cap\operatorname{Mov}(X)=\\{0\\}$. A fixed face is always
simplicial, and is generated by classes of fixed prime divisors; these
divisors are in very special relative positions (see §4.1 and 4.2).

Thus when $\operatorname{Eff}(X)$ has a movable face of small dimension, we
get a good bound on $\rho_{X}$; in particular if $\operatorname{Eff}(X)$ has a
one-dimensional movable face, then $\rho_{X}\leq 11$. Therefore we can reduce
to the case where every one-dimensional face of $\operatorname{Eff}(X)$ is
fixed, and this yields a lot of fixed prime divisors in $X$.

Strategy of the proof. Let $f\colon X\to Y$ be a small elementary contraction,
and assume that $\rho_{X}\geq 7$ and $\delta_{X}\leq 1$. We can also assume
that $\operatorname{Eff}(X)$ is generated by classes of fixed prime divisors,
and there exists one such divisor $D$ such that $D\cdot C<0$ for a curve
$C\subset X$ contracted by $f$, so that $D$ contains the exceptional locus of
$f$.

After the classification of fixed prime divisors, we have four possible types
for $D$; we can exclude that $D$ is of type $(3,0)^{sm}$. If $D$ were of type
$(3,2)$, $\operatorname{Exc}(f)\subset D$ would yield
$\dim\mathcal{N}_{1}(D,X)\leq 2$, which contradicts our assumptions on
$\rho_{X}$ and $\delta_{X}$. Therefore $D$ is of type $(3,1)^{sm}$ or
$(3,0)^{Q}$; for this reason we focus on these two types of fixed divisors,
which are the main characters of the paper (see Rem. 4.11).

Given a fixed prime divisor $E\subset X$ of type $(3,1)^{sm}$ or $(3,0)^{Q}$,
we construct from $E$ several families of lines, each covering a prime divisor
different from $E$; this is a key result that allows to produce many prime
divisors covered by lines with several good properties.

Using all these preliminary results, first we show that $\rho_{X}\leq 12$ when
$X$ has a fixed prime divisor of type $(3,1)^{sm}$ (Th. 8.1). The proof is
quite articulated and is contained in Section 8; we refer the reader to the
beginning of that section for an overview.

Then we are left to consider the case where $X$ has only fixed prime divisors
of type $(3,2)$ and $(3,0)^{Q}$; this is treated in Section 9, and again we
refer the reader to the beginning of that section for an overview.

A frequent strategy in the paper is to look for movable, non-big divisors on
$X$, in order to construct a rational contraction of fiber type, and then use
it to bound $\rho_{X}$. Such movable, non-big divisors are usually obtained
using fixed prime divisors and/or prime divisors covered by lines.

Summary. Section 2 contains the notation and recalls the preliminary results
on the birational geometry of Fano $4$-folds and on the Lefschetz defect.
Section 3 contains the results on families of lines and divisors covered by
lines in Fano $4$-folds. Section 4 is devoted to fixed prime divisors; in
particular in §4.4 and 4.5 we show several properties of fixed prime divisors
of type $(3,1)^{sm}$ and $(3,0)^{Q}$ that are needed in the sequel. Section 5
is devoted to rational contractions of fiber type; in particular in §5.3 we
show several results on rational contractions of fiber type onto surfaces and
onto $\mathbb{P}^{1}$, that are needed in the sequel.

Section 6 contains a key construction which, given a fixed prime divisor of
type $(3,1)^{sm}$ or $(3,0)^{Q}$, produces many prime divisors covered by
lines. Then we prove some properties of these divisors depending on the
different settings. Section 7 contains two results where we manage to
construct a rational contraction of fiber type on $X$, and use it to bound
$\rho_{X}$. Finally Sections 8 and 9 contain the actual proof of Th. 1.1,
first considering the case where $X$ has a fixed prime divisor of type
$(3,1)^{sm}$, and then the case where $X$ has only fixed prime divisors of
type $(3,2)$ and $(3,0)^{Q}$.

Acknowledgements. I thank the referee for useful comments.

###### Contents

  1. 1 Introduction
  2. 2 Notation and preliminaries
  3. 3 Lines in Fano $4$-folds
  4. 4 Fixed prime divisors and associated contractions
  5. 5 Rational contractions of fiber type
  6. 6 Constructing divisors covered by lines
  7. 7 Constructing rational contractions of fiber type
  8. 8 The case $(3,1)^{sm}$
  9. 9 The case $(3,0)^{Q}$

##  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 $\mathcal{N}^{*}$. 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.

We refer the reader to [HK00] for the notion and basic properties of a Mori
dream space; we recall that smooth Fano varieties are Mori dream spaces by
[BCHM10, Cor. 1.3.2]. We also refer to [Deb01, KM98] for the standard notions
in birational geometry.

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space.
If $D$ is a divisor and $C$ is a curve in $X$, we denote by
$[D]\in\mathcal{N}^{1}(X)$ and $[C]\in\mathcal{N}_{1}(X)$ their classes, and
we set $D^{\perp}:=[D]^{\perp}\subset\mathcal{N}_{1}(X)$ and
$C^{\perp}:=[C]^{\perp}\subset\mathcal{N}^{1}(X)$.

For every closed subset $Z\subset X$, we denote by $\mathcal{N}_{1}(Z,X)$ the
linear subspace of $\mathcal{N}_{1}(X)$ spanned by classes of curves contained
in $Z$. We will use the following simple property.

######  Remark 2.1.

Let $X$ be a smooth projective variety, $Z\subset X$ a closed subset, and
$D\subset X$ a prime divisor disjoint from $Z$. Then
$\mathcal{N}_{1}(Z,X)\subseteq D^{\perp}$, because $C\cdot D=0$ for every
curve $C\subset Z$.

A movable divisor is an effective divisor $D$ such that the stable base locus
of the linear system $|D|$ has codimension $\geq 2$. A fixed prime divisor is
a prime divisor $D$ which is the stable base locus of $|D|$, namely such that
$h^{0}(mD)=1$ for every $m\in\mathbb{Z}_{>0}$. We will consider the usual
cones of divisors and of curves:

$\operatorname{Nef}(X)\subseteq\operatorname{Mov}(X)\subseteq\operatorname{Eff}(X)\subset\mathcal{N}^{1}(X),\qquad\operatorname{mov}(X)\subseteq\operatorname{NE}(X)\subset\mathcal{N}_{1}(X),$

where all the notations are standard except $\operatorname{mov}(X)$, 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 closed, rational and
polyhedral. An extremal ray of $\operatorname{NE}(X)$ is a one-dimensional
face of this cone.

A contraction $f\colon X\to Y$ is a surjective map, with connected fibers,
where $Y$ is normal and projective. Given a divisor $D$ in $X$, $f$ is
$D$-negative if $D\cdot C<0$ for every curve $C\subset X$ such that
$f(C)=\\{pt\\}$. An extremal ray $R$ of $\operatorname{NE}(X)$ is $D$-negative
if $D\cdot\gamma<0$ for $\gamma\in R$, $\gamma\neq 0$. We do not assume that
contractions or flips are $K$-negative, unless specified.

A small $\mathbb{Q}$-factorial modification (SQM) is a birational map
$\varphi\colon X\dasharrow X^{\prime}$ which is an isomorphism in codimension
one, where $X^{\prime}$ is a normal and $\mathbb{Q}$-factorial projective
variety; then $X^{\prime}$ is a Mori dream space too, and $\varphi$ can be
factored as a finite sequence of flips.

A rational contraction (also called a contracting rational map) is a rational
map $f\colon X\dasharrow Y$ that can be factored as
$X\stackrel{{\scriptstyle\varphi}}{{\dasharrow}}X^{\prime}\stackrel{{\scriptstyle
f^{\prime}}}{{\to}}Y$, where $\varphi$ is a SQM and $f^{\prime}$ is a
contraction. As in the regular case, a rational contraction is: of fiber type
if $\dim Y<\dim X$, elementary if $\rho_{X}-\rho_{Y}=1$, and elementary
divisorial if $f^{\prime}$ is an elementary divisorial contraction.

######  Remark 2.2.

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space,
and $D$ a divisor such that $[D]\in\operatorname{Mov}(X)$, $[D]\neq 0$. Then
there is a prime divisor with class in $\mathbb{R}_{\geq 0}[D]$.

Indeed there is a SQM $X\dasharrow X^{\prime}$ such that the transform
$D^{\prime}$ of $D$ in $X^{\prime}$ is nef, and hence semiample. Therefore
$|mD^{\prime}|$ is base-point-free for $m\in\mathbb{N}$ large and divisible
enough; by Bertini the general member of $|mD^{\prime}|$ is irreducible,
unless the linear system yields a contraction $f\colon X^{\prime}\to C$ onto a
curve. In this last case we have $\mathbb{R}_{\geq
0}[D^{\prime}]=f^{*}\mathcal{N}^{1}(C)$, thus the general fiber of $f$ is an
irreducible divisor with class in $\mathbb{R}_{\geq 0}[D^{\prime}]$.

Assume now that $X$ is smooth of dimension $4$, and let $f\colon X\to Y$ be an
elementary divisorial contraction. We say that $f$ is:

  1. –

of type $(3,2)$ if $\dim(f(\operatorname{Exc}(f)))=2$;

  2. –

of type $(3,1)^{sm}$ if $Y$ is smooth and $f$ is the blow-up of a smooth
curve;

  3. –

of type $(3,0)^{Q}$ if $\operatorname{Exc}(f)$ is isomorphic to an irreducible
quadric $Q$, $f(\operatorname{Exc}(f))$ is a point, and
$\mathcal{N}_{\operatorname{Exc}(f)/X}\cong\mathcal{O}_{Q}(-1)$;

  4. –

of type $(3,0)^{sm}$ if $Y$ is smooth and $f$ is the blow-up of a point.

An exceptional plane is a closed subset $L\subset X$ such that
$L\cong\mathbb{P}^{2}$ and
$\mathcal{N}_{L/X}\cong\mathcal{O}_{\mathbb{P}^{2}}(-1)^{\oplus 2}$; we will
denote by $C_{L}\subset L$ a curve corresponding to a line in
$\mathbb{P}^{2}$. An exceptional curve is a closed subset $\ell\subset X$ such
that $\ell\cong\mathbb{P}^{1}$ and
$\mathcal{N}_{\ell/X}\cong\mathcal{O}_{\mathbb{P}^{1}}(-1)^{\oplus 3}$. We
have $-K_{X}\cdot C_{L}=1$ and $-K_{X}\cdot\ell=-1$.

######  Theorem 2.3 ([Kaw89]).

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

######  Lemma 2.4 ([Cas13a], Rem. 3.6).

Let $X$ be a smooth Fano $4$-fold and $\varphi\colon X\dasharrow\widetilde{X}$
a SQM. Then $\widetilde{X}$ is smooth and there are pairwise disjoint
exceptional planes $L_{1},\dotsc,L_{r}\subset X$ and exceptional curves
$\ell_{1},\dotsc,\ell_{r}\subset\widetilde{X}$ such that $\varphi$ factors as:

$\textstyle{{\widehat{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\widetilde{X}}$

where $f$ is the blow-up of $L_{1},\dotsc,L_{r}$, $g$ is the blow-up of
$\ell_{1},\dotsc,\ell_{r}$, and
$E_{i}:=f^{-1}(L_{i})=g^{-1}(\ell_{i})\cong\mathbb{P}^{2}\times\mathbb{P}^{1}$
for every $i=1,\dotsc,r$. Moreover:

  1. $(a)$

if $\widetilde{\Gamma}\subset\widetilde{X}$ is an irreducible curve different
from $\ell_{1},\dotsc,\ell_{r}$, with transforms $\Gamma\subset X$ and
$\widehat{\Gamma}\subset\widehat{X}$, then
$-K_{\widetilde{X}}\cdot\widetilde{\Gamma}=-K_{X}\cdot\Gamma+\sum_{i=1}^{r}E_{i}\cdot\widehat{\Gamma}>0$;

  2. $(b)$

if $\widetilde{\Gamma}$ intersects some exceptional curve, then
$-K_{X}\cdot\Gamma<-K_{\widetilde{X}}\cdot\widetilde{\Gamma}$, so that
$-K_{\widetilde{X}}\cdot\widetilde{\Gamma}\geq 2$.

######  Lemma 2.5 ([Cas13a], Rem. 3.7).

Let $X$ be a smooth Fano $4$-fold and $f\colon X\dasharrow Y$ a rational
contraction. Then one can factor $f$ as
$X\stackrel{{\scriptstyle\varphi}}{{\dasharrow}}X^{\prime}\stackrel{{\scriptstyle
f^{\prime}}}{{\to}}Y$, where $\varphi$ is a SQM, $X^{\prime}$ is smooth, and
$f^{\prime}$ is a $K$-negative contraction. Moreover $Y$ has rational
singularities.

Finally let us recall the following results on the Lefschetz defect (see
(1.3)).

######  Theorem 2.6 ([Cas12], Th. 3.3 and Cor. 1.3).

Let $X$ be a smooth Fano $4$-fold which is not a product of surfaces. Then
$\delta_{X}\leq 3$, and if $\delta_{X}=3$, then $\rho_{X}\leq 6$.

######  Theorem 2.7 ([Cas13b], Th. 1.2).

Let $X$ be a smooth Fano 4-fold with $\delta_{X}=2$. Then $\rho_{X}\leq 12$,
and if $\rho_{X}\geq 7$, then $X$ has a rational contraction onto a $3$-fold.

######  Remark 2.8 ([Cas12], Ex. 3.1).

If $X\cong S_{1}\times S_{2}$ where $S_{i}$ are del Pezzo surfaces with
$\rho_{S_{1}}\geq\rho_{S_{2}}$, then $\delta_{X}=\rho_{S_{1}}-1$, and
$\rho_{X}\leq 2\delta_{X}+2$.

##  3\. Lines in Fano $4$-folds

Let $X$ be a normal projective variety. We denote by
$\operatorname{Hom}_{bir}(\mathbb{P}^{1},X)$ the Hilbert scheme of morphisms
$\mathbb{P}^{1}\to X$ which are birational onto their image, and by
$\operatorname{Hom}_{bir}(\mathbb{P}^{1},X)^{n}$ its normalization; there is a
natural morphism
$\operatorname{Hom}_{bir}(\mathbb{P}^{1},X)^{n}\to\operatorname{Chow}(X)$
[Kol96, Cor. I.6.9]. By a family of rational curves in $X$ we mean an
irreducible subvariety $V$ of $\operatorname{Chow}(X)$ which is the closure of
the image of an irreducible component of
$\operatorname{Hom}_{bir}(\mathbb{P}^{1},X)^{n}$, see [Kol96, II.2.11]. For
$v\in V$ general, the corresponding cycle is an irreducible and reduced
rational curve, and every member of the family is an effective, connected one-
cycle with rational components. We denote by $[V]\in\mathcal{N}_{1}(X)$ the
numerical equivalence class of the general curve of the family. There is a
universal family:

(3.1)
$\textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{e\quad}$$\textstyle{X}$$\textstyle{V}$

Moreover $\operatorname{Locus}V:=e(\mathcal{C})\subseteq X$ is the union of
the curves of the family; it is an irreducible closed subset, and we say that
$V$ is covering if $\operatorname{Locus}V=X$.

######  Remark 3.2.

If $X$ is smooth and $V$ is covering, we have $-K_{X}\cdot[V]\geq 2$, indeed
the general member of $V$ corresponds to a free curve [Deb01, 4.10 and Example
4.7(1)].

######  Definition 3.3.

Let $X$ be a Gorenstein normal projective variety. A family of lines in $X$ is
a family of rational curves such that $-K_{X}\cdot[V]=1$.

We say that a prime divisor $D\subset X$ is covered by a family of lines if
there exists a family of lines $V$ with $\operatorname{Locus}V=D$.

We note that, for a smooth Fano variety $X$, lines are usually defined in
terms of the index $r_{X}$ of $X$: if $-K_{X}=r_{X}H$, then $V$ would be a
family of lines when $H\cdot[V]=1$. In this paper we are interested in Fano
$4$-folds $X$ of index $1$, indeed we will tipically assume that $\rho_{X}\geq
7$, while Fano $4$-folds with $r_{X}>1$ are classified and have $\rho_{X}\leq
4$, see [IP99, Theorems 3.3.1 and 7.2.15]. Thus for simplicity we define lines
directly as in Def. 3.3.

If $V$ is a family of lines and $X$ is Fano, then every member of the family
is irreducible and reduced, so that $V$ is an _unsplit_ family in the
terminology of [Kol96, Def. IV.2.1].

######  Remark 3.4.

Suppose that $X$ is smooth and Fano, and let $V$ be a family of lines in $X$.
If $Z\subset X$ is a closed subset, we set $\operatorname{Locus}V_{Z}$ to be
the union of the curves of the family meeting $Z$, namely:

$\operatorname{Locus}V_{Z}:=e\bigl{(}\pi^{-1}\bigl{(}\pi(e^{-1}(Z))\bigr{)}\bigr{)}.$

Thus $\operatorname{Locus}V_{Z}$ is a closed subset, and if it is non-empty,
then

(3.5)
$\mathcal{N}_{1}(\operatorname{Locus}V_{Z},X)=\mathcal{N}_{1}(Z,X)+\mathbb{R}[V],$

see [ACO04, Lemma 4.1]. If $Z=\\{x\\}$ is a point, we just write
$\operatorname{Locus}V_{x}$, and set $V_{x}:=\pi(e^{-1}(x))\subseteq V$ for
the subvariety of $V$ parametrizing curves containing $x$.

######  Example 3.6 (lines in products of surfaces).

Let $X=S\times T$ with $S$ and $T$ del Pezzo surfaces, and let $V$ be a family
of lines in $X$. It is not difficult to see that up to switching $S$ and $T$,
the curves of the family have the form $C\times\\{p\\}$ where $C\subset S$ is
a given rational curve with $-K_{S}\cdot C=1$, and $p$ varies in $T$, so that
$\dim V=2$, $V\cong T$ and $\operatorname{Locus}V\cong C\times T$.

If $C$ is smooth, then it is a $(-1)$-curve in $S$, and $[V]$ generates an
extremal ray of type $(3,2)$ of $\operatorname{NE}(X)$.

If $C$ is singular, then $\rho_{S}=9$, $C\in|-K_{S}|$, and
$D=\operatorname{Locus}V$ is a nef prime divisor.

######  Theorem 3.7.

Let $X$ be a smooth Fano $4$-fold and $V$ a family of lines in $X$. Then
$2\leq\dim V\leq 4$, and moreover:

  1. $(a)$

if $\dim V=4$, then $\dim\operatorname{Locus}V=3$ and $\rho_{X}\leq 3$;

  2. $(b)$

if $\dim V=3$, then $\dim\operatorname{Locus}V=3$ and $\rho_{X}\leq 5$;

  3. $(c)$

if $\dim V=2$, then one of the following holds:

    1. $(i)$

$\dim\operatorname{Locus}V=3$ and $V_{x}$ is finite for
$x\in\operatorname{Locus}V$ general;

    2. $(ii)$

$\dim\operatorname{Locus}V=2$ and
$\operatorname{Locus}V=\operatorname{Locus}V_{x}$ for every
$x\in\operatorname{Locus}V$.

If there are two curves of the family which are disjoint, then we are in case
$(i)$.

