# The Lefschetz defect of Fano varieties

C. Casagrande  Università di Torino, Dipartimento di Matematica, via Carlo
Alberto 10, 10123 Torino - Italy<EMAIL_ADDRESS>
(Date: November 30, 2022)

Smooth, complex Fano varieties are a very natural class of projective
varieties, and have been intensively studied both classically and in recent
times, due to their significance in the framework of the Minimal Model
Program.

The Lefschetz defect is an invariant of smooth Fano varieties that has been
recently introduced in [7]; it is related to the Picard number $\rho_{X}$ of
$X$, and to the Picard number of prime divisors in $X$. This paper is a survey
on this new invariant and its properties: we explain the definition of the
Lefschetz defect $\delta_{X}$ and its origin, the known results, and several
examples, especially in dimensions $3$ and $4$; we do not give new theoretical
results.

In particular, we explain how the results on the Lefschetz defect allow to
recover the classification of Fano $3$-folds with $\rho_{X}\geq 5$, due to
Mori and Mukai. We also review the known examples of Fano $4$-folds with
$\rho_{X}\geq 6$, that are remarkably few: apart products and toric examples,
there are only 6 known families, with $\rho_{X}\in\\{6,7,8,9\\}$.

###### Contents

  1. 1 Glossary
  2. 2 The definition of Lefschetz defect and the case $\delta_{X}\geq 4$
  3. 3 Studying $\operatorname{codim}\mathcal{N}_{1}(D,X)$ via birational geometry
  4. 4 Fano varieties with Lefschetz defect $3$
  5. 5 Fano varieties with Lefschetz defect $2$
  6. 6 An overview of known Fano $4$-folds with $\rho\geq 6$
  7. 7 Special Fano’s: index $>1$ and the toric case
  8. 8 Beyond smooth Fano’s

##  1\. Glossary

We collect here the notation and terminology used in the paper; we suggest the
reader to skip this section, and come back when needed.

Let $X$ be a projective variety.
$\mathcal{N}_{1}(X)$ (respectively $\mathcal{N}^{1}(X)$) is the vector space
of one-cycles (respectively Cartier divisors) in $X$, with coefficients in
$\mathbb{R}$, up to numerical equivalence.
$[\,\text{-}\,]$ stands for the numerical equivalence class of a curve or a
divisor.
$\rho_{X}=\dim\mathcal{N}_{1}(X)=\dim\mathcal{N}^{1}(X)$ is the Picard number
of $X$.
$\operatorname{NE}(X)$ is the cone of effective curves in
$\mathcal{N}_{1}(X)$, an extremal ray $R$ of $\operatorname{NE}(X)$ is a one-
dimensional face. If $D$ is a divisor, $D\cdot R>0$ means $D\cdot\gamma>0$ for
any non-zero class $\gamma\in R$.

A contraction is a surjective morphism with connected fibers $\varphi\colon
X\to Y$ between normal projective varieties; $\varphi$ is elementary if
$\rho_{X}-\rho_{Y}=1$.

A conic bundle is a contraction of fiber type such that every fiber is
isomorphic to a plane conic.

A small modification is a birational map $X\dasharrow X^{\prime}$ that is an
isomorphism in codimension one.

When $X$ is smooth and Fano, or more generally of Fano type,111A smooth
projective variety $X$ is of Fano type if there exists an effective
$\mathbb{Q}$-divisor $\Delta$ such that the pair $(X,\Delta)$ is log Fano,
namely $(X,\Delta)$ has klt singularities and $-(K_{X}+\Delta)$ is ample, see
[27]. then $\operatorname{NE}(X)$ is closed and polyhedral, and there is a
bijection between extremal rays $R$ of $\operatorname{NE}(X)$ and elementary
contractions $\varphi\colon X\to Y$, given by
$R=\operatorname{NE}(X)\cap\ker\varphi_{*}$, where
$\varphi_{*}\colon\mathcal{N}_{1}(X)\to\mathcal{N}_{1}(Y)$ is the pushforward.
This is because varieties of Fano type are Mori dream spaces by [3, Corollary
1.3.2]. We also set $\operatorname{Locus}(R):=\operatorname{Exc}(\varphi)$.