The bounds on $\rho_{X}$ in $(a)$ and $(b)$ are sharp, see Examples 3.9 and
3.10.

###### Proof.

The inequality $\dim V\geq 2$ follows from the standard lower bound on
$\dim\operatorname{Hom}(\mathbb{P}^{1},X)$, see [Kol96, Th. II.1.2 and Prop.
II.2.11]. We also have

(3.8) $\dim\operatorname{Locus}V+\dim\operatorname{Locus}V_{x}=\dim V+2$

for general $x\in\operatorname{Locus}V$, because the family is unsplit, see
[Kol96, Prop. IV.2.5], which yields $\dim\operatorname{Locus}V\geq 2$.

On the other hand $\operatorname{Locus}V\subsetneq X$ by Rem. 3.2, thus
$\dim\operatorname{Locus}V\leq 3$, and (3.8) yields $\dim V\leq 4$.

Suppose that $\dim V=4$. By (3.8) for general $x\in\operatorname{Locus}V$ we
have $\dim\operatorname{Locus}V+\dim\operatorname{Locus}V_{x}=6$, and
$\dim\operatorname{Locus}V\leq 3$, thus $D=\operatorname{Locus}V$ is a prime
divisor and $D=\operatorname{Locus}V_{x}$ for general $x\in D$. This implies
that $\dim\mathcal{N}_{1}(D,X)=1$ by (3.5), hence $\rho_{X}\leq 3$ by [Cas08,
Prop. 3.16].

Suppose that $\dim V=3$. By (3.8) for general $x\in\operatorname{Locus}V$ we
have $\dim\operatorname{Locus}V+\dim\operatorname{Locus}V_{x}=5$, thus
$D=\operatorname{Locus}V$ is a prime divisor and $\operatorname{Locus}V_{x}$
is a surface for general $x\in D$, while $\dim\operatorname{Locus}V_{x}\geq 2$
for every $x\in D$.

Choose $x_{0}$ such that $\dim\operatorname{Locus}V_{x_{0}}=2$. Since
$x_{0}\in\operatorname{Locus}V_{x_{0}}\subset D$ and $\dim D=3$, we can choose
an irreducible curve $C\subset D$ containing $x_{0}$ and not contained in
$\operatorname{Locus}V_{x_{0}}$. Then

$D=\bigcup_{x\in C}\operatorname{Locus}V_{x}=\operatorname{Locus}V_{C},$

thus $\mathcal{N}_{1}(D,X)=\mathbb{R}[C]+\mathbb{R}[V]$ by (3.5) and
$\dim\mathcal{N}_{1}(D,X)\leq 2$. This implies that
$\delta_{X}\geq\rho_{X}-2$. Let us notice that $X$ cannot be a product of
surfaces (see Ex. 3.6), so that $\delta_{X}\leq 3$ by Th. 2.6, and hence
$\rho_{X}\leq 5$.

Finally suppose that $\dim V=2$. By (3.8) for general
$x\in\operatorname{Locus}V$ we have
$\dim\operatorname{Locus}V+\dim\operatorname{Locus}V_{x}=4$.

If $\dim\operatorname{Locus}V=3$, then for $x\in\operatorname{Locus}V$ general
we have $\dim\operatorname{Locus}V_{x}=1$ and $\dim V_{x}=0$.

If $\dim\operatorname{Locus}V=2$, we get $\dim\operatorname{Locus}V_{x}=2$ for
$x\in\operatorname{Locus}V$ general, and in fact for every
$x\in\operatorname{Locus}V$ by upper semicontinuity. Since
$\operatorname{Locus}V$ is irreducible, we conclude that
$\operatorname{Locus}V=\operatorname{Locus}V_{x}$ for every
$x\in\operatorname{Locus}V$.

We show that in this last case two curves of the family always meet. Let
$\Gamma\subset\operatorname{Locus}V$ be a curve of the family and consider the
universal family as in (3.1). Then $e$ has fibers of dimension at least $1$,
so that $e^{-1}(\Gamma)$ has dimension $2$. Since $e(e^{-1}(\Gamma))=\Gamma$,
we have $\pi(e^{-1}(\Gamma))=V$. This means that $e^{-1}(\Gamma)$ meets every
fiber of $\pi$, hence $\Gamma$ meets every curve of the family $V$. ∎

######  Example 3.9 (a case with $\dim V=4$ and $\rho_{X}=3$).

Let $Y=\mathbb{P}_{\mathbb{P}^{3}}(\mathcal{O}\oplus\mathcal{O}(3))$,
$G\subset Y$ a section of the $\mathbb{P}^{1}$-bundle $Y\to\mathbb{P}^{3}$
with normal bundle $\mathcal{N}_{G/Y}\cong\mathcal{O}(3)$, and $X$ the blow-up
of $Y$ along a plane contained in $G\cong\mathbb{P}^{3}$. The $4$-fold $X$ is
toric, Fano, with $\rho_{X}=3$, and contains a divisor $D\cong\mathbb{P}^{3}$
with normal bundle $\mathcal{N}_{D/X}\cong\mathcal{O}(-3)$; this is $E_{1}$ in
[Bat99] and $X^{7}_{3,1}$ in [Sec21]. Let $V$ be the family of lines in $D$,
then $-K_{X}\cdot[V]=1$ and $\dim V=4$; in fact
$V\cong\operatorname{Gr}(1,\mathbb{P}^{3})$. Every curve of the family is
smooth with normal bundle $\mathcal{O}(1)^{\oplus 2}\oplus\mathcal{O}(-3)$,
and $D\cdot[V]=-3$.

More generally, Fano $4$-folds with $\rho_{X}=3$ and containing a prime
divisor $D$ with $\dim\mathcal{N}_{1}(D,X)=1$ are classified in [Sec21], there
are 28 families; these are the possible Fano $4$-folds with $\rho_{X}=3$ and
with a family of lines $V$ such that $\dim V=4$.

######  Example 3.10 (a case with $\dim V=3$ and $\rho_{X}=5$).

Let $X$ be the toric Fano $4$-fold $K_{1}$ in Batyrev’s list [Bat99]; we have
$\rho_{X}=5$ and $X$ has a smooth fibration onto $\mathbb{P}^{2}$, with fiber
the surface obtained by blowing-up $\mathbb{P}^{2}$ at $3$ non-collinear
points. The $4$-fold $X$ contains a prime divisor
$D\cong\mathbb{P}^{1}\times\mathbb{P}^{2}$ with normal bundle
$\mathcal{N}_{D/X}\cong\mathcal{O}(-1,-2)$. Let $V$ be the family of lines in
$\\{pt\\}\times\mathbb{P}^{2}\subset D$, then $-K_{X}\cdot[V]=1$ and $\dim
V=3$; in fact $V\cong\mathbb{P}^{1}\times\mathbb{P}^{2}$. Every curve of the
family is smooth with normal bundle
$\mathcal{O}(1)\oplus\mathcal{O}\oplus\mathcal{O}(-2)$, and $D\cdot[V]=-2$.
Let us note that $D$ is also the locus of another family of lines $W$, given
by the curves $\mathbb{P}^{1}\times\\{pt\\}\subset D$.

More generally, Fano $4$-folds with $\rho_{X}=5$ and containing a prime
divisor $D$ with $\dim\mathcal{N}_{1}(D,X)=2$ are classified in [CR22], there
are 6 families; these are the possible Fano $4$-folds with $\rho_{X}=5$ and
with a family of lines $V$ such that $\dim V=3$.

######  Lemma 3.11.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, and $D\subset X$ a
prime divisor covered by a family of lines.

  1. $(a)$

The family of lines $V$ such that $D=\operatorname{Locus}V$ is unique, so that
$D$ and $V$ determine each other.

  2. $(b)$

If $D$ contains an exceptional plane $L$, then $[V]=[C_{L}]$.

###### Proof.

We show $(a)$. Suppose by contradiction that there are two distinct families
$V$, $W$ of lines with $D=\operatorname{Locus}V=\operatorname{Locus}W$. We
prove that this yields $\dim\mathcal{N}_{1}(D,X)\leq 3$.

Let $x\in D$ and consider

$S_{x}:=\operatorname{Locus}W_{\,\operatorname{Locus}V_{x}}.$

We have $\dim S_{x}\geq 2$ and by (3.5)
$\mathcal{N}_{1}(S_{x},X)=\mathbb{R}[V]+\mathbb{R}[W]$. If $S_{x}=D$ for some
$x$, we are done. Otherwise take $x_{0}\in D$ a general point and let
$\Gamma\subset D$ be an irreducible curve through $x_{0}$ such that
$\Gamma\not\subset S_{x_{0}}$. Consider

$T:=\bigcup_{x\in\Gamma}S_{x}=\operatorname{Locus}W_{\operatorname{Locus}V_{\Gamma}}.$

Then $T=D$, so that by (3.5) we get

$\mathcal{N}_{1}(D,X)=\mathcal{N}_{1}(\operatorname{Locus}V_{\Gamma},X)+\mathbb{R}[W]=\mathbb{R}[\Gamma]+\mathbb{R}[V]+\mathbb{R}[W]$

and $\dim\mathcal{N}_{1}(D,X)\leq 3$. Since $\rho_{X}\geq 7$, this yields
$\delta_{X}\geq 4$ and hence $X$ is a product of surfaces by Th. 2.6, a
contradiction (see Ex. 3.6).

The proof of $(b)$ is similar and we leave it to the reader. ∎

######  Lemma 3.12 ([Cas17], Lemma 2.18).

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, and $D\subset X$ a
prime divisor covered by a family $V$ of lines. Then $D\cdot[V]\geq-1$, and
one of the following holds:

  1. $(i)$

$D$ is nef;

  2. $(ii)$

$D\cdot[V]=-1$ and $[V]$ generates an extremal ray of type $(3,2)$ of
$\operatorname{NE}(X)$.

The following result is similar to [MM86, Prop. 5.3] on Fano $3$-folds.

######  Lemma 3.13.

Let $X$ be a smooth Fano $4$-fold and $V$ a family of lines in $X$ with $\dim
V=2$ and $D:=\operatorname{Locus}V$ a prime divisor. Suppose that there exists
a curve $C$ belonging to the family $V$ such that $C\cong\mathbb{P}^{1}$,
$\mathcal{N}_{C/X}\cong\mathcal{O}^{\oplus 2}\oplus\mathcal{O}(-1)$, and $C$
does not intersect other curves of the family $V$. Then $D\cdot[V]=-1$.

###### Proof.

Since $C\cong\mathbb{P}^{1}$ and $h^{1}(C,\mathcal{N}_{C/X})=0$,
$\operatorname{Hilb}(X)$ and $\operatorname{Chow}(X)$ are smooth and locally
isomorphic at the point corresponding to $C$ [Kol96, Th. I.2.8 and I.6.3, Cor.
I.6.6.1], and $C$ corresponds to a smooth point $v_{0}$ of a unique
irreducible component $V$ of $\operatorname{Chow}(X)$; consider the universal
family (3.1) and set $C_{0}:=\pi^{-1}(v_{0})\subset\mathcal{C}$. Notice that
$\mathcal{C}$ is smooth around $C_{0}$, $\pi$ is a smooth fibration in
$\mathbb{P}^{1}$ around $C_{0}$, and $e_{|C_{0}}\colon C_{0}\to C$ is an
isomorphism.

We show that the differential of $e\colon\mathcal{C}\to X$ is injective at
every point of $C_{0}$; this is a standard argument. Restricting to the two
curves, $de\colon T_{\mathcal{C}\,|C_{0}}\to T_{X|C}$ restricts to an
isomorphism between $T_{C_{0}}$ and $T_{C}$, and induces
$\alpha\colon\mathcal{N}_{C_{0}/\mathcal{C}}\to\mathcal{N}_{C/X}$. Moreover
$\mathcal{N}_{C_{0}/\mathcal{C}}\cong
T_{v_{0}}V\otimes\mathcal{O}_{C_{0}}\cong
H^{0}(C,\mathcal{N}_{C/X})\otimes\mathcal{O}_{C_{0}}$, and $\alpha$ is
naturally identified with the evaluation of global sections (see e.g. [KPS18,
§2.2]), which is injective because $\mathcal{N}_{C/X}\cong\mathcal{O}^{\oplus
2}\oplus\mathcal{O}(-1)$.

Therefore $e$ is smooth at every point of $C_{0}$. If there exist $z_{1}\in
C_{0}$ and $z_{2}\in\mathcal{C}$ such that $e(z_{1})=e(z_{2})$, then
$e(\pi^{-1}(\pi(z_{2})))$ is a curve of the family which intersects $C$, so by
assumption $e(\pi^{-1}(\pi(z_{2})))=C$. This implies that $\pi(z_{2})=v_{0}$,
$z_{2}\in C_{0}$ and hence that $z_{1}=z_{2}$.

We conclude that $e\colon\mathcal{C}\to D$ is birational and that $C_{0}$ is
contained in the open subset where $e$ is an isomorphism. Therefore $D$ is
smooth around $C$, $-K_{D}\cdot C=2$, and finally $D\cdot C=-1$. ∎

##  4\. Fixed prime divisors and associated contractions

This section is devoted to fixed prime divisors. In §4.1 we introduce the
notions of “fixed face” of the effective cone of a Mori dream space, and of
“adjacent” fixed prime divisors; both will be very relevant in the rest of the
paper. In §4.2 we recall the classification of fixed prime divisors in Fano
$4$-folds with $\rho\geq 7$, and report some results on them. Finally in §4.3
we focus on fixed prime divisors of type $(3,1)^{sm}$ and $(3,0)^{Q}$, which
are the ones that are relevant for the proof of Th. 1.1 (see Rem. 4.11), and
prove many properties that we will need in the sequel.

###  4.1. Fixed faces of the effective cone

######  Definition 4.1.

Let $X$ be a projective, normal and $\mathbb{Q}$-factorial Mori dream space. A
face $\tau$ of $\operatorname{Eff}(X)$ is called a fixed face if
$\tau\cap\operatorname{Mov}(X)=\\{0\\}$.

A fixed face is generated by classes of fixed prime divisors. There is a
bijection between fixed prime divisors of $X$ and one-dimensional fixed faces
of $\operatorname{Eff}(X)$, via $D\mapsto\mathbb{R}_{\geq 0}[D]$ (see [Cas13a,
Rem. 2.19]).

######  Lemma 4.2.

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space.
Every fixed face of $\operatorname{Eff}(X)$ is simplicial.

###### Proof.

Let $f\colon X\dasharrow Y$ be a birational (rational) contraction such that
$Y$ is $\mathbb{Q}$-factorial, and $\varepsilon_{f}\subset\mathcal{N}^{1}(X)$
the convex cone generated by the classes of all exceptional prime divisors of
$f$. Then $\varepsilon_{f}$ is a face of $\operatorname{Eff}(X)$, because if
$D_{1},D_{2}$ are effective $\mathbb{Q}$-divisors on $X$ such that
$[D_{1}+D_{2}]\in\varepsilon_{f}$, then $0=f_{*}(D_{1})+f_{*}(D_{2})$ and
$f_{*}(D_{1})$, $f_{*}(D_{2})$ are effective $\mathbb{Q}$-divisors on $Y$; we
conclude that $f_{*}(D_{1})=f_{*}(D_{2})=0$ so that $D_{1}$ and $D_{2}$ are
exceptional and $[D_{1}],[D_{2}]\in\varepsilon_{f}$. Thus $\varepsilon_{f}$ is
a fixed face, and it is a simplicial cone by [Oka16, Lemma 2.7].

Consider the cone
$\sigma_{f}:=\varepsilon_{f}+f^{*}\operatorname{Nef}(Y)\subset\mathcal{N}^{1}(X)$;
then every face of $\sigma_{f}$ is generated by a face of $\varepsilon_{f}$
and a face of $f^{*}\operatorname{Nef}(Y)$. Recall that
$\operatorname{Eff}(X)$ is the union of the cones $\sigma_{f}$ when $f$ varies
[HK00, Prop. 1.11(2)], and such cones intersect each other along common faces
[Oka16, Prop. 2.9].

Now if $\tau$ is a fixed face of $\operatorname{Eff}(X)$, there exists some
$f$ such that $\sigma_{f}$ has a face $\eta$ with $\eta\subseteq\tau$ and
$\dim\eta=\dim\tau$. Since $\tau$ is fixed, we have $\eta\cap
f^{*}\operatorname{Nef}(Y)=\\{0\\}$ and $\eta\subseteq\varepsilon_{f}$,
therefore $\eta$ is a simplicial face of $\operatorname{Eff}(X)$ and
$\eta=\tau$. ∎

######  Definition 4.3.

We say that two fixed prime divisors are adjacent if their classes in
$\mathcal{N}^{1}(X)$ generate a fixed face of $\operatorname{Eff}(X)$.

######  Lemma 4.4.

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space,
$f\colon X\dasharrow Y$ an elementary divisorial rational contraction, and
$D:=\operatorname{Exc}(f)$, so that $D$ is a fixed prime divisor.

  1. $(a)$

For every $r\in\mathbb{Z}_{>0}$ there is a bijection between
$(r+1)$-dimensional faces of $\operatorname{Eff}(X)$ containing $[D]$, and
$r$-dimensional faces of $\operatorname{Eff}(Y)$, given by $\tau\mapsto
f_{*}\tau$;

  2. $(b)$

$\tau$ is fixed if and only if $f_{*}\tau$ is fixed;

  3. $(c)$

there is a bijection between fixed prime divisors $E_{Y}\subset Y$ and fixed
prime divisors $E_{X}\subset X$ adjacent to $D$ (and different from $D$); here
$E_{X}$ is the transform of $E_{Y}$ in $X$.

###### Proof.

[Cas17, Lemma 2.21] shows $(a)$ for $r=1$, as well as $(c)$. The same proof
yields $(a)$ for any $r$, because under the push-forward of divisors
$f_{*}\colon\mathcal{N}^{1}(X)\to\mathcal{N}^{1}(Y)$, the cone
$\operatorname{Eff}(Y)$ can be seen as the “quotient cone” of
$\operatorname{Eff}(X)$ modulo the one-dimensional face $\langle[D]\rangle$,
see for instance [Ewa96, Def. V.2.8 and Th. V.2.9].

For $(b)$, if there exists a non-zero movable divisor $M$ with $[M]\in\tau$,
then $f_{*}M$ is a non-zero movable divisor with class in $f_{*}\tau$.
Conversely, if there exists a non-zero movable divisor $M_{Y}$ with
$[M_{Y}]\in f_{*}\tau$, then there exists $\mu\in\mathbb{Q}_{\geq 0}$ such
that $M:=f^{*}M_{Y}-\mu D$ is non-zero and movable, and since
$f_{*}([M])=[M_{Y}]\in f_{*}\tau$, there exists $\lambda\in\mathbb{Q}$ such
that $[M+\lambda D]\in\tau$. Since $\tau$ is a face of
$\operatorname{Eff}(X)$, there exists a class
$\gamma\in\operatorname{Eff}(X)^{\vee}$ such that
$\tau=\operatorname{Eff}(X)\cap\gamma^{\perp}$. Then $M\cdot\gamma=(M+\lambda
D)\cdot\gamma-\lambda D\cdot\gamma=0$ and $[M]\in\operatorname{Eff}(X)$, so
that $[M]\in\tau$. ∎

###  4.2. Fixed prime divisors of Fano $4$-folds

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$. After [Cas13a, Cas17]
there are four possible types of fixed prime divisors in $X$. In this
subsection we recall this classification.

######  Theorem - Definition 4.5 ([Cas17], Th. 5.1, Def. 5.3, Cor. 5.26, Def.
5.27).

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, and $D$ a fixed prime
divisor in $X$.

  1. $(a)$

There exists a unique diagram:

$X\stackrel{{\scriptstyle\varphi}}{{\dasharrow}}\widetilde{X}\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}Y$

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

  2. $(b)$

$\sigma$ is of type $(3,0)^{sm}$, $(3,0)^{Q}$, $(3,1)^{sm}$, or $(3,2)$, and
we define $D$ to be of type $(3,0)^{sm}$, $(3,0)^{Q}$, $(3,1)^{sm}$, or
$(3,2)$, respectively.

  3. $(c)$

If $D$ is of type $(3,2)$, then $X=\widetilde{X}$.

  4. $(d)$

We define $C_{D}\subset D\subset X$ to be the transform of a general
irreducible curve $C_{\widetilde{D}}\subset\widetilde{X}$ contracted by
$\sigma$, of minimal anticanonical degree. Then $C_{D}\cong\mathbb{P}^{1}$,
$D\cdot C_{D}=-1$, and $C_{D}$ is contained in the open subset where $\varphi$
is an isomorphism.

  5. $(e)$

Given a SQM $X\dasharrow X^{\prime}$ and an elementary divisorial contraction
$k\colon X^{\prime}\to Y$ with $\operatorname{Exc}(k)$ the transform of $D$,
then $k$ has the same type as $\sigma$.

We will frequently use the notations $C_{D}\subset D$ and
$C_{\widetilde{D}}\subset\widetilde{D}$ introduced above.

######  Example 4.6.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$. If $X$ is a product
of surfaces, then every elementary birational contraction of $X$ has the form
$S\times T\to S^{\prime}\times T$, where $S\to S^{\prime}$ is the blow-up of a
point. In particular $X$ has no small contraction, and every fixed prime
divisor of $X$ is of type $(3,2)$.

Fano $4$-folds with a fixed prime divisor of type $(3,0)^{sm}$ have been
treated in [Cas17]:

######  Theorem 4.7 ([Cas17], Th. 5.40).

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, having a fixed prime
divisor of type $(3,0)^{sm}$. Then $\rho_{X}\leq 12$, and if $\rho_{X}=12$,
then $X$ has a rational contraction onto a $3$-fold.

Concerning fixed prime divisors of type $(3,2)$, we recall the following
results, that will be relevant in the sequel.

######  Theorem 4.8 ([Cas17], Prop. 5.32).

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $\delta_{X}\leq
1$, having a fixed prime divisor $D$ of type $(3,2)$ such that
$\mathcal{N}_{1}(D,{X})\subsetneq\mathcal{N}_{1}({X})$. Then $\rho_{X}\leq
12$, and if $\rho_{X}=12$, then $X$ has a rational contraction onto a
$3$-fold.

######  Remark 4.9 ([Cas17], Rem. 2.17(2)).

Let $X$ be a Fano $4$-fold with $\rho_{X}\geq 7$ and $D\subset X$ a fixed
prime divisor of type $(3,2)$. Then $D$ does not contain exceptional planes.

######  Lemma 4.10.

Let $X$ be a Fano $4$-fold with $\rho_{X}\geq 7$, $D\subset X$ a fixed prime
divisor of type $(3,2)$, $X\dasharrow\widetilde{X}$ a SQM, and
$\widetilde{D}\subset\widetilde{X}$ the transform of $X$. Then
$\dim\mathcal{N}_{1}(D,X)=\dim\mathcal{N}_{1}(\widetilde{D},\widetilde{X})$.

###### Proof.

This follows from Rem. 4.9 and [Cas13a, Cor. 3.14]. ∎

Let us note that Lemma 4.10 extends the applicability of Th. 4.8 also to SQM’s
of $X$. This is a special property of fixed prime divisors of type $(3,2)$, as
for a general prime divisor $D$ in a Fano $4$-fold $X$, if
$X\dasharrow\widetilde{X}$ is a SQM and $\widetilde{D}\subset\widetilde{X}$ is
the transform of $D$, then $\dim\mathcal{N}_{1}(\widetilde{D},\widetilde{X})$
may be smaller than $\dim\mathcal{N}_{1}(D,X)$ (see for instance §4.3).

######  Remark 4.11 (fixed prime divisors and small contractions).

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$. If $X$ has a fixed
prime divisor $D$ not of type $(3,2)$, then $X$ has a small elementary
contraction, indeed by [Cas17, Th. 5.1] the map $\varphi$ in Th.-Def. 4.5
factors as a sequence of at least $\rho_{X}-4$ $K$-negative flips.

Conversely, it is not difficult to show (see Rem. 5.7) that if $X$ has a small
elementary contraction $f\colon X\to Y$, then either $\rho_{X}\leq 11$, or $X$
has a fixed prime divisor $D$ such that $D\cdot\operatorname{NE}(f)<0$, and
$D$ cannot be of type $(3,2)$ by Th. 2.3 and Rem. 4.9. Moreover, if $D$ is of
type $(3,0)^{sm}$, we can apply Th. 4.7.

Therefore Th. 1.1 can be seen as a statement on Fano $4$-folds with
$\rho_{X}\geq 7$ having a fixed prime divisor of type $(3,1)^{sm}$ or
$(3,0)^{Q}$, and we will focus on these two types.

We will need some further properties of fixed prime divisors.

######  Lemma 4.12.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and
$D_{1},D_{2}\subset X$ two distinct fixed prime divisors.

  1. $(a)$

If $D_{1}\cdot C_{D_{2}}=0$, then $D_{1}$ and $D_{2}$ are adjacent.

  2. $(b)$

If $D_{1}\cdot C_{D_{2}}>0$ and $D_{2}\cdot C_{D_{1}}>0$, then $D_{1}+D_{2}$
is movable and $[C_{D_{1}}+C_{D_{2}}]\in\operatorname{mov}(X)$.

  3. $(c)$

$D_{1}\cdot C_{D_{2}}=D_{2}\cdot C_{D_{1}}=1$ if and only if
$\dim(\langle[D_{1}],[D_{2}]\rangle\cap\operatorname{Mov}(X))=1$; in this case
$\langle[D_{1}],[D_{2}]\rangle\cap\operatorname{Mov}(X)=\langle[D_{1}+D_{2}]\rangle$,
and $D_{1}+D_{2}$ is movable and non-big.

###### Proof.

This is [Cas20, Lemma 4.6], except $(b)$ which follows from the same proof. ∎

######  Lemma 4.13 ([Cas20], Lemma 4.9).

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

  1. $(a)$

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

  2. $(b)$

if $D$ and $E$ are not of type $(3,2)$, then $D\cdot C_{E}=E\cdot C_{D}=0$,
and $D\cap E$ is either empty or a disjoint union of exceptional planes.

######  Corollary 4.14.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and
$D_{1},D_{2}\subset X$ two fixed prime divisors, both of type $(3,2)$ or
neither. If $D_{1}\cdot C_{D_{2}}>0$, then $D_{2}\cdot C_{D_{1}}>0$, $D_{1}$
and $D_{2}$ are not adjacent, $D_{1}+D_{2}$ is movable, and
$[C_{D_{1}}+C_{D_{2}}]\in\operatorname{mov}(X)$.

###### Proof.

By Lemma 4.13 $D_{1}$ and $D_{2}$ are not adjacent, and the rest follows from
Lemma 4.12. ∎

######  Lemma 4.15.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $E_{1},E_{2}$ two
fixed prime divisors of type $(3,2)$ such that $E_{1}\cdot C_{E_{2}}=0$ and
$E_{1}\cap E_{2}\neq\emptyset$. Then $E_{2}\cdot C_{E_{1}}=0$ and every
connected component of $E_{1}\cap E_{2}$ is isomorphic to
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ with normal bundle
$\mathcal{O}(-1,0)\oplus\mathcal{O}(0,-1)$.

###### Proof.

By Cor. 4.14 we have $E_{2}\cdot C_{E_{1}}=0$ and $E_{1}$ and $E_{2}$ are
adjacent. Moreover $\langle[C_{E_{i}}]\rangle$ is the unique $E_{i}$-negative
extremal ray of $\operatorname{NE}(X)$, for $i=1,2$, by [Cas17, Rem. 2.17].
Then $-K_{X}+E_{1}+E_{2}$ is nef and
$(-K_{X}+E_{1}+E_{2})^{\perp}\cap\operatorname{NE}(X)=\langle[C_{E_{1}}],[C_{E_{2}}]\rangle$
is a face of $\operatorname{NE}(X)$. The associated contraction $f\colon X\to
Z$ is birational, has exceptional locus $E_{1}\cup E_{2}$, and the general
fiber of $f_{|E_{i}}$ is one-dimensional. The possible two-dimensional fibers
of $f$ are classified in [AW97, Th. 4.7] and are $\mathbb{P}^{2}$ or a
(possibly singular/reducible) quadric surface.

If $F$ is an irreducible component of $E_{1}\cap E_{2}$, then
$\mathcal{N}_{1}(F,X)=\mathbb{R}[C_{E_{1}}]\oplus\mathbb{R}[C_{E_{2}}]$ and
$f(F)=\\{pt\\}$. Thus $F$ cannot be isomorphic to $\mathbb{P}^{2}$ nor to a
quadric cone, and we conclude that $F\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$
and $F$ is a fiber of $f$. The normal bundle is given in [AW97], and $F$ must
be a connected component of $E_{1}\cap E_{2}$. ∎

###  4.3. Contraction of a fixed prime divisor of type $(3,1)^{sm}$ or
$(3,0)^{Q}$

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $D\subset X$ a
fixed prime divisor of type $(3,1)^{sm}$ or $(3,0)^{Q}$. Let us consider the
diagram:

$X\stackrel{{\scriptstyle\varphi}}{{\dasharrow}}\widetilde{X}\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}Y$

given by Th.-Def. 4.5$(a)$, where $\varphi$ is a SQM, $\sigma$ is an
elementary divisorial contraction with exceptional divisor the transform
$\widetilde{D}\subset\widetilde{X}$ of $D$, and $Y$ is Fano. We will refer to
this diagram as _the contraction associated to $D$_. Let us sum up here its
main properties and fix the related notation; see [Cas17, §5.1] for more
details.

Recall from Lemma 2.4 that the indeterminacy locus of $\varphi$ is a disjoint
union of exceptional planes.

For every exceptional plane $L\subset X$ in the indeterminacy locus of
$\varphi$ we have $L\subset D$, $D\cdot C_{L}<0$, and if
$\ell\subset\widetilde{X}$ is the corresponding exceptional curve, then
$\widetilde{D}\cdot\ell=-D\cdot C_{L}>0$. Conversely, every exceptional plane
contained in $D$ is in the indeterminacy locus of $\varphi$, so that the
exceptional planes in $D$ are pairwise disjoint. No exceptional line of
$\widetilde{X}$ is contained in $\widetilde{D}$ [Cas17, Rem. 5.6]. We have
$-K_{X}\cdot C_{D}=-K_{\widetilde{X}}\cdot C_{\widetilde{D}}=2$.

If $D$ is of type $(3,1)^{sm}$, then $Y$ is smooth, $\sigma$ is the blow-up of
a smooth irreducible curve $C\subset Y$, $\widetilde{D}$ is a
$\mathbb{P}^{2}$-bundle over $C$, and $C_{\widetilde{D}}\subset\widetilde{D}$
is a line in a fiber of $\sigma$. Every fiber of the $\mathbb{P}^{2}$-bundle
$\sigma_{|\widetilde{D}}$ meets the union of the exceptional lines of
$\widetilde{X}$ in at most one point [Cas17, Rem. 5.5].

If $D$ is of type $(3,0)^{Q}$, then $Y$ has an isolated terminal and locally
factorial singularity at $p:=\sigma(\widetilde{D})$, and $\widetilde{D}$ is
either a smooth quadric, or the cone over a smooth $2$-dimensional quadric
[Cas17, Lemma 2.19]; moreover $C_{\widetilde{D}}\subset\widetilde{D}$ is a
line.

######  Remark 4.16 ([Cas17], Lemmas 5.10 and 5.20).

Let $\Gamma\subset Y$ be an irreducible curve such that
$\Gamma\cap\sigma(\widetilde{D})\neq\emptyset$ and
$\Gamma\neq\sigma(\widetilde{D})$. Then $-K_{Y}\cdot\Gamma\neq 2$, and if
$-K_{Y}\cdot\Gamma=1$ we have $\Gamma=\sigma(\ell)$, where
$\ell\subset\widetilde{X}$ is an exceptional curve such that
$\widetilde{D}\cdot\ell=1$. If $D$ is $(3,1)^{sm}$, then $C$ cannot meet any
exceptional plane.

######  Remark 4.17.

In the above setting set

$m:=\dim\mathcal{N}_{1}(D,X)-\dim\mathcal{N}_{1}(\widetilde{D},\widetilde{X})=\begin{cases}\dim\mathcal{N}_{1}(D,X)-1\,\text{
if $D$ is of type $(3,0)^{Q}$}\\\ \dim\mathcal{N}_{1}(D,X)-2\,\text{ if $D$ is
of type $(3,1)^{sm}$.}\end{cases}$

It follows from [Cas13a, Rem. 3.13 and Cor. 3.14] that $D$ contains at least
$m$ exceptional planes $L_{1},\dotsc,L_{m}$ such that the classes
$[C_{L_{1}}],\dotsc,[C_{L_{m}}]$ are linearly independent in
$\mathcal{N}_{1}(X)$. Let us note that $X$ cannot be a product of surfaces,
thus $\delta_{X}\leq 2$ by Th. 2.6,
$\dim\mathcal{N}_{1}(D,X)\geq\rho_{X}-2\geq 5$, and $m\geq 3$.

###  4.4. Additional properties for the case $(3,1)^{sm}$

######  4.18.

In this subsection $X$ is a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and
$D\subset X$ is a fixed prime divisor of type $(3,1)^{sm}$. We keep the same
notation as in §4.3.

######  Lemma 4.19.

In Setting 4.18, let $\ell\subset\widetilde{X}$ an exceptional curve and
$\Gamma:=\sigma(\ell)\subset Y$. Then $\Gamma$ cannot meet any curve of
anticanonical degree one, except possibly $C$ (and $\Gamma$ itself).

###### Proof.

Let $C_{0}\subset Y$ be an irreducible curve with $-K_{Y}\cdot C_{0}=1$ and
$C_{0}\neq C$, $C_{0}\neq\Gamma$, and let
$\widetilde{C}_{0}\subset\widetilde{X}$ be its transform. Note that
$\widetilde{C}_{0}\neq\ell$, as $C_{0}\neq\Gamma$.

If $C\cap C_{0}\neq\emptyset$, then $\widetilde{C}_{0}$ is an exceptional
curve by Rem. 4.16, and $\widetilde{C}_{0}\neq\ell$, so that
$\widetilde{C}_{0}$ and $\ell$ are disjoint and meet $\widetilde{D}$ in
different fibers of $\sigma$. Therefore $\Gamma\cap C_{0}=\emptyset$.

If instead $C\cap C_{0}=\emptyset$, then
$-K_{\widetilde{X}}\cdot\widetilde{C}_{0}=1$ and
$\widetilde{D}\cap\widetilde{C}_{0}=\emptyset$, therefore
$\ell\cap\widetilde{C}_{0}=\emptyset$ by Lemma 2.4$(b)$, and their images stay
disjoint in $Y$. ∎

######  Lemma 4.20.

In Setting 4.18, let $\Gamma\subset Y$ be an integral curve such that
$-K_{Y}\cdot\Gamma=1$, $\Gamma\cap C\neq\emptyset$, and $\Gamma\neq C$. Then
$\Gamma\cong\mathbb{P}^{1}$ and
$\mathcal{N}_{\Gamma/Y}\cong\mathcal{O}(-1)\oplus\mathcal{O}^{\oplus 2}$.

###### Proof.

By Rem. 4.16 we have $\Gamma=\sigma(\ell)$ where $\ell\subset\widetilde{X}$ is
an exceptional curve with $\widetilde{D}\cdot\ell=1$; therefore
$\sigma_{|\ell}\colon\ell\to\Gamma$ is an isomorphism, and
$\Gamma\cong\mathbb{P}^{1}$.

We have $\mathcal{N}_{\ell/\widetilde{X}}\cong\mathcal{O}(-1)^{\oplus 3}$ and
$\mathcal{N}_{\Gamma/Y}\cong\mathcal{O}(a)\oplus\mathcal{O}(b)\oplus\mathcal{O}(-1-a-b)$
with $a,b\in\mathbb{Z}$. The differential of $\sigma$ induces a morphism of
sheaves
$\zeta\colon\mathcal{N}_{\ell/\widetilde{X}}\to\mathcal{N}_{\Gamma/Y}$, which
is generically an isomorphism. Then $\zeta$ is given by a $3\times 3$ matrix
$(s_{ij})_{i,j=1,2,3}$, where for $j=1,2,3$

$\displaystyle
s_{1j}\in\operatorname{Hom}(\mathcal{O}(-1),\mathcal{O}(a))=H^{0}(\mathbb{P}^{1},\mathcal{O}(a+1)),$
$\displaystyle s_{2j}\in H^{0}(\mathbb{P}^{1},\mathcal{O}(b+1)),\quad
s_{3j}\in H^{0}(\mathbb{P}^{1},\mathcal{O}(-a-b)),$

and the matrix is generically invertible. Thus for every $i=1,2,3$ there
exists $j\in\\{1,2,3\\}$ such that $s_{ij}\neq 0$, and this implies that
$a\geq-1$, $b\geq-1$, and $a+b\leq 0$, giving two possibilities for the normal
bundle of $\Gamma$: either $\mathcal{O}(-1)\oplus\mathcal{O}^{\oplus 2}$ or
$\mathcal{O}(-1)^{\oplus 2}\oplus\mathcal{O}(1)$.

Suppose that $\mathcal{N}_{\Gamma/Y}\cong\mathcal{O}(-1)^{\oplus
2}\oplus\mathcal{O}(1)$. Up to reordering we have $a=b=-1$ and hence $s_{1j}$
and $s_{2j}$ are constant for every $j$, namely the first and second column of
the matrix are constant. Since the matrix is generically invertible, these
columns must be linearly independent, so that the rank of $\zeta$ is at least
$2$ at every point of $\ell$. To rule out this case, we show that for
$x\in\widetilde{D}\cap\ell$, the rank of $\zeta$ at $x$ is $1$.

Set $y:=\sigma(x)\in C\cap\Gamma$. With a local computation one checks that
$d_{x}\sigma\colon T_{x}\widetilde{X}\to T_{y}Y$ has rank 2. On the other hand
$x\in\ell$ and $d_{x}\sigma(T_{x}\ell)=T_{y}\Gamma$, thus
$\operatorname{Im}d_{x}\sigma$ contains $T_{y}\Gamma$. Then the image of
$\operatorname{Im}d_{x}\sigma$ in
$(\mathcal{N}_{\Gamma/Y})_{y}=T_{y}Y/T_{y}\Gamma$ is one dimensional, and this
is the image of $\zeta$. ∎

######  Lemma 4.21.

In Setting 4.18:

  1. $(a)$

if $Y$ contains a nef prime divisor $H$ covered by a family $V$ of lines, such
that $H\cap C\neq\emptyset$, then $C$ is a curve of the family $V$;

  2. $(b)$

if $Y$ is covered by a family $W$ of rational curves with $-K_{Y}\cdot[W]=2$,
then $C$ is a component of a curve of the family $W$.