For any closed subset $Z\subset X$ we set
$\mathcal{N}_{1}(Z,X):=\iota_{*}(\mathcal{N}_{1}(Z))$, where $\iota\colon
Z\hookrightarrow X$ is the inclusion and
$\iota_{*}\colon\mathcal{N}_{1}(Z)\to\mathcal{N}_{1}(X)$ is the induced
pushforward.

$\operatorname{Bl}_{m}\mathbb{P}^{2}$ is the blow-up of $\mathbb{P}^{2}$ in
$m$ points in general linear position.

A $\mathbb{P}^{r}$-bundle is the projectivization of a vector bundle.

We work over the field of complex numbers.

##  2\. The definition of Lefschetz defect and the case $\delta_{X}\geq 4$

Let $X$ be a smooth Fano variety. We introduce an invariant of $X$, the
Lefschetz defect, which relates the Picard number of $X$ to that of its prime
divisors.

Given a prime divisor $D$ in $X$, let us consider the natural pushforward
$\iota_{*}\colon\mathcal{N}_{1}(D)\to\mathcal{N}_{1}(X)$ given by the
inclusion $\iota\colon D\hookrightarrow X$, and set

$\mathcal{N}_{1}(D,X):=\iota_{*}(\mathcal{N}_{1}(D))\subseteq\mathcal{N}_{1}(X).$

This is the linear span, in $\mathcal{N}_{1}(X)$, of classes of curves
contained in $D$. The Lefschetz defect of $X$ is defined as:

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

Remarks

  1. 1.

The linear subspace $\mathcal{N}_{1}(D,X)$ is always non-zero, because if
$C\subset D$ is any curve, we have $[C]\neq 0$ in $\mathcal{N}_{1}(X)$.
Therefore $\delta_{X}\in\\{0,\dotsc,\rho_{X}-1\\}$; in particular we see that
$\delta_{X}=0$ whenever $\rho_{X}=1$.

  2. 2.

When $\dim X=2$, a prime divisor $D\subset X$ is an irreducible curve, and
$\mathcal{N}_{1}(D,X)=\mathbb{R}[D]$, thus $\delta_{X}=\rho_{X}-1$.

  3. 3.

We can consider the dual picture: let $D\subset X$ be a prime divisor, and
consider the restriction $r_{D}\colon H^{2}(X,\mathbb{R})\to
H^{2}(D,\mathbb{R})$; then we have
$\operatorname{codim}\mathcal{N}_{1}(D,X)=\dim\ker r_{D}$.

Indeed there is a commutative diagram:

${\mathcal{N}^{1}(X)}$${{\mathcal{N}^{1}(D)}}$${{H^{2}(X,\mathbb{R})}}$${{H^{2}(D,\mathbb{R})}}$$\scriptstyle{\tilde{r}_{D}}$$\scriptstyle{\wr}$$\scriptstyle{r_{D}}$

where $\tilde{r}_{D}$ is dual to $\iota_{*}$, and $\mathcal{N}^{1}(X)\cong
H^{2}(X,\mathbb{R})$ because $X$ is Fano.

  4. 4.

One can also define the Lefschetz defect as:

$\delta_{X}=\max\bigl{\\{}\dim\ker\bigl{(}r_{D}\colon H^{2}(X,\mathbb{R})\to
H^{2}(D,\mathbb{R})\bigr{)}\,|\,D\text{ an effective divisor in }X\bigr{\\}}$

(where with a slight abuse of notation we identify $D$ with its support).

Indeed let $D$ be an effective divisor, and $D_{1}$ an irreducible component
of $D$. Then $r_{D_{1}}$ factors as $H^{2}(X,\mathbb{R})\to
H^{2}(D,\mathbb{R})\to H^{2}(D_{1},\mathbb{R})$, so that $\ker
r_{D}\subseteq\ker r_{D_{1}}$, and the maximum above is always attained for
some prime divisor.

If $\dim X\geq 3$ and $D$ is ample, then $r_{D}$ is injective by Lefschetz’s
theorem on hyperplane sections, so that the Lefschetz defect depends on non-
ample effective divisors, and measures the failure of this instance of
Lefschetz’s theorem.