###### Proof.

$(a)\,$ Let $q\in C\cap H$, and let $\Gamma$ be a curve of the family $V$
containing $q$. If $\Gamma=C$, we are done. Otherwise, since
$-K_{Y}\cdot\Gamma=1$ and $\Gamma\cap C\neq\emptyset$, we have that
$\Gamma\cong\mathbb{P}^{1}$,
$\mathcal{N}_{\Gamma/Y}\cong\mathcal{O}(-1)\oplus\mathcal{O}^{\oplus 2}$, and
$\Gamma$ is the image of an exceptional curve, by Lemma 4.20 and Rem. 4.16.
Since $H$ is nef, by Lemma 3.13 $\Gamma$ must intersect some other curve of
the family $V$. On the other hand by Lemma 4.19 $\Gamma$ cannot intersect any
curve of anticanonical degree one, except $C$; this implies that $C$ belongs
to the family $V$.

$(b)\,$ Let $\Gamma_{2}$ be a one-cycle of the family $W$ intersecting $C$. If
$\Gamma_{2}$ is irreducible and reduced, then $\Gamma_{2}=C$ by Rem. 4.16.

If $\Gamma_{2}$ is reducible, let $\Gamma_{2}^{\prime}$,
$\Gamma_{2}^{\prime\prime}$ be its irreducible components, both of
anticanonical degree one. We can assume that $\Gamma_{2}^{\prime}\neq C$ and
$\Gamma_{2}^{\prime}\cap C\neq\emptyset$. Then $\Gamma_{2}^{\prime\prime}=C$
by Lemma 4.19.

Finally if $\Gamma_{2}$ is non-reduced, either it is supported on $C$, or
$\Gamma_{2}=2\Gamma_{2}^{\prime}$ where $-K_{Y}\cdot\Gamma_{2}^{\prime}=1$,
$\Gamma_{2}^{\prime}\neq C$, and $\Gamma_{2}^{\prime}\cap C\neq\emptyset$.
Then again by Rem. 4.16 $\Gamma_{2}^{\prime}$ is the image of an exceptional
curve of $\widetilde{X}$. In this last case, $\Gamma_{2}$ meets $C$ in the
finite set $T$ given by the intersection with the images of the finitely many
exceptional curves of $\widetilde{X}$. Thus if $\Gamma_{3}$ is a one-cycle of
the family $W$ meeting $C$ outside $T$, it must have $C$ as a component. ∎

######  Lemma 4.22.

In Setting 4.18, suppose moreover that $\rho_{X}\geq 8$. Let $E_{Y}\subset Y$
be a fixed prime divisor, and $E\subset X$ its transform, so that $E$ is a
fixed prime divisor adjacent to $D$ (see Lemma 4.4$(c)$). Then $E$ and $E_{Y}$
have the same type.

###### Proof.

Suppose first that $E$ is not of type $(3,2)$; then, by Lemma 4.13$(b)$,
$D\cap E$ is either empty of a disjoint union of exceptional planes, and for
every such $L$, we have $D\cdot C_{L}<0$ and $E\cdot C_{L}<0$ (see §4.3).

There is a SQM $\psi\colon X\dasharrow\widehat{X}$ whose indeterminacy locus
is the union of all exceptional planes contained in $D\cup E$ (this can be
obtained by flipping consecutively all small $K$-negative extremal rays of
$\operatorname{NE}(X)$ having negative intersection with $D+E$). The
transforms $\widehat{D},\widehat{E}\subset\widehat{X}$ are disjoint, and the
SQM $\psi\circ\varphi^{-1}\colon\widetilde{X}\dasharrow\widehat{X}$ induces an
isomorphism between $\widetilde{D}$ and $\widehat{D}$. Similarly, if
$X\stackrel{{\scriptstyle\varphi_{E}}}{{\dasharrow}}\widetilde{X}_{E}\to
Y_{E}$ is the contraction associated to $E$, the SQM
$\psi\circ\varphi_{E}^{-1}\colon\widetilde{X}_{E}\dasharrow\widehat{X}$
induces an isomorphism between $\widetilde{E}\subset\widetilde{X}_{E}$ and
$\widehat{E}$.

It is not difficult to see that $\widehat{D},\widehat{E}$ are the loci of two
divisorial extremal rays $R_{D},R_{E}$ of $\operatorname{NE}(\widehat{X})$,
such that $R_{D}+R_{E}$ is a face of $\operatorname{NE}(\widehat{X})$. In
fact, if $m:=-K_{X}\cdot C_{E}$ ($m\in\\{2,3\\}$ depending on the type of
$E$), the divisor $-K_{\widehat{X}}+2\widehat{D}+m\widehat{E}$ is nef and
$(-K_{\widehat{X}}+2\widehat{D}+m\widehat{E})^{\perp}\cap\operatorname{NE}(\widehat{X})=R_{D}+R_{E}$.
Thus we have a diagram:

$\textstyle{{\widehat{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{D}}$$\scriptstyle{\sigma_{E}}$$\textstyle{{\widehat{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\textstyle{{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W}$

where $\sigma_{D}$ and $\sigma_{E}$ are elementary divisorial contractions
with exceptional divisors $\widehat{D}$ and $\widehat{E}$ respectively,
$\widehat{Y}$ is a SQM of $Y$, and $k$ is an elementary divisorial contraction
with exceptional divisor the transform of $E_{Y}$. Since
$\widehat{D}\cap\widehat{E}=\emptyset$, $\sigma_{E}$ and $k$ coincide in a
neighborhood of their exceptional divisors. Moreover, by Th.-Def. 4.5$(e)$,
$\sigma_{E}$ and $E$ have the same type, and similarly for $k$ and $E_{Y}$.
Therefore $E$ and $E_{Y}$ have the same type.

Suppose now that $E$ is of type $(3,2)$; then by Lemma 4.13$(a)$ we have
$D\cdot C_{E}=0$, so that the general curve $C_{E}$ is disjoint from $D$,
hence it is contained in the open subset where the birational map $X\dasharrow
Y$ is an isomorphism. Let $\Gamma\subset E_{Y}\subset Y$ be the image of the
general $C_{E}$. Then $-K_{Y}\cdot\Gamma=1$ and $E_{Y}\cdot\Gamma=-1$, so that
$E_{Y}$ is covered by a family of lines and it is not nef; then $E_{Y}$ is of
type $(3,2)$ by Lemma 3.12. ∎

######  Lemma 4.23.

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

###### Proof.

Let $X\dasharrow\widetilde{X}\to Y$ be the contraction of $D$ as in §4.3, and
$E_{\widetilde{X}}\subset\widetilde{X}$, $E_{Y}\subset Y$ the transforms of
$E$, so that $E_{Y}$ is a fixed prime divisor by Lemma 4.4$(c)$. Since
$E_{\widetilde{X}}\cdot C_{\widetilde{D}}=E\cdot C_{D}>0$, $E_{\widetilde{X}}$
intersects every nontrivial fiber of $\sigma$, thus $E_{Y}$ contains $C$. By
[Cas17, Lemma 5.11] we deduce that $[C]$ generates an extremal ray of type
$(3,2)$ of $\operatorname{NE}(Y)$, $C\cong\mathbb{P}^{1}$ is a fiber of the
associate contraction, $E_{Y}$ is a smooth $\mathbb{P}^{1}$-bundle around $C$,
$\mathcal{N}_{C/Y}\cong\mathcal{O}^{\oplus 2}\oplus\mathcal{O}(-1)$, and
$\widetilde{D}\cong\mathbb{P}_{\mathbb{P}^{1}}(\mathcal{O}^{\oplus
2}\oplus\mathcal{O}(1))$ is isomorphic to the blow-up of $\mathbb{P}^{3}$
along a line. Moreover $E_{\widetilde{X}}$ and $\widetilde{D}$ intersect
transversally along a smooth irreducible surface and
$\sigma^{*}E_{Y}=E_{\widetilde{X}}+\widetilde{D}$, so that $E\cdot
C_{D}=E_{\widetilde{X}}\cdot
C_{\widetilde{D}}=(\sigma^{*}E_{Y}-\widetilde{D})\cdot C_{\widetilde{D}}=1$.

If $\Gamma\subset\widetilde{D}$ is a non-trivial fiber of the blow-up
$\widetilde{D}\to\mathbb{P}^{3}$, it is not difficult to see that
$\widetilde{D}\cdot\Gamma=0$, using that
$\mathcal{O}_{\widetilde{X}}(-\widetilde{D})_{|\widetilde{D}}$ is the
tautological line bundle. Then
$-K_{\widetilde{X}}\cdot\Gamma=-K_{\widetilde{D}}\cdot\Gamma=1$, and
$E_{\widetilde{X}}\cdot\Gamma=\sigma^{*}E_{Y}\cdot\Gamma=E_{Y}\cdot C=-1$, so
that $\Gamma\subset E_{\widetilde{X}}$. This implies that $\widetilde{D}\cap
E_{\widetilde{X}}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ is the exceptional
divisor of the blow-up $\widetilde{D}\to\mathbb{P}^{3}$, and it is covered by
lines. On the other hand also
$E_{\widetilde{X}}\smallsetminus\widetilde{D}\cong E_{Y}\smallsetminus C$ is
covered by lines. Thus $E_{\widetilde{X}}$ cannot meet any exceptional curve
of $\widetilde{X}$ by Lemma 2.4$(b)$, and $E$ cannot meet any exceptional
plane contained in $D$. ∎

###  4.5. Additional properties for the case $(3,0)^{Q}$

######  4.24.

In this subsection $X$ is a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and
$D\subset X$ is a fixed prime divisor of type $(3,0)^{Q}$. We keep the same
notation as in §4.3.

######  Remark 4.25.

In Setting 4.24, suppose that $\widetilde{D}\subset\widetilde{X}$ is a quadric
cone. Then no exceptional curve can intersect $\widetilde{D}$ at the vertex of
the cone.

Indeed, suppose otherwise: then for every line $\Gamma\subset\widetilde{D}$
through the vertex of the cone, we have $-K_{\widetilde{X}}\cdot\Gamma=2$, so
if $\Gamma_{X}\subset X$ is the transform of $\Gamma$, we have
$-K_{X}\cdot\Gamma_{X}=1$ by Lemma 2.4$(b)$. This yields that $D$ is covered
by a family of lines, it is not nef and of type $(3,0)^{Q}$, contradicting
Lemma 3.12.

######  Remark 4.26.

Let $B\subset X$ be a prime divisor, different from $D$, and such that $B\cap
D\neq\emptyset$. Then either $B\cap D$ is a disjoint union of exceptional
planes, or $B\cdot C_{D}>0$.

Indeed let $\widetilde{B}\subset\widetilde{X}$ be the transform of $B$. If
$\widetilde{B}\cap\widetilde{D}=\emptyset$, then $B\cap D$ is contained in the
indeterminacy locus of $\varphi\colon X\dasharrow\widetilde{X}$, thus it is a
disjoint union of exceptional planes. If
$\widetilde{B}\cap\widetilde{D}\neq\emptyset$, then $B\cdot
C_{D}=\widetilde{B}\cdot C_{\widetilde{D}}>0$.

The proof of the next lemma is analogous to that of Lemma 4.19, thus we omit
it.

######  Lemma 4.27.

In Setting 4.24, let $\ell\subset\widetilde{X}$ be an exceptional curve and
$\Gamma:=\sigma(\ell)\subset Y$. Then $\Gamma$ cannot meet any curve of
anticanonical degree one outside $p$ (except possibly $\Gamma$ itself).

######  Lemma 4.28.

In Setting 4.24, let $E\subset X$ be a fixed prime divisor adjacent to $D$,
and $E_{Y}\subset Y$ its transform. If
$\mathcal{N}_{1}(E_{Y},Y)\subsetneq\mathcal{N}_{1}(Y)$, then one of the
following holds:

  1. $(i)$

$X$ has a rational contraction onto a $3$-fold;

  2. $(ii)$

$X$ has a fixed prime divisor $G$ of type $(3,2)$, adjacent to $D$, and such
that $E\cdot C_{G}>0$.

###### Proof.

We recall that $Y$ is Fano and has one isolated terminal and locally factorial
singularity. Suppose that
$\mathcal{N}_{1}(E_{Y},Y)\subsetneq\mathcal{N}_{1}(Y)$; we apply [Del14, Th.
3.1 and Lemma 3.3] to $Y$ and $E_{Y}$. This yields a diagram:

$Y=Y_{0}\stackrel{{\scriptstyle\sigma_{0}}}{{\dasharrow}}Y_{1}\dasharrow\cdots\dasharrow
Y_{k-1}\stackrel{{\scriptstyle\sigma_{k-1}}}{{\dasharrow}}Y_{k}\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}Z$

where each $\sigma_{i}$ is either an elementary divisorial contraction or a
flip, and $\psi$ is an elementary contraction of fiber type [Del14, Th.
3.1(2)]. The divisor $E_{Y}$ is not exceptional for any $\sigma_{i}$; let
$E_{Y}^{i}\subset Y_{i}$ be its transform and set
$c_{i}:=\operatorname{codim}\mathcal{N}_{1}(E_{Y}^{i},Y_{i})$. We have
$c_{i+1}\in\\{c_{i},c_{i}-1\\}$ for $i=0,\dotsc,k-1$ and $c_{k}\in\\{0,1\\}$
[Del14, Th. 3.1(3)].

If $c_{k}=1$, then
$\operatorname{NE}(\psi)\not\subset\mathcal{N}_{1}(E_{Y}^{k},Y_{k})$ and
$E_{Y}^{k}\cdot\operatorname{NE}(\psi)>0$ [Del14, Th. 3.1(2) and (3)]. This
implies that if $F\subset Y_{k}$ is a fiber of $\psi$, then $\dim(F\cap
E_{Y}^{k})\leq 0$ and $F\cap E_{Y}^{k}\neq\emptyset$, thus $\dim F=1$ and
$\dim Z=3$, and we get $(i)$.

If $c_{k}=0$, since $c_{0}=\operatorname{codim}\mathcal{N}_{1}(E_{Y},Y)>0$,
there exists some $i\in\\{0,\dotsc,k-1\\}$ such that $c_{i+1}=c_{i}-1$. Then
$\sigma_{i}$ is a $K$-negative elementary divisorial contraction of type
$(3,2)$ [Del14, Lemma 3.3(2)], and $\operatorname{Exc}(\sigma_{i})$ is
contained in the open subset where the birational map $Y\dasharrow Y_{i}$ is
an isomorphism [Del14, Lemma 3.3(3)]; let $G_{Y}\subset Y$ and $G\subset X$ be
the transforms of $\operatorname{Exc}(\sigma_{i})\subset Y_{i}$.

Then $G_{Y}$ is covered by irreducible curves $\Gamma$ such that
$E_{Y}\cdot\Gamma>0$, $-K_{Y}\cdot\Gamma=1$, and $G_{Y}\cdot\Gamma=-1$ [Del14,
Lemma 3.3(5)]; the general $\Gamma$ does not contain
$p=\sigma(\widetilde{D})$, so by Lemma 4.27 it is contained in the open subset
where the birational map $X\dasharrow Y$ is an isomorphism. Therefore
$G\subset X$ is covered by a family of lines $V$ such that $G\cdot[V]=-1$ and
$E\cdot[V]>0$; $G$ is a fixed prime divisor of type $(3,2)$ with $[V]=[C_{G}]$
by Lemma 3.12, and it is adjacent to $D$ by Lemma 4.4$(c)$, so we have $(ii)$.
∎

##  5\. Rational contractions of fiber type

###  5.1. Rational contractions of fiber type in Mori dream spaces

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space.
We need to introduce two special notions of rational contractions of fiber
type, namely “quasi-elementary” and “special” contractions; we refer the
reader to [Cas13a, §2.2] and [Cas20, §2] respectively for more details.

######  Definition 5.1.

Let $f\colon X\to Z$ be a contraction of fiber type. We say that $f$ is quasi-
elementary if $Z$ is $\mathbb{Q}$-factorial and for every prime divisor
$B\subset Z$ the pull-back $f^{*}B$ is irreducible (but possibly non-reduced).

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

We will be interested mainly in the cases where $\dim Z\leq 2$: if $Z$ is a
curve, then every contraction of fiber type is special. If $Z$ is a surface,
then $f$ is special if and only if it is equidimensional: indeed if $f$ is
special and $\dim Z=2$, then $f$ must be equidimensional; the converse follows
from [Cas20, Lemma 2.7].

Consider now a rational contraction of fiber type $f\colon X\dasharrow Z$. We
say that $f$ is quasi-elementary, respectively special, if given a
factorization of $f$ as a SQM $X\dasharrow X^{\prime}$ followed by a regular
contraction $f^{\prime}\colon X^{\prime}\to Z$, then $f^{\prime}$ is quasi-
elementary, respectively special; this does not depend on the choice of the
factorization.

######  Lemma 5.2.

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space,
and $f\colon X\dasharrow S$ a rational contraction onto a surface. Then $S$ is
$\mathbb{Q}$-factorial.

###### Proof.

Suppose first that $X$ is a surface, so that $f$ is birational and regular. By
dimensional reasons, $\operatorname{Exc}(f)$ is a divisor. If $f$ is
elementary, the statement is well known. In general, we can factor $f$ as
$X\stackrel{{\scriptstyle\sigma}}{{\to}}X^{\prime}\stackrel{{\scriptstyle
f^{\prime}}}{{\to}}S$, where $\sigma$ is elementary; then $X^{\prime}$ is a
$\mathbb{Q}$-factorial Mori dream space, and we conclude by induction on
$\rho_{X}-\rho_{S}$.

If $\dim X\geq 3$, then $f$ is of fiber type and by [Cas20, Prop. 2.13] it can
be factored as

$X\stackrel{{\scriptstyle g}}{{\dasharrow}}T\stackrel{{\scriptstyle
h}}{{\longrightarrow}}S,$

where $g$ is a special rational contraction of fiber type and $h$ is
birational. Then $T$ is $\mathbb{Q}$-factorial, so that $S$ is
$\mathbb{Q}$-factorial by the first part of the proof. ∎

######  Lemma 5.3.

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space.
Let $f\colon X\dasharrow Z$ be a rational contraction of fiber type, and
$\tau$ the smallest face of $\operatorname{Eff}(X)$ containing
$f^{*}\operatorname{Eff}(Z)$. The following are equivalent:

  1. $(i)$

$f$ is special;

  2. $(ii)$

$\tau\cap\operatorname{Mov}(X)=f^{*}\operatorname{Mov}(Z)$;

  3. $(iii)$

$\dim(\tau\cap\operatorname{Mov}(X))=\rho_{Z}$.

###### Proof.

We show $(i)\Rightarrow(ii)$. Up to composing with a SQM of $X$, we can assume
that $f$ is regular. Let $F\subset X$ be a general fiber of $f$; by [Cas13a,
Lemma 2.21] we have
$\tau=\operatorname{Eff}(X)\cap\mathcal{N}_{1}(F,X)^{\perp}$, thus
$\tau\cap\operatorname{Mov}(X)=\operatorname{Mov}(X)\cap\mathcal{N}_{1}(F,X)^{\perp}$.

If $B$ is a movable divisor on $Z$, then $f^{-1}(B)\cap F=\emptyset$, thus
$f^{*}B\cdot C=0$ for every curve $C\subset F$, and
$[f^{*}B]\in\mathcal{N}_{1}(F,X)^{\perp}$. Moreover if $B_{0}\subset Z$ is the
stable base locus of the linear system $|B|$, then
$\operatorname{codim}B_{0}\geq 2$, and since $f$ is special we have
$\operatorname{codim}f^{-1}(B_{0})\geq 2$. Since the stable base locus of
$|f^{*}B|$ is contained in $f^{-1}(B_{0})$, we have
$[f^{*}B]\in\operatorname{Mov}(X)\cap\mathcal{N}_{1}(F,X)^{\perp}$. Hence
$f^{*}\operatorname{Mov}(Z)\subseteq\operatorname{Mov}(X)\cap\mathcal{N}_{1}(F,X)^{\perp}$.

Conversely, let $D$ be a divisor on $X$ with
$[D]\in\operatorname{Mov}(X)\cap\mathcal{N}_{1}(F,X)^{\perp}$. By Rem. 2.2, up
to replacing $D$ with a divisor with class in $\mathbb{R}_{\geq 0}[D]$, we can
assume that $D$ is a prime divisor. Since $D\cdot C=0$ for every curve
$C\subset F$, we must have $D\cap F=\emptyset$ and $f(D)\subsetneq Z$; since
$f$ is special, $f(D)$ is a prime divisor, and it is movable. If
$f^{-1}(f(D))$ is reducible, then by [Cas20, Cor. 2.18] every irreducible
component of $f^{-1}(f(D))$ is a fixed prime divisor; since $D$ is movable and
is a component of $f^{-1}(f(D))$, we conclude that $f^{-1}(f(D))=D$ and
$D=\lambda f^{*}(f(D))$ for some $\lambda\in\mathbb{Q}_{>0}$, so that $[D]\in
f^{*}\operatorname{Mov}(Z)$, and we get $(ii)$.

The implication $(ii)\Rightarrow(iii)$ is clear. We show
$(iii)\Rightarrow(i)$. By [Cas20, Prop. 2.13] $f$ can be factored as
$X\stackrel{{\scriptstyle g}}{{\dasharrow}}T\stackrel{{\scriptstyle
h}}{{\to}}Z$, where $g$ is a special rational contraction of fiber type and
$h$ is birational. Up to composing with a SQM of $X$, we can assume that $g$
is regular; then $g$ and $f$ have the same general fiber $F\subset X$, and
again $\tau=\operatorname{Eff}(X)\cap\mathcal{N}_{1}(F,X)^{\perp}$. By the
first part of the proof we have
$\operatorname{Mov}(X)\cap\mathcal{N}_{1}(F,X)^{\perp}=g^{*}\operatorname{Mov}(T)$,
thus

$\rho_{Z}=\dim\bigl{(}\tau\cap\operatorname{Mov}(X)\bigr{)}=\dim\bigl{(}\operatorname{Mov}(X)\cap\mathcal{N}_{1}(F,X)^{\perp}\bigr{)}=\rho_{T},$

therefore $h$ is an isomorphism and $f$ is special. ∎

###  5.2. Movable faces of the effective cone

Let $X$ be a projective, normal, and $\mathbb{Q}$-factorial Mori dream space.

######  Definition 5.4.

Let $\tau$ be a proper face of $\operatorname{Eff}(X)$. We say that $\tau$ is
a movable face if the relative interior of $\tau$ intersects
$\operatorname{Mov}(X)$.

There exists a movable face of $\operatorname{Eff}(X)$ if and only if
$\partial\operatorname{Eff}(X)\cap\operatorname{Mov}(X)\neq\\{0\\}$, if and
only if $X$ has some non-zero, non-big movable divisor.

If a (non-zero) face of $\operatorname{Eff}(X)$ is not fixed, it always
contains a movable face.

We say that a movable face $\tau$ is minimal if every proper face of $\tau$ is
not movable, equivalently if every proper face of $\tau$ is fixed.

We recall that the Mori chamber decomposition of $X$ is a fan in
$\mathcal{N}^{1}(X)$, supported on the cone $\operatorname{Mov}(X)$, given by
the cones $f^{*}\operatorname{Nef}(Z)$ for all rational contractions $f\colon
X\dasharrow Z$; the cones of the Mori chamber decomposition are in bijection
with the rational contractions of $X$.

######  5.5.

Let $\tau$ be a movable face of $\operatorname{Eff}(X)$. We associate to
$\tau$ a special rational contraction of fiber type of $X$, as follows.

The cone $\tau\cap\operatorname{Mov}(X)$ is a non-zero face of the movable
cone, so it is a union of cones of the Mori chamber decomposition of $X$. Let
us choose a cone $\eta$ of the Mori chamber decomposition such that
$\eta\subseteq\tau\cap\operatorname{Mov}(X)$ and
$\dim\eta=\dim(\tau\cap\operatorname{Mov}(X))$; note that $\eta$ must
intersect the relative interior of $\tau$, because $\tau$ is movable.

Then $\eta$ yields a rational contraction $f\colon X\dasharrow Z$, of fiber
type because $\eta$ is contained in the boundary of the effective cone. We
have:

$\eta=f^{*}\operatorname{Nef}(Z),\quad\rho_{Z}=\dim\eta=\dim\bigl{(}\tau\cap\operatorname{Mov}(X)\bigr{)},\quad\text{and
}\ \tau\cap\operatorname{Mov}(X)\subset f^{*}\mathcal{N}^{1}(Z).$

We show that $f^{*}\operatorname{Eff}(Z)\subseteq\tau$. Indeed since $\tau$ is
a face of $\operatorname{Eff}(X)$, there exists
$\gamma\in\operatorname{Eff}(X)^{\vee}$ such that
$\tau=\operatorname{Eff}(X)\cap\gamma^{\perp}$. Then
$f^{*}\operatorname{Nef}(Z)=\eta\subseteq\tau\subseteq\gamma^{\perp}$, thus
$f^{*}\mathcal{N}^{1}(Z)\subseteq\gamma^{\perp}$ and
$f^{*}\operatorname{Eff}(Z)\subseteq\gamma^{\perp}\cap\operatorname{Eff}(X)=\tau$.

On the other hand $\eta\subseteq f^{*}\operatorname{Eff}(Z)$, therefore
$f^{*}\operatorname{Eff}(Z)$ intersects the relative interior of $\tau$, and
$\tau$ is the minimal face of $\operatorname{Eff}(X)$ containing
$f^{*}\operatorname{Eff}(Z)$. In particular $f$ is special by Lemma 5.3.

Let $X\dasharrow\widetilde{X}\stackrel{{\scriptstyle\tilde{f}}}{{\to}}Z$ be a
factorization of $f$ as a SQM followed by a regular contraction, and
$F\subset\widetilde{X}$ a general fiber of $\tilde{f}$. By [Cas13a, Lemma
2.21] we deduce that, under the natural identification
$\operatorname{Eff}(X)\cong\operatorname{Eff}(\widetilde{X})$, we have:

(5.6)
$\tau=\operatorname{Eff}(\widetilde{X})\cap\mathcal{N}_{1}(F,\widetilde{X})^{\perp}\quad\text{and}\quad\rho_{X}=\dim\tau+\dim\mathcal{N}_{1}(F,\widetilde{X})\leq\dim\tau+\rho_{F}.$

Thus given a prime divisor $D\subset X$, we have $[D]\in\tau$ if and only if
the transform $\widetilde{D}\subset\widetilde{X}$ does not dominate $Z$ under
$\tilde{f}$. Indeed
$[\widetilde{D}]\in\mathcal{N}_{1}(F,\widetilde{X})^{\perp}$ is equivalent to
$\widetilde{D}\cap F=\emptyset$.

######  Remark 5.7.

Let $X$ be a smooth Fano $4$-fold with a small elementary contraction $g\colon
X\to Y$. Then
$\operatorname{NE}(g)\not\subset\operatorname{mov}(X)=\operatorname{Eff}(X)^{\vee}$,
hence there exists a one-dimensional face $\tau$ of $\operatorname{Eff}(X)$
such that $\tau\cdot\operatorname{NE}(g)<0$.

If $\tau$ is movable, we apply the construction above; by Lemma 2.5 we can
take $\widetilde{X}$ smooth and $\tilde{f}$ $K$-negative. Then $F$ is smooth,
Fano, with $\dim F\leq 3$, therefore $\rho_{F}\leq 10$ (see [IP99, §12.6]) and
$\rho_{X}\leq 11$ by (5.6).

If instead $\tau$ is fixed, then there exists a fixed prime divisor $D\subset
X$ such that $D\cdot\operatorname{NE}(g)<0$.

###  5.3. Rational contractions of fiber type of Fano $4$-folds

Fano $4$-folds with a rational contraction of fiber type have been studied in
[Cas20]; in particular we will need the following.

######  Theorem 5.8 ([Cas20], Th. 6.1).

Let $X$ be a smooth Fano $4$-fold which is not a product of surfaces. If $X$
has a rational contraction onto a $3$-fold, then $\rho_{X}\leq 12$.

In the rest of this subsection we focus on Fano $4$-folds having a rational
contraction onto a surface or $\mathbb{P}^{1}$; in this case we do not have an
analog of Th. 5.8, and we will prove several results which we will use in the
rest of the article to tackle this situation.

######  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.

###### Proof.

By [Cas13a, Prop. 4.1 and its proof] we know that $S$ is smooth, rational, and
a Mori dream space. We show that $-K_{S}\cdot\operatorname{NE}(g)>0$ for every
elementary contraction $g\colon S\to S_{1}$; this implies that $-K_{S}$ is
ample. If $g$ is of fiber type, then either $S\cong\mathbb{P}^{2}$, or $S$ is
isomorphic to a Hirzebruch surface and $g$ is a $\mathbb{P}^{1}$-bundle, hence
$-K_{S}\cdot\operatorname{NE}(g)>0$.

Let assume that $g$ is birational, so that $C:=\operatorname{Exc}(g)$ is an
irreducible curve, and $D:=f^{*}C$ is a fixed prime divisor, see [Cas13a, Rem.
3.10]. Moreover $S_{1}$ has (at most) a rational singularity at the point
$g(C)$ (see Lemma 2.5). Consider a factorization of $f$ as in Lemma 2.5:

$X\stackrel{{\scriptstyle\psi}}{{\dasharrow}}X^{\prime}\stackrel{{\scriptstyle
f^{\prime}}}{{\longrightarrow}}S$

where $\psi$ is a SQM. We know that $X\smallsetminus\operatorname{dom}\psi$ is
a finite union of exceptional planes (see Lemma 2.4). Then in $X$ there exists
a curve $\Gamma\equiv C_{D}$, $\Gamma\subset D\cap\operatorname{dom}\psi$;
this follows from Rem. 4.9 if $D$ is of type $(3,2)$, and from the description
in Th.-Def. 4.5 in the other cases.

Let $\Gamma^{\prime}\subset D^{\prime}\subset X^{\prime}$ be the transforms of
$\Gamma\subset D\subset X$. Then

$-1=D\cdot\Gamma=D^{\prime}\cdot\Gamma^{\prime}=(f^{\prime})^{*}C\cdot\Gamma^{\prime}=C\cdot(f^{\prime})_{*}\Gamma^{\prime}.$

This implies that $f^{\prime}(\Gamma^{\prime})$ is not a point, thus
$f^{\prime}(\Gamma^{\prime})=C$ and $(f^{\prime})_{*}\Gamma^{\prime}=mC$ with
$m\in\mathbb{Z}_{\geq 1}$. We get $mC^{2}=-1$, hence $m=1$ and $C^{2}=-1$. We
conclude that $g$ is the blow-up of a smooth point and
$-K_{S}\cdot\operatorname{NE}(g)>0$. ∎

######  Lemma 5.10 ([Cas13a], proof of Cor. 3.9).

Let $X$ be a smooth Fano $4$-fold, $\varphi\colon X\dasharrow\widetilde{X}$ a
SQM, and $f\colon\widetilde{X}\to Z$ a $K$-negative contraction of fiber type.
If the general fiber $F$ of $f$ is either a del Pezzo surface with
$\rho_{F}=9$, or a product $S\times\mathbb{P}^{1}$ with $S$ a del Pezzo
surface with $\rho_{S}=9$, then $\varphi$ is an isomorphism.

######  Lemma 5.11.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, $f\colon X\dasharrow
Z$ a rational contraction of fiber type, and $H$ an effective divisor in $X$
such that $[H]\in f^{*}\mathcal{N}^{1}(Z)$ and $\operatorname{Supp}H$ has a
component $E$ which is a fixed prime divisor of type $(3,2)$. Then one of the
following holds:

  1. $(i)$

$\mathcal{N}_{1}(E,X)\subsetneq\mathcal{N}_{1}(X)$;

  2. $(ii)$

$\dim Z=3$;

  3. $(iii)$

$Z\cong\mathbb{P}^{2}$ and $f$ is equidimensional.

###### Proof.

Let $\widetilde{X}\dasharrow X$ be a SQM such that the composition
$\tilde{f}\colon\widetilde{X}\to Z$ is regular and $K$-negative (see Lemma
2.5), and $\widetilde{E}\subset\widetilde{X}$ the transform of $E$.

We have
$\mathcal{N}_{1}(\tilde{f}(\widetilde{E}),Z)=\tilde{f}_{*}(\mathcal{N}_{1}(\widetilde{E},\widetilde{X}))$,
thus
$\operatorname{codim}\mathcal{N}_{1}(\widetilde{E},\widetilde{X})\geq\operatorname{codim}\mathcal{N}_{1}(\tilde{f}(\widetilde{E}),Z)$.
Since $H$ is the pullback of an effective $\mathbb{Q}$-divisor of $Z$,
$\tilde{f}(\widetilde{E})\subseteq\tilde{f}(\operatorname{Supp}H)\subsetneq
Z$. If $\dim Z=1$, or $\dim Z=2$ and $\rho_{Z}>1$, this easily implies that
$\operatorname{codim}\mathcal{N}_{1}(\tilde{f}(\widetilde{E}),Z)>0$, therefore
$\operatorname{codim}\mathcal{N}_{1}(\widetilde{E},\widetilde{X})>0$ and we
get $(i)$ by Lemma 4.10.

If $\dim Z=2$ and $f$ is not equidimensional, by [Cas20, Prop. 2.13] we can
factor $f$ as $X\stackrel{{\scriptstyle g}}{{\dasharrow}}T\to Z$ where $T$ is
a surface with $\rho_{T}>\rho_{Z}$, and again $E$ cannot dominate $T$, so we
get $(i)$ as before.

Finally if $\dim Z=2$, $\rho_{Z}=1$, and $f$ is equidimensional, then $Z$ is
smooth by [Cas20, Lemma 4.3], and being a rational surface, we have $(iii)$. ∎

######  Lemma 5.12.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $f\colon
X\dasharrow S$ an equidimensional rational contraction onto a surface. If $f$
is not quasi-elementary and $\rho_{S}>1$, then $X$ has a fixed prime divisor
$E$ of type $(3,2)$ such that
$\mathcal{N}_{1}(E,X)\subsetneq\mathcal{N}_{1}(X)$.

###### Proof.

Since $f$ is not quasi-elementary, there exists an irreducible curve
$\Gamma\subset S$ such that $f^{*}\Gamma$ is reducible. By [Cas20, Lemma 5.2]
there is a fixed prime divisor $E\subset X$ of type $(3,2)$ which is a
component of $f^{*}\Gamma$. Then the statement follows from Lemma 5.11. ∎

######  Lemma 5.13.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $f\colon
X\dasharrow S$ a rational contraction onto a surface with $\rho_{S}=1$.
Suppose that there is a unique prime divisor $D\subset X$ contracted to a
point by $f$, and that $D$ is fixed not of type $(3,2)$. Then one of the
following holds:

  1. $(i)$

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

  2. $(ii)$

$\rho_{X}\leq 10$.

###### Proof.

By [Cas20, Prop. 2.13] we can factor $f$ as $X\stackrel{{\scriptstyle
g}}{{\dasharrow}}T\stackrel{{\scriptstyle h}}{{\to}}S$ where $g$ is
equidimensional and $h$ is birational. The surfaces $T$ and $S$ are
$\mathbb{Q}$-factorial by Lemma 5.2, so that $\rho_{T}-\rho_{S}$ is the number
of irreducible components of $\operatorname{Exc}(h)$.

Let $X^{\prime}\dasharrow X$ be a SQM such that the composition
$g^{\prime}\colon X^{\prime}\to T$ is regular and $K$-negative (see Lemma
2.5), $F\subset X^{\prime}$ a general fiber of $g^{\prime}$, and
$D^{\prime}\subset X^{\prime}$ the transform of $D$. Since $D^{\prime}$ is the
unique prime divisor contracted to a point by $h\circ g^{\prime}\colon
X^{\prime}\to S$, we deduce that
$D^{\prime}=(g^{\prime})^{-1}(\operatorname{Exc}(h))$, $\operatorname{Exc}(h)$
is irreducible, and $\rho_{T}=2$.

Let $R$ be an extremal ray of $\operatorname{NE}(X^{\prime})$ such that
$(g^{\prime})_{*}R=\operatorname{NE}(h)$ (see [Cas08, §2.5]). Then for every
curve $C\subset X^{\prime}$ with $[C]\in R$ we have
$g^{\prime}(C)=\operatorname{Exc}(h)$, therefore
$\operatorname{Locus}(R)\subseteq D^{\prime}$. Moreover if $F\subset
X^{\prime}$ is a non-trivial fiber of the contraction of $R$, $g^{\prime}$ is
finite on $F$, and $g^{\prime}(F)=\operatorname{Exc}(h)$, so that $\dim F=1$.
If $-K_{X^{\prime}}\cdot R>0$, then $R$ is of type $(3,2)$ by [Wiś91, Th.
1.2], and $D^{\prime}=\operatorname{Locus}(R)$, a contradiction by Th.-Def.
4.5$(e)$. Therefore $-K_{X^{\prime}}\cdot R\leq 0$ and $X^{\prime}$ is not
Fano. This implies that $\rho_{F}\leq 8$ by Lemma 5.10.

Now if $g$ is quasi-elementary, by [Cas20, Cor. 2.16] we get
$\rho_{X}\leq\rho_{F}+\rho_{T}\leq 10$, namely $(ii)$. Instead if $g$ is not
quasi-elementary, then we get $(i)$ by Lemma 5.12. ∎

######  Lemma 5.14.

Let $X$ be a smooth Fano $4$-fold with $\delta_{X}\leq 1$ and $f\colon X\to S$
a contraction onto a surface. Then either $\rho_{S}=2$ and $f$ is
equidimensional, or $\rho_{S}=1$ and there is at most one prime divisor
contracted to a point.