######  Lemma 5 (the Lefschetz defect of a product variety).

If $X\cong S\times T$, then $\delta_{X}=\max\\{\delta_{S},\delta_{T}\\}$.

In particular, if $S$ is a surface, then
$\delta_{X}=\max\\{\rho_{S}-1,\delta_{T}\\}$.

###### Proof.

We can assume that $\delta_{S}\geq\delta_{T}$. Let $D_{S}\subset S$ be a prime
divisor with $\operatorname{codim}\mathcal{N}_{1}(D_{S},S)=\delta_{S}$. Then
$\operatorname{codim}\mathcal{N}_{1}(D_{S}\times T,X)=\delta_{S}$, so that
$\delta_{X}\geq\delta_{S}$.

Conversely, let $D\subset X$ be a prime divisor, and $\pi\colon X\to S$ the
projection.

If $\pi(D)\subsetneq S$, then $D=\pi(D)\times T$ and
$\operatorname{codim}\mathcal{N}_{1}(D,X)=\operatorname{codim}\mathcal{N}_{1}(\pi(D),S)\leq\delta_{S}$.

Suppose instead that $\pi(D)=S$, and consider the pushforward
$\pi_{*}\colon\mathcal{N}_{1}(X)\to\mathcal{N}_{1}(S)$. Then we have
$\pi_{*}(\mathcal{N}_{1}(D,X))=\mathcal{N}_{1}(S)$, and hence
$\dim\mathcal{N}_{1}(D,X)=\rho_{S}+\dim(\mathcal{N}_{1}(D,X)\cap\ker\pi_{*})$.

Let $p\in S$; we have a natural identification
$\mathcal{N}_{1}(T)\cong\mathcal{N}_{1}(\pi^{-1}(p),X)=\ker\pi_{*}$. Moreover
$\pi^{-1}(p)\cap D$ is either $T$ or a divisor in $T$; in both cases
$\dim(\mathcal{N}_{1}(D,X)\cap\ker\pi_{*})\geq\dim\mathcal{N}_{1}(\pi^{-1}(p)\cap
D,T)\geq\rho_{T}-\delta_{T}\geq\rho_{T}-\delta_{S}$, so that
$\dim\mathcal{N}_{1}(D,X)\geq\rho_{X}-\delta_{S}$, and we conclude that
$\delta_{X}=\delta_{S}$.

Finally if $\dim S=2$ we have $\delta_{S}=\rho_{S}-1$ by Remark 2. ∎

######  Example 6 ($\delta_{X}$ and disjoint divisors).

Let $D,E_{1},\dotsc,E_{r}\subset X$ be prime divisors, and assume that
$D\cap(E_{1}\cup\cdots\cup E_{r})=\emptyset$. If the classes
$[E_{1}],\dotsc,[E_{r}]\in\mathcal{N}^{1}(X)$ are linearly independent, then
$\delta_{X}\geq\operatorname{codim}\mathcal{N}_{1}(D,X)\geq r$.

Indeed for every curve $C\subset D$ we have $E_{i}\cdot C=0$ for every
$i=1,\dotsc,r$, so that $[C]\in E_{1}^{\perp}\cap\cdots\cap E_{r}^{\perp}$
(where $E_{i}^{\perp}\subset\mathcal{N}_{1}(X)$ is the hyperplane of classes
orthogonal to $E_{i}$), and finally $\mathcal{N}_{1}(D,X)\subseteq
E_{1}^{\perp}\cap\cdots\cap E_{r}^{\perp}$.

For instance, as soon as $X$ contains two disjoint prime divisors, we have
$\delta_{X}>0$.

The next example concerns the case $\dim X=3$, and shows how birational
geometry can be used to find prime divisors $D$ with small
$\dim\mathcal{N}_{1}(D,X)$.

######  Example 7 ([17], Lemma 5.1).

Let $X$ be a Fano $3$-fold. We show that $X$ always contains a prime divisor
$D$ with $\dim\mathcal{N}_{1}(D,X)\leq 2$, so that
$\delta_{X}\in\\{\rho_{X}-2,\rho_{X}-1\\}$.

Let $\varphi\colon X\to Y$ be an elementary contraction. It is well-known that
$\varphi$ cannot be small: it is either divisorial, or of fiber type. In any
case there exists a prime divisor $D\subset X$ such that $\dim\varphi(D)\leq
1$.