###### Proof.

For any prime divisor $D\subset X$ such that $f(D)\subsetneq S$ we have

$1\geq\operatorname{codim}\mathcal{N}_{1}(D,X)\geq\operatorname{codim}\mathcal{N}_{1}(f(D),S)\geq\rho_{S}-1,$

hence $\rho_{S}\leq 2$. If $\rho_{S}=2$, then $f(D)$ must always be a curve,
and $f$ is equidimensional.

Suppose that $\rho_{S}=1$ and $f$ is not equidimensional, so there exists a
prime divisor $D$ contracted to a point.

We note that in our setting $f(D)=\\{pt\\}$ is equivalent to the pushforward
of $D$ (as a cycle) being zero, and this is invariant under linear
equivalence. Since there cannot be a positive dimensional linear system of
divisors contracted to points, $D$ must be a fixed divisor.

Let us take a general very ample curve $A\subset S$ and consider the prime
divisor $f^{-1}(A)$. We have $f^{-1}(A)\cap D=\emptyset$, hence
$\mathcal{N}_{1}(f^{-1}(A),X)\subseteq D^{\perp}$ (see Rem. 2.1) and
$\operatorname{codim}\mathcal{N}_{1}(f^{-1}(A),X)\leq\delta_{X}\leq 1$, so
that $\mathcal{N}_{1}(f^{-1}(A),X)=D^{\perp}$.

We show that $D$ is unique. Indeed, if $B\subset X$ is another prime divisor
contracted to a point, as above we get
$\mathcal{N}_{1}(f^{-1}(A),X)=B^{\perp}$, hence $D^{\perp}=B^{\perp}$. This
means that the classes $[D]$ and $[B]$ are multiples in $\mathcal{N}^{1}(X)$,
and being $D$ a fixed divisor, it implies that $B=D$. ∎

######  Lemma 5.15.

Let $X$ be a smooth Fano $4$-fold, $X\dasharrow\widetilde{X}$ a SQM, and
$f\colon\widetilde{X}\to\mathbb{P}^{1}$ a $K$-negative contraction with
general fiber $F$. Suppose that $F\cong\mathbb{P}^{1}\times S$ and that
$\dim\mathcal{N}_{1}(F,\widetilde{X})=\rho_{F}$. Then $X$ has a rational
contraction onto a $3$-fold.

###### Proof.

Set $C:=\mathbb{P}^{1}\times\\{pt\\}\subset F$. Then
$\mathcal{N}_{C/F}\cong\mathcal{O}_{\mathbb{P}^{1}}^{\oplus 2}$,
$(\mathcal{N}_{F/\widetilde{X}})_{|C}\cong\mathcal{O}_{\mathbb{P}^{1}}$, and
$\mathcal{N}_{C/\widetilde{X}}\cong\mathcal{O}_{\mathbb{P}^{1}}^{\oplus 3}$.
This implies that $\operatorname{Hilb}(\widetilde{X})$ is smooth of dimension
$3$ at the point $[C]$; let $W$ be the irreducible component of
$\operatorname{Hilb}(\widetilde{X})$ containing $[C]$.

Any point in $W$ yields an effective and connected one-cycle $\Gamma$ in
$\widetilde{X}$ such that $\Gamma\equiv C$. We show that if $\Gamma\cap
C\neq\emptyset$, then $\Gamma=C$.

We have $f(\Gamma)=\\{pt\\}$, and $\Gamma$ intersects the fiber $F$, so that
$\Gamma\subset F$. Let us consider the natural linear map
$\mathcal{N}_{1}(F)\twoheadrightarrow\mathcal{N}_{1}(F,\widetilde{X})\subseteq\mathcal{N}_{1}(\widetilde{X})$:
since $\dim\mathcal{N}_{1}(F,\widetilde{X})=\rho_{F}$, this map is injective.
Therefore $\Gamma\equiv_{F}C$, thus $\Gamma$ is supported on a fiber of the
projection $F\to S$. Since $\Gamma$ intersects $C$, we conclude that
$\Gamma=C$.

Let $\widetilde{X}_{0}\subset\widetilde{X}$ be the open subset where the
fibers of $f$ satisfy the same assumptions as $F$, and $W_{0}\subset W$ the
open subset parametrizing the curves $\mathbb{P}^{1}\times\\{pt\\}$ in the
fibers of $f_{|\widetilde{X}_{0}}$. Let $\mathcal{C}_{0}\to W_{0}$ be the
universal family and $e\colon\mathcal{C}_{0}\to\widetilde{X}$ the natural map.
Then $W_{0}$ and $\mathcal{C}_{0}$ are smooth and
$e\colon\mathcal{C}_{0}\to\widetilde{X}_{0}$ is bijective, and since
$\widetilde{X}$ is smooth, we conclude that $e$ is an isomorphism and there is
a projective morphism $g_{0}\colon\widetilde{X}_{0}\to W_{0}$. Then $X$ has a
rational contraction onto a $3$-fold, see [Cas20, proof of Th. 1.2]. ∎

######  Proposition 5.16.

Let $X$ be a smooth Fano 4-fold with $\rho_{X}\geq 7$ and $\delta_{X}\leq 1$,
$\tau$ a movable face of $\operatorname{Eff}(X)$, and $f\colon X\dasharrow Z$
an associated rational contraction as in 5.5. Then one of the following holds:

  1. $(i)$

$X$ has a rational contraction onto a $3$-fold;

  2. $(ii)$

$X$ has a fixed prime divisor $D$ of type $(3,2)$ such that
$\mathcal{N}_{1}(D,X)\subsetneq\mathcal{N}_{1}(X)$;

  3. $(iii)$

$\rho_{X}\leq 8+\dim\tau$, $Z\cong\mathbb{P}^{1}$, and $\tau$ does not contain
classes of fixed prime divisors of type $(3,2)$;

  4. $(iv)$

$\rho_{X}=10$ and $f\colon X\to\mathbb{P}^{1}$ is regular with general fiber
$S\times\mathbb{P}^{1}$, where $S$ is a del Pezzo surface with $\rho_{S}=9$;

  5. $(v)$

$\rho_{X}\leq 9+\dim\tau$, $Z\cong\mathbb{P}^{2}$, and every fixed prime
divisor with class in $\tau$ is of type $(3,2)$ or $(3,1)^{sm}$;

  6. $(vi)$

$\tau$ is not minimal, $f$ is quasi-elementary, and $\dim Z=2$.

Moreover in cases $(i)$ and $(ii)$ we have $\rho_{X}\leq 12$, and if
$\rho_{X}=12$, then $X$ has a rational contraction onto a $3$-fold.

###### Proof.

We apply the set-up as in 5.5, and keep the same notation. We factor $f$ as
$X\dasharrow\widetilde{X}\stackrel{{\scriptstyle\tilde{f}}}{{\to}}Z$ where
$\tilde{f}$ is $K$-negative (see Lemma 2.5), so that the general fiber
$F\subset\widetilde{X}$ is a smooth Fano variety. If $\dim Z=3$ we get $(i)$.

Suppose that $Z=\mathbb{P}^{1}$. If there exists a fixed prime divisor $D$ of
type $(3,2)$ such that $[D]\in\tau$, then its transform
$\widetilde{D}\subset\widetilde{X}$ is contained in a fiber of $\tilde{f}$.
Thus
$\mathcal{N}_{1}(\widetilde{D},\widetilde{X})\subseteq\ker\tilde{f}_{*}\subsetneq\mathcal{N}_{1}(\widetilde{X})$,
and we get $(ii)$ by Lemma 4.10.

Assume that $\tau$ does not contain classes of fixed prime divisors of type
$(3,2)$. If $\dim\mathcal{N}_{1}(F,\widetilde{X})\leq 8$, we get $(iii)$ by
(5.6). If $\dim\mathcal{N}_{1}(F,\widetilde{X})\geq 9$ and
$\dim\mathcal{N}_{1}(F,\widetilde{X})=\rho_{F}$, then
$F\cong\mathbb{P}^{1}\times S$ (see [IP99, §12.6]), and we get $(i)$ by Lemma
5.15.

The last possibility is $9=\dim\mathcal{N}_{1}(F,\widetilde{X})<\rho_{F}=10$,
and $F\cong\mathbb{P}^{1}\times S$ where $\rho_{S}=9$. In this case
$X\cong\widetilde{X}$ by Lemma 5.10, and $f\colon X\to\mathbb{P}^{1}$ is
regular. Thus $0<\operatorname{codim}\mathcal{N}_{1}(F,X)\leq\delta_{X}=1$,
and we conclude that $\operatorname{codim}\mathcal{N}_{1}(F,X)=1$ and
$\rho_{X}=10$, so we get $(iv)$.

Finally suppose that $\dim Z=2$, so that $f$ is an equidimensional rational
contraction, and $Z$ is a smooth rational surface by [Cas20, Lemma 4.3].

Suppose first that $\rho_{Z}>1$. If $f$ is not quasi-elementary, we get $(ii)$
by Lemma 5.12.

If $f$ is quasi-elementary, then $f^{*}\operatorname{Eff}(Z)$ is a face of
$\operatorname{Eff}(X)$ by [Cas13a, Prop. 2.22], thus
$f^{*}\operatorname{Eff}(Z)=\tau$. On the other hand there is a contraction
$Z\to\mathbb{P}^{1}$, so that the boundary of $\operatorname{Eff}(Z)$ contains
some non-zero movable divisor, therefore the boundary of $\tau$ contains some
non-zero movable divisor. Hence $\tau$ is not minimal, and we get $(vi)$.

Finally if $\rho_{Z}=1$, then $Z\cong\mathbb{P}^{2}$. If $D\subset X$ is a
fixed prime divisor with $[D]\in\tau$, then the image of $D$ in
$\mathbb{P}^{2}$ is an irreducible curve $\Gamma$, and $D$ is a component of
$f^{*}\Gamma$. Since $f^{*}\Gamma$ is movable, while $D$ is fixed,
$f^{*}\Gamma$ is reducible. By [Cas20, Lemma 4.10] $D$ must be of type $(3,2)$
or $(3,1)^{sm}$. Moreover $F$ is a del Pezzo surface, so that $\rho_{F}\leq 9$
and $\rho_{X}\leq 9+\dim\tau$ by (5.6), so we get $(v)$.

The last statement follows from Theorems 5.8 and 4.8 (note that $X$ is not a
product of surfaces, see Rem. 2.8). ∎

##  6\. Constructing divisors covered by lines

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$. In this section we
show that, given a fixed prime divisor $D\subset X$ of type $(3,1)^{sm}$ or
$(3,0)^{Q}$,111Note in particular that $X$ is not a product of surfaces, see
Ex. 4.6. and $L\subset D$ an exceptional plane, then there exists a prime
divisor $B$ covered by a family of lines $V$ such that $[V]+C_{L}\equiv C_{D}$
(Prop. 6.1). The construction of this family of lines if based on the explicit
geometry of the divisor $D$, and we also use the results of Section 3. By Rem.
4.17 $D$ contains at least $\rho_{X}-4$ exceptional planes with distinct
classes $[C_{L}]$, so that we obtain many distinct prime divisors covered by
lines. This construction will be important in the rest of the paper. Then we
give some properties of $D$ and $B$ depending on the different settings.

######  Proposition 6.1.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, $D\subset X$ a fixed
prime divisor of type $(3,1)^{sm}$ or $(3,0)^{Q}$, and $L\subset D$ an
exceptional plane. Then there exists a family of lines $V$ such that
$B:=\operatorname{Locus}V$ is a divisor different from $D$, $[V]+C_{L}\equiv
C_{D}$, and $B\cdot C_{L}>0$. Moreover $L^{\prime}\not\subset B$ for every
exceptional plane $L^{\prime}\subset D$.

###### Proof.

Let
$X\stackrel{{\scriptstyle\varphi}}{{\dasharrow}}\widetilde{X}\stackrel{{\scriptstyle\sigma}}{{\to}}Y$
be the contraction associated to $D$ as in §4.3, and
$\ell\subset\widetilde{X}$ the exceptional curve corresponding to $L$ (see
Lemma 2.4), so that $\widetilde{D}\cdot\ell>0$ and
$\ell\not\subset\widetilde{D}$. Let $q\in\ell\cap\widetilde{D}$.

If $D$ is $(3,1)^{sm}$, then $\sigma$ is the blow-up of a smooth curve, and
$q$ belongs to a fiber $S_{0}$ of $\sigma$. Let $\Gamma_{0}$ be a line in
$S_{0}\cong\mathbb{P}^{2}$ through $q$.

If $D$ is $(3,0)^{Q}$, then $\widetilde{D}$ is an irreducible quadric, either
smooth or with one singular point, which cannot be $q$ by Rem. 4.25. Let
$\Gamma_{0}$ be a line in $\widetilde{D}$ through $q$, and let
$S_{0}\subset\widetilde{D}$ be the union of the lines through $q$, which is a
quadric cone (if $\widetilde{D}$ is smooth) or a reducible quadric surface,
singular at $q$ (if $\widetilde{D}$ is singular).

In both cases $-K_{\widetilde{X}}\cdot\Gamma_{0}=2$, so that in the
factorization of $\varphi$ given in Lemma 2.4:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{{\widehat{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{f}$$\textstyle{{\widetilde{X},}}$

if $E\subset\widehat{X}$ is the exceptional divisor over $L$ and $\ell$, by
Lemma 2.4$(a)$ we have $E\cdot\widehat{\Gamma}=1$ for the transform
$\widehat{\Gamma}\subset\widehat{X}$ of $\Gamma_{0}$. This implies that
$\Gamma_{0}$ intersects $\ell$ only at $q$, transversally, and does not
intersect other exceptional curves. Moreover the transform $\Gamma\subset X$
of $\Gamma_{0}$ is a smooth rational curve with $-K_{X}\cdot\Gamma=1$, so that
there is a family of lines $V$ in $X$ containing the general $\Gamma$. The
transform $S\subset D\subset X$ of $S_{0}$ yields a surface contained in
$\operatorname{Locus}V$.

The curve $C_{D}\subset D$ is the transform of a general line in $S_{0}$ (if
$D$ is $(3,1)^{sm}$), or of a general line in $\widetilde{D}$ (if $D$ is
$(3,0)^{Q}$), so we must have $C_{D}\equiv[V]+mC_{L}$ with $m\in\mathbb{Z}$.
Intersecting with $-K_{X}$ we get $m=1$ and $C_{D}\equiv[V]+C_{L}$.

We claim that, varying $\Gamma_{0}$ in $S_{0}$, we can find two disjoint
$\Gamma$’s in $S$. Indeed if $D$ is $(3,1)^{sm}$, then the plane
$T_{q}S_{0}\subset T_{q}\widetilde{X}$ is the union of $T_{q}\Gamma_{0}$ for
all $\Gamma_{0}$’s, and since $\Gamma_{0}$ and $\ell$ are transverse at $q$,
$T_{q}S_{0}$ does not contain $T_{q}\ell$, namely $S_{0}$ and $\ell$ meet
transversally at $q$. Moreover $S_{0}\cap\ell=\\{q\\}$ and $S_{0}$ does not
meet other exceptional curves (see §4.3). Locally
$g\colon\widehat{X}\to\widetilde{X}$ is the blow-up of $\ell$, therefore the
strict transform of $S_{0}$ is isomorphic to the blow-up of $S_{0}$ at $q$,
and the lines $\Gamma_{0}$ get separated.

If $D$ is $(3,0)^{Q}$ and $\widetilde{D}$ is singular, we can just repeat the
same argument with each irreducible component of $S_{0}$. Finally if $D$ is
$(3,0)^{Q}$ and $\widetilde{D}$ is smooth, then $T_{q}\widetilde{D}\subset
T_{q}\widetilde{X}$ is a hyperplane, and when $\Gamma_{0}$ varies,
$T_{q}\Gamma_{0}$ describes a quadric cone surface in $T_{q}\widetilde{D}$,
not containing $T_{q}\ell$. If we choose $\Gamma_{0}$ and
$\Gamma_{0}^{\prime}$ such that the plane generated by $T_{q}\Gamma_{0}$ and
$T_{q}\Gamma_{0}^{\prime}$ in $T_{q}\widetilde{X}$ does not contain
$T_{q}\ell$, then the curves $\widehat{\Gamma}$ and
$\widehat{\Gamma}^{\prime}$ in $\widehat{X}$ must meet
$g^{-1}(q)\cong\mathbb{P}((T_{q}\widetilde{X}/T_{q}\ell)^{\vee})$ at different
points. Since $f_{|g^{-1}(q)}\colon g^{-1}(q)\to L$ is an isomorphism, the
curves $\Gamma$ and $\Gamma^{\prime}$ are disjoint in $X$.

By Th. 3.7 we deduce that $\dim V=2$ and $\dim\operatorname{Locus}V=3$.

Since $D\cdot[V]+D\cdot C_{L}=D\cdot C_{D}=-1$, we have $[V]\neq[C_{L}]$, and
$L\not\subset B:=\operatorname{Locus}V$ by Lemma 3.11$(b)$. On the other hand
$B\cap L\neq\emptyset$ because $S\subset B$, so that $B\cdot C_{L}>0$. This
also implies that $B\neq D$, because $D\cdot C_{L}<0$, hence $D\cdot[V]\geq
0$. If $L^{\prime}\subset D$ is any exceptional plane, we have $D\cdot
C_{L^{\prime}}<0$, thus $[V]\neq[C_{L^{\prime}}]$, and $L^{\prime}\not\subset
B$ again by Lemma 3.11$(b)$. ∎

######  Lemma 6.2.

Notation as in Prop. 6.1. Let $L^{\prime}\subset D$ be an exceptional plane
with $C_{L^{\prime}}\not\equiv C_{L}$ , and $V^{\prime}$ the corresponding
family of lines given by Prop. 6.1, with locus $B^{\prime}$. Suppose moreover
that $D$ is of type $(3,0)^{Q}$ and that both $B$ and $B^{\prime}$ are fixed
of type $(3,2)$. Then $B\cdot[V^{\prime}]>0$ and $B^{\prime}\cdot[V]>0$.

###### Proof.

Since $C_{D}\equiv[V]+C_{L}\equiv[V^{\prime}]+C_{L^{\prime}}$, we have
$[V]\neq[V^{\prime}]$ and hence $B\neq B^{\prime}$ by Lemma 3.11$(a)$.
Moreover by the same lemma we have $[V]=[C_{B}]$ and
$[V^{\prime}]=[C_{B^{\prime}}]$.

Assume by contradiction that $B\cdot[V^{\prime}]=0$. Then by Lemma 4.15 we
have $B^{\prime}\cdot[V]=0$ and, if $B\cap B^{\prime}\neq\emptyset$, every
connected component of $B\cap B^{\prime}$ is irreducible.

We keep the same notation as in the proof of Prop. 6.1. Let us consider the
transforms $\widetilde{B}$, $\widetilde{B}^{\prime}$ in $\widetilde{X}$, and
let $\ell^{\prime}\subset\widetilde{X}$ be the exceptional curve corresponding
to $L^{\prime}$. By construction $\widetilde{B}^{\prime}$ contains the surface
$S_{0}^{\prime}$, given by the union of the lines $\Gamma_{0}^{\prime}$ in
$\widetilde{D}$ through a point $q^{\prime}\in\ell^{\prime}\cap\widetilde{D}$.
The surfaces $S_{0}$ and $S_{0}^{\prime}$ meet along a plane conic $\Lambda$,
which does contain neither $q$ nor $q^{\prime}$; in fact $\Lambda$ is
contained in the open subset where $\varphi\colon X\dasharrow\widetilde{X}$ is
an isomorphism.

If some line $\Gamma_{0}$ is not contained in $\widetilde{B}^{\prime}$, then
its transform $\Gamma$ in $X$ is a curve of the family $V$ such that
$\Gamma\cap B^{\prime}\neq\emptyset$ and $\Gamma\not\subset B^{\prime}$,
contradicting $B^{\prime}\cdot[V]=0$. Therefore every $\Gamma_{0}$ must be
contained in $\widetilde{B}^{\prime}$, so that
$S_{0}\subseteq\widetilde{B}^{\prime}$ and $S\subseteq B\cap B^{\prime}$.
Similarly $S^{\prime}\subseteq B\cap B^{\prime}$, where $S^{\prime}$ is the
transform of $S_{0}^{\prime}$. Hence we get $S\cup S^{\prime}\subseteq B\cap
B^{\prime}$ and $S\cap S^{\prime}\neq\emptyset$; this gives a reducible
connected component of $B\cap B^{\prime}$, a contradiction. ∎

######  Lemma 6.3.

Notation as in Prop. 6.1. Suppose moreover that $D$ is of type $(3,0)^{Q}$ and
that $D\cdot C_{L}=-1$. Then $B\cap L^{\prime}=\emptyset$ for every
exceptional plane $L^{\prime}\subset D$ with $C_{L^{\prime}}\not\equiv C_{L}$.

###### Proof.

Since $D\cdot C_{D}=D\cdot C_{L}=-1$ and $C_{D}\equiv C_{L}+[V]$, we have
$D\cdot[V]=0$, thus every line of the family $V$ that meets $D$ must be
contained in $D$. On the other hand $D\neq B$, thus the general line of the
family $V$ is disjoint from $D$, and $V$ yields a family of lines
$V_{\widetilde{X}}$ in $\widetilde{X}$. We have
$[V_{\widetilde{X}}]\equiv\ell+C_{\widetilde{D}}$,
$\widetilde{D}\cdot[V_{\widetilde{X}}]=0$, and
$\operatorname{Locus}V_{\widetilde{X}}=\widetilde{B}$, the transform of $B$.
Since $-K_{\widetilde{X}}$ is not ample, the family $V_{\widetilde{X}}$ can
have reducible members, containing some exceptional curve.

Let us consider all the exceptional planes $L_{1}:=L,L_{2},\dotsc,L_{d}\subset
D$ such that $C_{L_{i}}\equiv C_{L}\equiv C_{D}-[V]$, and let
$\ell_{1},\dotsc,\ell_{d}\subset\widetilde{X}$ be the corresponding
exceptional curves. We show that for every one-cycle $\Lambda$ of the family
$V_{\widetilde{X}}$ one of the following holds:

  1. $(i)$

$\Lambda$ is integral and disjoint from $\widetilde{D}$ and from every
exceptional curve of $\widetilde{X}$;

  2. $(ii)$

$\Lambda=\Gamma_{0}+\ell_{i}$ for some $i\in\\{1,\dotsc,d\\}$, where
$\Gamma_{0}\subset\widetilde{D}$ is a line meeting $\ell_{i}$, and $\Lambda$
does not meet exceptional curves except $\ell_{i}$.

Indeed, let us consider
$-K_{\widetilde{X}}+2\widetilde{D}=\sigma^{*}(-K_{Y})$, so that
$-K_{\widetilde{X}}+2\widetilde{D}$ is nef and

$(-K_{\widetilde{X}}+2\widetilde{D})^{\perp}\cap\operatorname{NE}(\widetilde{X})=\mathbb{R}_{\geq
0}[C_{\widetilde{D}}].$

We have $(-K_{\widetilde{X}}+2\widetilde{D})\cdot\Lambda=1$, so we can write
$\Lambda=\Lambda_{1}+\Lambda_{0}$ where $\Lambda_{1}$ is an integral curve
with $(-K_{\widetilde{X}}+2\widetilde{D})\cdot\Lambda_{1}=1$, and
$\Lambda_{0}$ is an effective one-cycle with
$(-K_{\widetilde{X}}+2\widetilde{D})\cdot\Lambda_{0}=0$. In particular
$\Lambda_{1}\not\subset\widetilde{D}$, while $\Lambda_{0}$ is supported in
$\widetilde{D}$.

If $\Lambda_{0}=0$, then $\Lambda=\Lambda_{1}$ is integral and disjoint from
$\widetilde{D}$; moreover $\Lambda$ cannot intersect any exceptional curve
(see Lemma 2.4$(b)$), and we get $(i)$.

If $\Lambda_{0}\neq 0$, then $\widetilde{D}\cdot\Lambda_{0}<0$. Moreover we
have
$0=\widetilde{D}\cdot\Lambda=\widetilde{D}\cdot\Lambda_{1}+\widetilde{D}\cdot\Lambda_{0}$,
therefore $\widetilde{D}\cdot\Lambda_{1}>0$. Together with
$(-K_{\widetilde{X}}+2\widetilde{D})\cdot\Lambda_{1}=1$, this implies that
$-K_{\widetilde{X}}\cdot\Lambda_{1}<0$, so that $\Lambda_{1}$ is an
exceptional curve (see Lemma 2.4$(a)$),
$-K_{\widetilde{X}}\cdot\Lambda_{1}=-1$, $\widetilde{D}\cdot\Lambda_{1}=1$,
and finally $\widetilde{D}\cdot\Lambda_{0}=-1$. Hence $\Lambda_{0}\equiv
C_{\widetilde{D}}$ is integral and is a line in the quadric $\widetilde{D}$;
moreover $\Lambda_{1}\equiv[V_{\widetilde{X}}]-C_{\widetilde{D}}\equiv\ell$,
so that $\Lambda_{1}=\ell_{i}$ for some $i\in\\{1,\dotsc,d\\}$. Since the
1-cycle $\Lambda$ is connected, $\Lambda_{0}$ must meet $\ell_{i}$. Finally
$\Lambda_{0}$ cannot meet other exceptional curves (as shown in the proof of
Prop. 6.1), and neither can $\Lambda_{1}$ (by Lemma 2.4$(b)$), so that we get
$(ii)$.

Now let $L^{\prime}\subset D$ be an exceptional plane such that
$C_{L^{\prime}}\not\equiv C_{L}$, and $\ell^{\prime}\subset\widetilde{X}$ the
corresponding exceptional curve, so that $\ell^{\prime}\neq\ell_{i}$ for every
$i=1,\dotsc,d$. By what precedes, we have
$\ell^{\prime}\cap\widetilde{B}=\emptyset$, thus $L^{\prime}\cap B=\emptyset$.
∎

######  Proposition 6.4.

Notation as in Prop. 6.1. Suppose moreover that $B$ is fixed of type $(3,2)$
and adjacent to $D$, let $L^{\prime}\subset D$ be an exceptional plane with
$C_{L^{\prime}}\not\equiv C_{L}$, and $V^{\prime}$ the corresponding family
given by Prop. 6.1. Then $D\cdot C_{L}=-1$, $D\cdot[V]=0$, and:

  1. $\bullet$

$B\cdot C_{D}=0$, $B\cdot C_{L}=1$, $B\cdot[V^{\prime}]=0$, if $D$ is of type
$(3,1)^{sm}$;

  2. $\bullet$

$B\cdot C_{D}=1$, $B\cdot C_{L}=2$, $B\cdot[V^{\prime}]=1$, if $D$ is of type
$(3,0)^{Q}$.

###### Proof.

We keep the same notation as in the proof of Prop. 6.1. Note that
$[C_{B}]=[V]$ by Lemma 3.11$(a)$.

######  6.5.

By Lemma 4.13$(a)$ we have $D\cdot[V]=0$; since $C_{D}\equiv C_{L}+[V]$, this
also yields $D\cdot C_{L}=-1$. Thus $\widetilde{D}\cdot\ell=1$, and
$\widetilde{D}$ and $\ell$ meet transversally at the unique point $q$. This
implies that $D$ is smooth around $L$ and $\varphi_{|D}\colon
D\dasharrow\widetilde{D}$ is regular around $L$ and is just the blow-up of
$q$, with exceptional divisor $L$.

The transform $S\subset D\subset X$ of the surface
${S}_{0}\subset\widetilde{D}\subset\widetilde{X}$ is isomorphic to either
$\mathbb{F}_{1}$ (if $D$ is $(3,1)^{sm}$), or to $\mathbb{F}_{2}$ (if
$\widetilde{D}$ is a smooth quadric), or to the union of two copies of
$\mathbb{F}_{1}$ (if $\widetilde{D}$ is a singular quadric).

######  6.6.

Let $f\colon X\to X^{\prime}$ be the contraction of $\mathbb{R}_{\geq
0}[C_{B}]$, so that $\operatorname{Exc}(f)=B$, $f$ can have at most finitely
many $2$-dimensional fibers, and $B$ is a smooth $\mathbb{P}^{1}$-bundle
outside these fibers.

We show that $D$ is disjoint from the possible $2$-dimensional fibers of $f$.
Indeed if $F$ is such a fiber and $F\cap D\neq\emptyset$, since $D\cdot
C_{B}=0$, we have $D\cdot\Gamma=0$ for every curve $\Gamma\subset F$, so that
$F\subset D$. For every exceptional plane $L^{\prime}\subset D$ we have
$D\cdot C_{L^{\prime}}<0$, therefore the classes $[C_{L^{\prime}}]$ and
$[C_{B}]$ cannot be proportional. Since
$\mathcal{N}_{1}(F,X)=\mathbb{R}[C_{B}]$ and
$\mathcal{N}_{1}(L^{\prime},X)=\mathbb{R}[C_{L^{\prime}}]$, we deduce that
$\dim(L^{\prime}\cap F)\leq 0$. Therefore there exists an irreducible curve
$\Gamma_{2}\subset F$ disjoint from every exceptional plane $L^{\prime}\subset
D$. Then $\Gamma_{2}$ is contained in the open subset where $\varphi\colon
X\dasharrow\widetilde{X}$ is an isomorphism, and we conclude that
$\Gamma_{2}\equiv mC_{D}$ for some $m\in\mathbb{Z}_{\geq 0}$, which is
impossible because $[\Gamma_{2}]\in\operatorname{NE}(f)$.

We deduce that $B\cap D$ is covered by one-dimensional fibers of $f$ and
$B\cap D\subset B_{reg}$. The irreducible components of $B\cap D$ are
$f^{-1}(K)$ where $K$ is an irreducible component of the curve $f(B\cap D)$,
and distinct irreducible components intersect at most along fibers of $f$.

######  6.7.

Suppose that $D$ is $(3,1)^{sm}$. We show that $S$ is a connected component of
$B\cap D$. By contradiction, let $T$ be an irreducible component of $B\cap D$
such that $T\neq S$ and $T\cap S\neq\emptyset$. Then $T$ must contain some
curve $\Gamma$, and $\Gamma\cap L\neq\emptyset$, so that $T\cap
L\neq\emptyset$ and $T\cap L$ is a curve (as $T\subset D$ and $L\subset
D_{reg}$). Since $f(T)$ is an irreducible curve and $f_{|T}$ is a
$\mathbb{P}^{1}$-bundle, we also have $\rho_{T}=2$ and
$\mathcal{N}_{1}(T)=\mathbb{R}[T\cap L]\oplus\mathbb{R}[\Gamma]$.

If $\widetilde{T}\subset\widetilde{D}$ is the transform of $T$, then
$\widetilde{T}$ cannot contain exceptional curves, and
$\rho_{\widetilde{T}}=1$. We have $S_{0}\cap\widetilde{T}\neq\emptyset$,
$S_{0}\neq\widetilde{T}$, $S_{0}$ is a fiber of the $\mathbb{P}^{2}$-bundle
$\sigma_{|\widetilde{D}}$, and $\rho_{\widetilde{T}}=1$, which yields a
contradiction.

Thus every irreducible component of $\widetilde{B}\cap\widetilde{D}$ is a
fiber of $\sigma$, therefore $\widetilde{B}$ is disjoint from the general
fiber of $\sigma_{|\widetilde{D}}$; this yields $B\cdot
C_{D}=\widetilde{B}\cdot C_{\widetilde{D}}=0$. Then $C_{D}\equiv C_{B}+C_{L}$
and $B\cdot C_{B}=-1$ imply that $B\cdot C_{L}=1$.

Finally set $B^{\prime}:=\operatorname{Locus}V^{\prime}$. We have
$[V^{\prime}]\neq[V]$, thus $B^{\prime}\neq B$ by Lemma 3.11$(a)$, and
$B\cdot[V^{\prime}]\geq 0$. On the other hand we also have $B\cdot
C_{L^{\prime}}\geq 0$ because $L^{\prime}\not\subset B$ by Prop. 6.1, hence
$0=B\cdot C_{D}=B\cdot C_{L^{\prime}}+B\cdot[V^{\prime}]$ implies that $B\cdot
C_{L^{\prime}}=B\cdot[V^{\prime}]=0$.

######  6.8.

Suppose now that $D$ is $(3,0)^{Q}$. The proof of Lemma 6.3 shows that
$\widetilde{B}\cap\widetilde{D}=S^{1}_{0}\cup\cdots\cup S^{d}_{0}$, where
$S^{i}_{0}$ is the union of the lines in $\widetilde{D}$ through the point
$q_{i}:=\widetilde{D}\cap\ell_{i}$ for $i=1,\dotsc,d$, and
$\ell_{1},\dotsc,\ell_{d}$ are all the exceptional curves of $\widetilde{X}$
numerically equivalent to $\ell=\ell_{1}$, so that
$\widetilde{D}\cdot\ell_{i}=1$ for every $i$. In $X$ we have $B\cap
D=S^{1}\cup\cdots\cup S^{d}$, where $S^{i}$ is the transform of $S_{0}^{i}$.

We show that $d=1$. Otherwise, for $i\neq j$ the surfaces $S_{0}^{i}$ and
$S_{0}^{j}$ meet along a plane conic contained in the open subset where
$\varphi\colon X\dasharrow\widetilde{X}$ is an isomorphism, so that $S^{i}$
and $S^{j}$ meet along a curve with class in $\mathbb{R}_{\geq 0}[C_{D}]$.
This is impossible, because they should intersect along fibers of $f\colon
X\to X^{\prime}$. We conclude that $d=1$ and that $D\cap B=S$.

Suppose that $\widetilde{D}$ is a smooth quadric, so that $S$ is irreducible
and contained in $D_{reg}$ (see 6.5). We have $B_{|D}=mS$ for some
$m\in\mathbb{Z}_{\geq 1}$. If $\Gamma\subset S$ is a curve of the family $V$,
we have $-1=B\cdot\Gamma=mS\cdot\Gamma$, where the last intersection is in
$D$. This implies that $m=1$ and $B\cdot C_{L}=S\cdot C_{L}=2$ (again the last
intersection is in $D$), because $S\cap L$ is a conic in
$L\cong\mathbb{P}^{2}$.

Suppose now that $\widetilde{D}$ is a quadric cone, with vertex $v_{0}$; then
$S_{0}$ is a Cartier divisor in $\widetilde{D}$. In $D$ we have
$(\varphi_{|D})^{*}(S_{0})=S+2L$, so that again $S$ is Cartier in $D$.
Moreover $S=H_{1}+H_{2}$ where $H_{i}\cong\mathbb{F}_{1}$ and $H_{1}\cap
H_{2}$ is a common fiber of the $\mathbb{P}^{1}$-bundles, which contains the
singular point $v:=\varphi^{-1}(v_{0})$. Note that $v\not\in L$ and that
$L\cup(S\smallsetminus\\{v\\})\subset D_{reg}$.

Write $B_{|D}=m_{1}H_{1}+m_{2}H_{2}$, with $m_{i}\in\mathbb{Z}_{\geq 1}$, and
let $\Gamma_{1}\subset H_{1}$ be a curve of the family $V$ disjoint from
$H_{2}$, so that $\Gamma_{1}\subset D_{reg}$. Then
$-1=B\cdot\Gamma_{1}=m_{1}H_{1}\cdot\Gamma_{1}=-m_{1}$, so that $m_{1}=1$.
Similarly we see that $m_{2}=1$, and finally that $B_{|D}=S$; as before this
yields $B\cdot C_{L}=2$.

We have $C_{D}\equiv C_{B}+C_{L}$ and $B\cdot C_{B}=-1$, therefore $B\cdot
C_{D}=1$. Finally we have $D\cap L^{\prime}=\emptyset$ by Lemma 6.3, thus
$1=B\cdot C_{D}=B\cdot[V^{\prime}]+B\cdot C_{L^{\prime}}=B\cdot[V^{\prime}]$.∎

######  Lemma 6.9.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$, $D\subset X$ a fixed
prime divisor of type $(3,1)^{sm}$ or $(3,0)^{Q}$, and $E\subset X$ a fixed
prime divisor of type $(3,2)$ such that $D\cap E\neq\emptyset$ and $D\cdot
C_{E}=0$. If $D$ is of type $(3,1)^{sm}$, assume moreover that $E\cdot
C_{D}=0$.

Then there exists an exceptional plane $L\subset D$ such that $D\cdot
C_{L}=-1$ and $C_{E}$ is the family of lines given by $D$ and $L$ as in Prop.
6.1.

###### Proof.

Let $X\dasharrow\widetilde{X}\stackrel{{\scriptstyle\sigma}}{{\to}}Y$ be the
contraction associated to $D$ as in §4.3, and
$E_{\widetilde{X}}\subset\widetilde{X}$, $E_{Y}\subset Y$ the transforms of
$E$. We have $D\cap E\neq\emptyset$ and $E$ does not contain exceptional
planes (see Rem. 4.9), therefore $\widetilde{D}\cap
E_{\widetilde{X}}\neq\emptyset$ and $\sigma(\widetilde{D})\cap
E_{Y}\neq\emptyset$.

Since $D\cdot C_{E}=0$, $D$ is disjoint from the general curve $C_{E}$, so
that $Y$ has a family of lines $V_{Y}$ with locus $E_{Y}$. If $D$ is
$(3,1)^{sm}$ and $[C]=[V_{Y}]$, then $C\subset E_{Y}$ and $E\cdot
C_{D}=E_{\widetilde{X}}\cdot C_{\widetilde{D}}>0$, against our assumptions.
Thus $[C]\neq[V_{Y}]$.

Let $\Gamma\subset Y$ be a curve of the family $V_{Y}$ such that
$\Gamma\cap\sigma(\widetilde{D})\neq\emptyset$ and
$\Gamma\neq\sigma(\widetilde{D})$. By Rem. 4.16 we have $\Gamma=\sigma(\ell)$
where $\ell\subset\widetilde{X}$ is an exceptional curve; moreover if
$\widetilde{C}_{E}\subset\widetilde{X}$ is the transform of the general
$C_{E}\subset X$, we have $\widetilde{C}_{E}\equiv\ell+C_{\widetilde{D}}$ in
$\widetilde{X}$. Then in $X$ we get $C_{D}\equiv C_{L}+C_{E}$ where $L\subset
D$ is the exceptional plane corresponding to $\ell$. Moreover $-1=D\cdot
C_{D}=D\cdot(C_{L}+C_{E})=D\cdot C_{L}$, and we get the statement. ∎

######  Corollary 6.10.

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

###### Proof.

By Lemma 6.9 there exists an exceptional plane $L\subset D$ such that $C_{E}$
is the family of lines given by $D$ and $L$ as in Prop. 6.1, so that $E\cdot
C_{D}=1$ by Prop. 6.4. ∎

##  7\. Constructing rational contractions of fiber type

In this section we consider two situations where we can construct a rational
contraction of fiber type on $X$, and then prove that $\rho_{X}\leq 12$ using
the results of Section 5.