Consider the pushforward
$\varphi_{*}\colon\mathcal{N}_{1}(X)\to\mathcal{N}_{1}(Y)$; this is a
surjective linear map with one-dimensional kernel. We have
$\varphi_{*}(\mathcal{N}_{1}(D,X))=\mathcal{N}_{1}(\varphi(D),Y)$; on the
other hand $\mathcal{N}_{1}(\varphi(D),Y)=\\{0\\}$ if $\varphi(D)=\\{pt\\}$,
and $\mathcal{N}_{1}(\varphi(D),Y)=\mathbb{R}[\varphi(D)]$ if $\varphi(D)$ is
a curve. In any case we have $\dim\mathcal{N}_{1}(\varphi(D),Y)\leq 1$ and
hence $\dim\mathcal{N}_{1}(D,X)\leq 2$.

The following is the main property of the Lefschetz defect.

######  Theorem 8 ([7], Th. 3.3).

For every smooth Fano variety $X$ we have $\delta_{X}\in\\{0,\dotsc,8\\}$.

Moreover, if $\delta_{X}\geq 4$, then $X\cong S\times T$ where $S$ is a del
Pezzo surface with $\rho_{S}=\delta_{X}+1$, and $\delta_{T}\leq\delta_{X}$.

Thus, if a Fano variety $X$ is not a product, then
$\delta_{X}\in\\{0,1,2,3\\}$.

######  Remark 9 (bounding the Picard number of $X$).

Let $X$ be a smooth Fano variety and $D\subset X$ a prime divisor. We have
$\dim\mathcal{N}_{1}(D,X)\leq\rho_{D}$ and hence
$\delta_{X}\geq\rho_{X}-\rho_{D}$. Thus Theorem 8 implies that
$\rho_{X}\leq\rho_{D}+8$, and if $\rho_{X}\geq\rho_{D}+4$, then $X$ is a
product.

Note however that this result does not immediately give an inductive bound on
$\rho_{X}$ in terms of the dimension of $X$, because we do not know whether
$X$ contains a prime divisor $D$ which is smooth and Fano.

It is indeed expected that $\rho_{X}\leq\frac{9}{2}\dim X$; this is attained
in every even dimension $2m$ by products $X=S_{1}\times\cdots\times S_{m}$ of
degree one del Pezzo surfaces, which have $\rho_{X}=9m$ and $\delta_{X}=8$. We
will discuss the maximal Picard number for Fano $3$-folds and $4$-folds in
Corollary 14 and §6 respectively.

######  Remark 10 (the origin of the notion of Lefschetz defect).

Bonavero, Campana, and Wiśniewski [4] have classified smooth Fano varieties
obtained by blowing-up a point in a smooth variety; this has been later
generalized by Tsukioka [31] to smooth Fano varieties containing a divisor
$D\cong\mathbb{P}^{n-1}$ with negative normal bundle. In both papers, the main
strategy is to consider the divisor $D\cong\mathbb{P}^{n-1}$, an extremal ray
$R$ of $\operatorname{NE}(X)$ such that $D\cdot R>0$, and the associated
elementary contraction $\varphi\colon X\to Y$.

The pushforward $\varphi_{*}\colon\mathcal{N}_{1}(X)\to\mathcal{N}_{1}(Y)$ is
a surjective linear map with one-dimensional kernel. Since
$D\cong\mathbb{P}^{n-1}$, the space $\mathcal{N}_{1}(D,X)$ is one-dimensional,
and we have two possibilities: either $\ker\varphi_{*}=\mathcal{N}_{1}(D,X)$
and $\varphi(D)=\\{pt\\}$, or
$\ker\varphi_{*}\cap\mathcal{N}_{1}(D,X)=\\{0\\}$ and $\varphi$ is finite on
$D$. In this last case, if $F\subset X$ is a non-trivial fiber of $\varphi$,
we have $D\cap F\neq\emptyset$ (because $D$ has positive intersection with
every curve contracted by $\varphi$) and $\dim(D\cap F)=0$, so that $\dim
F=1$. On the other hand, elementary contractions of smooth Fano varieties
having only fibers of dimension $\leq 1$ are classified [1, 32]: they are
either a conic bundle, or a blow-up of a smooth, codimension $2$ subvariety in
a smooth variety.