######  Proposition 7.1.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 8$ and $\delta_{X}\leq
1$, $D$ a fixed prime divisor of type $(3,1)^{sm}$ or $(3,0)^{Q}$, and
$X\stackrel{{\scriptstyle\varphi}}{{\dasharrow}}\widetilde{X}\stackrel{{\scriptstyle\sigma}}{{\to}}Y$
the associated contraction as in §4.3. Suppose that $Y$ contains a nef prime
divisor $H$ covered by a family of lines and such that
$H\cap\sigma(\operatorname{Exc}(\sigma))=\emptyset$. Then one of the following
holds:

  1. $(i)$

$X$ has a contraction onto a $3$-fold;

  2. $(ii)$

$X$ contains a fixed prime divisor $E$ of type $(3,2)$ with
$\mathcal{N}_{1}(E,X)\subsetneq\mathcal{N}_{1}(X)$;

  3. $(iii)$

$\rho_{X}\leq 10$.

Moreover $\rho_{X}\leq 12$, and if $\rho_{X}=12$, then $X$ has a rational
contraction onto a $3$-fold.

###### Proof.

Let $g\colon Y\to Z$ be the contraction defined by $mH$ for $m\in\mathbb{N}$,
$m\gg 0$, so that $H=g^{*}A$ where $A\subset Z$ is an ample prime Cartier
divisor, and $\operatorname{NE}(g)=\operatorname{NE}(Y)\cap H^{\perp}$. If
$\ell\subset\widetilde{X}$ is an exceptional curve, then by Lemma 4.19
$\sigma(\ell)$ cannot meet any line in $Y$ outside
$\sigma(\operatorname{Exc}(\sigma))$, so that $\sigma(\ell)\cap H=\emptyset$
and $[\sigma(\ell)]\in\operatorname{NE}(g)$. Moreover $H$ is contained in the
open subset of $Y$ where the birational map $X\dasharrow Y$ is an isomorphism;
let $H_{X}\subset X$ and $\widetilde{H}\subset\widetilde{X}$ be the transforms
of $H$. Then $\widetilde{H}=\sigma^{*}(g^{*}A)$, and $H_{X}$ is still nef, so
that $h:=g\circ\sigma\circ\varphi\colon X\to Z$ is regular, $H_{X}=h^{*}A$,
and $\operatorname{NE}(h)=\operatorname{NE}(X)\cap H_{X}^{\perp}$. Moreover
$H_{X}\cap D=\emptyset$, therefore $h(D)=\\{pt\\}$.

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{\varphi}$$\textstyle{{\widetilde{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Z}$

We show that $g$ and $h$ are of fiber type.

If $D$ is $(3,1)^{sm}$, have $H\cdot C=0$ hence $[C]\in\operatorname{NE}(g)$.
If $[C]\in\operatorname{mov}(Y)$, then $g$ is of fiber type. Otherwise, $[C]$
generates an extremal ray of type $(3,2)$ of $\operatorname{NE}(Y)$ by [Cas17,
Lemma 5.11(3)], let $E_{1}$ be its locus. By [Cas17, Prop. 5.8 and its proof],
there is fixed prime divisor $E_{2}\subset Y$ of type $(3,2)$, such that
$E_{2}\cdot C>0$, so that $[C]+[C_{E_{2}}]\in\operatorname{mov}(X)$ by Cor.
4.14. If $\Gamma\subset E_{2}$ is an irreducible curve such that $\Gamma\equiv
C_{E_{2}}$ and $\Gamma\cap C\neq\emptyset$, then by Rem. 4.16
$\Gamma=\sigma(\ell)$ where $\ell\subset\widetilde{X}$ is an exceptional
curve, thus $[\Gamma]=[C_{E_{2}}]\in\operatorname{NE}(g)$,
$[C]+[C_{E_{2}}]\in\operatorname{NE}(g)$, and again $g$ is of fiber type.

If $D$ is $(3,0)^{Q}$, let $V$ be a family of lines in $X$ given by $D$ and an
exceptional plane $L\subset D$ as in Prop. 6.1, so that
$\operatorname{Locus}V$ is a divisor and $C_{D}\equiv[V]+C_{L}$. Since $H_{X}$
is nef and $H_{X}\cdot C_{D}=0$, we have $H_{X}\cdot[V]=0$ and
$[V]\in\operatorname{NE}(h)$. If $[V]\in\operatorname{mov}(X)$, then $h$ is of
fiber type. Otherwise there exists a prime divisor $B$ such that
$B\cdot[V]<0$, hence $B=\operatorname{Locus}V$ and by Lemma 3.12 $[V]$
generates an extremal ray of type $(3,2)$. By Rem. 4.17 we can choose an
exceptional plane $L^{\prime}\subset D$ with $C_{L^{\prime}}\not\equiv C_{L}$
and this yields another family of lines $V^{\prime}$ with
$[V^{\prime}]\neq[V]$. As before, $[V^{\prime}]\in\operatorname{NE}(h)$, and
either $[V^{\prime}]\in\operatorname{mov}(X)$ and $h$ is of fiber type, or
$[V^{\prime}]$ generates an extremal ray of type $(3,2)$, with locus
$B^{\prime}$. Then $B\cdot[V^{\prime}]>0$ and $B^{\prime}\cdot[V]>0$ by Lemma
6.2, so that $[V]+[V^{\prime}]\in\operatorname{mov}(X)$ by Lemma 4.12$(b)$,
and again $h$ is of fiber type.

If $\dim Z=3$, we get $(i)$.

We show that $\dim Z>1$. Otherwise, let $F\subset X$ be a general fiber of
$h$. We have $\mathcal{N}_{1}(F,X)\subseteq\ker
h_{*}\subsetneq\mathcal{N}_{1}(X)$, and since $\delta_{X}\leq 1$, we deduce
that $\mathcal{N}_{1}(F,X)=\ker h_{*}$, and similarly
$\mathcal{N}_{1}(D,X)=\ker h_{*}$. On the other hand $F\cap D=\emptyset$, thus
$\mathcal{N}_{1}(F,X)\subseteq D^{\perp}$, which is impossible because $D\cdot
C_{D}=-1$ and $[C_{D}]\in\mathcal{N}_{1}(D,X)=\mathcal{N}_{1}(F,X)$.

Finally suppose that $\dim Z=2$. Since $h$ is not equidimensional, by Lemma
5.14 we have that $\rho_{Z}=1$ and $D$ is the unique prime divisor contracted
to a point. Then Lemma 5.13 yields $(ii)$ or $(iii)$.

The last statement follows from Th. 5.8 and 4.8 (note that $X$ is not a
product of surfaces, see Ex. 4.6 or Rem. 2.8). ∎

######  Lemma 7.2.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $\delta_{X}\leq
1$, 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 $\rho_{X}\leq 12$, and if $\rho_{X}=12$, then $X$ has a
rational contraction onto a $3$-fold.

###### Proof.

Let $i\in\\{1,2\\}$; we have $D\cdot C_{E_{i}}=0$ by Lemma 4.13$(a)$. If
$E_{i}\cap D=\emptyset$ for some $i\in\\{1,2\\}$, then
$\mathcal{N}_{1}(E_{i},X)\subseteq D^{\perp}\subsetneq\mathcal{N}_{1}(X)$, and
we conclude by Th. 4.8. Thus we can assume that $E_{i}\cap D\neq\emptyset$ for
$i=1,2$, so that by Lemma 6.9 there exists an exceptional plane $L_{i}\subset
D$ such that $C_{E_{i}}$ is the family of lines given by $D$ and $L_{i}$ as in
Prop. 6.1. Then Prop. 6.4 yields:

$E_{i}\cdot C_{D}=E_{1}\cdot C_{E_{2}}=E_{2}\cdot C_{E_{1}}=1.$

Consider now $H:=E_{1}+E_{2}+2D$; we show that $H$ is movable. By [Cas17,
Lemma 5.29(2)], $[H]\in\operatorname{Mov}(X)$ if and only if $H\cdot C_{G}\geq
0$ for every fixed prime divisor $G\subset X$. This is clear if $G\neq
E_{1},E_{2},D$; moreover $H\cdot C_{E_{1}}=H\cdot C_{E_{2}}=H\cdot C_{D}=0$,
therefore $H$ is movable. We also have $H\cdot(C_{E_{1}}+C_{E_{2}})=0$ and
$[C_{E_{1}}+C_{E_{2}}]\in\operatorname{mov}(X)=\operatorname{Eff}(X)^{\vee}$
by Lemma 4.12$(b)$, so that $[H]\in\partial\operatorname{Eff}(X)$, namely $H$
is not big.

Let $f\colon X\dasharrow Z$ be the rational contraction of fiber type defined
by $mH$ for $m\in\mathbb{N}$, $m\gg 0$. Since $H\cdot C_{D}=0$, $f$ contracts
$D$ to a point, so that if $\dim Z=2$ $f$ is not equidimensional. Therefore by
Lemma 5.11 we have either $\dim Z=3$, or
$\mathcal{N}_{1}(E_{1},X)\subsetneq\mathcal{N}_{1}(X)$, and we get the
statement by Th. 5.8 and 4.8 respectively (note that $X$ is not a product of
surfaces, see Ex. 4.6 or Rem. 2.8). ∎

##  8\. The case $(3,1)^{sm}$

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $\delta_{X}\leq
1$; in this section we prove the following result, which implies Th. 1.1 when
$X$ has a fixed prime divisor $D$ of type $(3,1)^{sm}$.

######  Theorem 8.1.

Let $X$ be a smooth Fano $4$-fold with $\rho_{X}\geq 7$ and $\delta_{X}\leq
1$, having a fixed divisor of type $(3,1)^{sm}$. Then $\rho_{X}\leq 12$, and
if $\rho_{X}=12$, then $X$ has a rational contraction onto a $3$-fold.

Let us give an outline of the proof. We consider the contraction $X\dasharrow
Y$ associated to $D$, and work in the smooth Fano $4$-fold $Y$. We show that
we can reduce to the following situation:

  1. –

except possibly one, the fixed prime divisors $E_{1},\dotsc,E_{r}$ of type
$(3,2)$ of $Y$ satisfy $E_{i}\cdot C_{E_{j}}=0$ for $i\neq j$;

  2. –

$Y$ contains at most one nef prime divisor covered by lines;

  3. –

$\operatorname{Eff}(Y)$ is generated by classes of fixed prime divisors, none
of type $(3,0)^{sm}$, and there is at least one not of type $(3,2)$, say $B$.

  4. –

The divisors covered by lines given by $B$ and its exceptional planes, as in
Section 3, are all fixed of type $(3,2)$, except at most one. From the
properties of these families of lines we deduce that $B$ is $(3,1)^{sm}$, so
that every fixed prime divisor of $Y$ is of type $(3,2)$ or $(3,1)^{sm}$.

Then we use these fixed prime divisors to construct some movable, non-big
divisors in $Y$, which yield some rational contractions of fiber type on $Y$
and $X$; these also allow to describe some facets of $\operatorname{Eff}(Y)$.
Finally we conclude applying the results in §5.3.

###### Proof of Th. 8.1.

We assume that $\rho_{X}\geq 8$, and note that $X$ is not a product of
surfaces (see Ex. 4.6 or Rem. 2.8). We also assume the following:

  1. $(a)$

$X$ has no rational contraction onto a $3$-fold;

  2. $(b)$

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

  3. $(c)$

$X$ has no fixed prime divisor of type $(3,0)^{sm}$.

Otherwise, the statement follows from Th. 5.8, 4.8, or 4.7, respectively.

######  8.2.

Let $X\dasharrow\widetilde{X}\stackrel{{\scriptstyle\sigma}}{{\to}}Y$ be the
contraction associated to $D$ as in §4.3, 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 $C\subset Y$. We have the following:

  1. $(a^{\prime})$

$Y$ has no rational contraction onto a $3$-fold;

  2. $(b^{\prime})$

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

  3. $(c^{\prime})$

$Y$ has no fixed prime divisor of type $(3,0)^{sm}$.

###### Proof.

The implication $(a)\Rightarrow(a^{\prime})$ is clear. For
$(b)\Rightarrow(b^{\prime})$, let $E\subset Y$ be a fixed prime divisor of
type $(3,2)$, and let us consider its transforms $E_{X}\subset X$ and
$E_{\widetilde{X}}\subset\widetilde{X}$. Then $E_{X}$ is a fixed prime divisor
of type $(3,2)$ by Lemma 4.22, and
$\dim\mathcal{N}_{1}(E_{X},X)=\dim\mathcal{N}_{1}(E_{\widetilde{X}},\widetilde{X})$
by Lemma 4.10, thus $(a)$ yields
$\mathcal{N}_{1}(E_{\widetilde{X}},\widetilde{X})=\mathcal{N}_{1}(\widetilde{X})$.
On the other hand $E=\sigma(E_{\widetilde{X}})$, hence
$\mathcal{N}_{1}(E,Y)=\sigma_{*}(\mathcal{N}_{1}(E_{\widetilde{X}},\widetilde{X}))=\mathcal{N}_{1}(Y)$.

Finally if $Y$ has a fixed prime divisor of type $(3,0)^{sm}$, its transform
in $X$ is a fixed prime divisor of type $(3,0)^{sm}$ by Lemma 4.22, therefore
$(c)\Rightarrow(c^{\prime})$. ∎

######  8.3.

We note that $Y$ is not a product of surfaces, because a Fano $4$-fold
$S_{1}\times S_{2}$ with $\rho_{S_{1}}>1$ has a contraction onto
$\mathbb{P}^{1}\times S_{2}$, contradicting $(a^{\prime})$. Then we have
$\delta_{Y}\leq 1$ by $(a^{\prime})$ and Theorems 2.6 and 2.7.

######  8.4.

Every fixed prime divisor $E$ of type $(3,2)$ in $Y$ meets $C$.

Indeed if $E\cap C=\emptyset$, then $E$ is disjoint from the images of all
exceptional curves in $\widetilde{X}$, by Lemma 4.19, thus $E$ is contained in
the open where the birational map $X\dasharrow Y$ is an isomorphism. Its
transform $E_{X}\subset X$ is a fixed prime divisor of type $(3,2)$ by Lemma
4.22, and it is disjoint from $D$, so that $\mathcal{N}_{1}(E_{X},X)\subseteq
D^{\perp}\subsetneq\mathcal{N}_{1}(X)$ (see Rem. 2.1), contradicting $(b)$.

######  8.5.

We introduce the following notation: if $C$ belongs to a family of lines which
cover a divisor, we denote such divisor by $E_{0}$; by Lemma 3.12 it could be
nef, or fixed of type $(3,2)$; in this last case $C_{E_{0}}\equiv C$.

######  8.6.

If $E\subset Y$ is a fixed prime divisor of type $(3,2)$, then $E\cap
C\neq\emptyset$ by 8.4, and applying [Cas17, Lemma 5.11] we know that either
$C\not\subset E$ and $E\cdot C>0$, or $E=E_{0}$ and $C_{E_{0}}\equiv C$.

######  8.7.

Let $E_{1},\dotsc,E_{r}$ be the fixed prime divisors of $Y$ of type $(3,2)$
with $E_{i}\cdot C>0$, so that by 8.6 the fixed prime divisors of type $(3,2)$
of $Y$ are either $r$ (namely $E_{1},\dotsc,E_{r}$) or $r+1$ (namely
$E_{0},E_{1},\dotsc,E_{r}$).

We show that $E_{i}\cdot C_{E_{j}}=0$ for every $i,j\in\\{1,\dotsc,r\\}$ with
$i\neq j$.

Take for simplicity $i=1$ and $j=2$, and let $\Gamma$ be a curve such that
$\Gamma\equiv C_{E_{2}}$ and $C\cap\Gamma\neq\emptyset$. Then by Rem. 4.16
$\Gamma$ is the image of an exceptional curve of $\widetilde{X}$, hence by
Lemma 4.19 $\Gamma$ cannot intersect any curve of anticanonical degree one
(different from $C$ and from $\Gamma$ itself). Therefore $\Gamma\cap
E_{1}=\emptyset$ and $E_{1}\cdot C_{E_{2}}=0$.

This implies that the classes $[E_{1}],\dotsc,[E_{r}]$ are linearly
independent in $\mathcal{N}^{1}(Y)$, hence $r\leq\rho_{Y}$. In fact
$r<\rho_{Y}$, otherwise given an ample divisor $A$, we can write
$A\equiv\sum_{i=1}^{\rho_{Y}}\lambda_{i}E_{i}$, and $A\cdot
C_{E_{j}}=-\lambda_{j}>0$ for every $j=1,\dotsc,\rho_{Y}$, which gives a
contradiction.

######  8.8.

Suppose that $E_{0}$ is a fixed prime divisor of type $(3,2)$. Then $E_{0}$
and $E_{i}$ are not adjacent, $E_{0}\cdot C_{E_{i}}>0$, and $E_{0}+E_{i}$ is
movable, for every $i=1,\dotsc,r$. This follows from Cor. 4.14, because
$E_{i}\cdot C_{E_{0}}=E_{i}\cdot C>0$.

######  8.9.

Suppose that there exists a movable face $\tau$ of $\operatorname{Eff}(Y)$
such that, if $f\colon Y\dasharrow Z$ is an associated rational contraction of
fiber type as in 5.5, we have $Z\cong\mathbb{P}^{2}$.

Then either $\rho_{X}\leq 10$, or $\tau$ is a facet and the general fiber of
$f$ is $\mathbb{P}^{2}$.

###### Proof.

Let $\zeta\colon\widetilde{Y}\dasharrow Y$ be a SQM such that
$\tilde{f}:=f\circ\zeta\colon\widetilde{Y}\to\mathbb{P}^{2}$ is regular and
$K$-negative (see Lemma 2.5), and let $F\subset\widetilde{Y}$ be a general
fiber of $\tilde{f}$, so that $F$ is a smooth del Pezzo surface. Note that $F$
is contained in the open subset where $\zeta$ is an isomorphism, because the
indeterminacy locus of $\zeta$ has dimension at most one, see Lemma 2.4.

If $\rho_{F}=1$, then $F\cong\mathbb{P}^{2}$ and $\tau$ is a facet by (5.6).
Suppose that $\rho_{F}>1$. Then $F$ is covered by rational curves of
anticanonical degree $2$, so that $\widetilde{Y}$ and $Y$ have a covering
family of rational curves of anticanonical degree $2$. By Lemma 4.21$(b)$ $C$
must be a component of a curve of this family in $Y$; moreover $C$ cannot meet
any exceptional plane (see Rem. 4.16) and thus it is contained in the open
subset where $\zeta^{-1}$ is an isomorphism and $f$ is regular, so that $f(C)$
is a point.

Let us consider now the composition $X\dasharrow\mathbb{P}^{2}$. It contracts