In this way one gets a very special elementary contraction of $X$, and this
argument may be iterated in the target $Y$ of $\varphi$. Here the relevant
property is that $\dim\mathcal{N}_{1}(D,X)=1$, more than
$D\cong\mathbb{P}^{n-1}$.

Later, in [6], the author has studied Fano varieties having a birational
elementary contraction $f\colon X\to X^{\prime}$ sending the exceptional
divisor $E$ to a curve (for instance the blow-up of a smooth curve in
$X^{\prime}$). It is not difficult to see that $\dim\mathcal{N}_{1}(E,X)=2$ in
this case (see Example 7), and prime divisors $D$ with
$\dim\mathcal{N}_{1}(D,X)=2$ play a special role in [6]. This assumption is
still applied by studying extremal rays $R$ such that $D\cdot R>0$, and the
associated elementary contractions.

Finally this led the author to generalize this strategy in [7], using
birational geometry and the MMP. We give an overview in the following section.

##  3\. Studying $\operatorname{codim}\mathcal{N}_{1}(D,X)$ via birational
geometry

Let $X$ be a smooth Fano variety and $D\subset X$ a prime divisor. In this
section we explain a construction, based on birational geometry, which allows
to get information on $X$ as soon as
$\operatorname{codim}\mathcal{N}_{1}(D,X)>0$. We refer the reader to [20, 18]
for the terminology in birational geometry and Mori dream spaces, and to [7,
§2] for more details.

Since Fano varieties are Mori dream spaces [3, Corollary 1.3.2], one can run a
MMP for any divisor [7, Proposition 2.2]. We consider here a MMP for $-D$,
which means that we consider _extremal rays having positive intersection with
$D$_, as in Remark 10. Since $-D$ can never become semiample in the MMP, the
program must end with a fiber type contraction. This means that we get a
sequence

$X=X_{0}\stackrel{{\scriptstyle\sigma_{0}}}{{\dasharrow}}X_{1}\dasharrow\quad\cdots\quad\dasharrow
X_{k-1}\stackrel{{\scriptstyle\sigma_{k-1}}}{{\dasharrow}}X_{k}\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}Y$

such that:

  1. $\bullet$

every $X_{i}$ is a normal and $\mathbb{Q}$-factorial projective variety, and
the transform $D_{i}\subset X_{i}$ of $D$ is a prime divisor in $X_{i}$;

  2. $\bullet$

for every $i=0,\dotsc,k-1$ there exists an extremal ray $R_{i}$ of $X_{i}$
such that $D_{i}\cdot R_{i}>0$, $\operatorname{Locus}(R_{i})\subsetneq X_{i}$,
and $\sigma_{i}$ is either the contraction of $R_{i}$ (if $R_{i}$ is
divisorial), or its flip (if $R_{i}$ is small);

  3. $\bullet$

there exists an extremal ray $R_{k}$ in $X_{k}$, with a fiber type contraction
$\varphi\colon X_{k}\to Y$, such that $D_{k}\cdot R_{k}>0$.

Moreover, since $X$ is Fano, using an MMP with scaling of $-K_{X}$, in the
sequence above we can also assume that $-K_{X_{i}}\cdot R_{i}>0$ for every
$i=0,\dotsc,k$ [7, Proposition 2.4].

Then we analyse how $\operatorname{codim}\mathcal{N}_{1}(D_{i},X_{i})$ varies
along the steps of the MMP. It is not difficult to show that, for every
$i=0,\dotsc,k-1$, we have:

$\operatorname{codim}\mathcal{N}_{1}(D_{i+1},X_{i+1})=\begin{cases}\operatorname{codim}\mathcal{N}_{1}(D_{i},X_{i})\quad&\text{if
$R_{i}\subset\mathcal{N}_{1}(D_{i},X_{i})$,}\\\
\operatorname{codim}\mathcal{N}_{1}(D_{i},X_{i})-1\quad&\text{if
$R_{i}\not\subset\mathcal{N}_{1}(D_{i},X_{i})$}\end{cases}$

(see [7, Lemma 2.6(3)]).

Let us consider now the last step, given by the fiber type contraction
$\varphi\colon X_{k}\to Y$. Since $D_{k}\cdot R_{k}>0$, the prime divisor
$D_{k}\subset X_{k}$ intersects every fiber of $\varphi$, namely
$\varphi(D_{k})=Y$. Let
$\varphi_{*}\colon\mathcal{N}_{1}(X_{k})\to\mathcal{N}_{1}(Y)$ be the
pushforward; then
$\varphi_{*}(\mathcal{N}_{1}(D_{k},X_{k}))=\mathcal{N}_{1}(\varphi(D_{k}),Y)=\mathcal{N}_{1}(Y)$.
On the other hand $\varphi$ is elementary, thus $\dim\ker\varphi_{*}=1$, and
we get:

$\operatorname{codim}\mathcal{N}_{1}(D_{k},X_{k})\leq 1.$

In conclusion: _we have at least $\operatorname{codim}\mathcal{N}_{1}(D,X)-1$
steps where $\operatorname{codim}\mathcal{N}_{1}(D_{i},X_{i})$ drops, namely
where $R_{i}\not\subset\mathcal{N}_{1}(D_{i},X_{i})$._ These steps are very
special, let us describe them.

Fix $i\in\\{0,\dotsc,k-1\\}$ such that
$R_{i}\not\subset\mathcal{N}_{1}(D_{i},X_{i})$, and let $\psi\colon X_{i}\to
Z$ be the associated elementary birational contraction (divisorial or small).

Let $F\subset X_{i}$ be a non-trivial fiber of $\psi$. We have $D_{i}\cdot
R_{i}>0$, thus $D_{i}$ has positive intersection with every curve contracted
by $\psi$, so that $D_{i}\cap F\neq\emptyset$. On the other hand, a curve
$C\subset D_{i}\cap F$ would have class
$[C]\in\mathcal{N}_{1}(D_{i},X_{i})\cap R_{i}$, which is impossible because
$R_{i}\not\subset\mathcal{N}_{1}(D_{i},X_{i})$ by assumption. We conclude that
$\dim(D_{i}\cap F)=0$ and $\dim F=1$.

It turns out that since $-K_{X_{i}}\cdot R_{i}>0$, the property of having
fibers of dimension $\leq 1$ makes $\psi$ very special. If $X_{i}$ is smooth,
then [1, 32] yield that $\psi$ is divisorial (so that $\psi=\sigma_{i}\colon
X_{i}\to X_{i+1}$) and is the blow-up of a smooth, codimension $2$ subvariety.

In general $X_{i}$ is singular, but a careful analysis shows that
$\psi=\sigma_{i}\colon X_{i}\to X_{i+1}$ is a divisorial contraction, that
$\operatorname{Exc}(\sigma_{i})$ is contained in the open subset where the
birational map $X\dasharrow X_{i}$ is an isomorphism (in particular, it is
contained in the smooth locus of $X_{i}$), and that $\sigma_{i}$ is the blow-
up of a smooth, codimension $2$ subvariety contained in the smooth locus of
$X_{i+1}$ [7, Lemma 2.7(1)]. For this argument, it is crucial to have the
assumption that $-K_{X_{j}}\cdot R_{j}>0$ for every $j$.

Let $E_{i}\subset X$ be the transform of
$\operatorname{Exc}(\sigma_{i})\subset X_{i}$. Then $E_{i}$ is a smooth prime
divisor, with a $\mathbb{P}^{1}$-bundle structure, such that if $f_{i}\subset
E_{i}$ is a fiber, we have $E_{i}\cdot f_{i}=-1$, $D\cdot f_{i}>0$, and
$[f_{i}]\not\in\mathcal{N}_{1}(D,X)$ [7, Lemma 2.7(3)].

In the end we get the following.

######  Proposition 11 ([7], Proposition 2.5).

Let $X$ be a smooth Fano variety and $D\subset X$ a prime divisor with
$\operatorname{codim}\mathcal{N}_{1}(D,X)>0$.

Then there exist pairwise disjoint smooth prime divisors
$E_{1},\dotsc,E_{s}\subset X$, with
$s=\operatorname{codim}\mathcal{N}_{1}(D,X)-1$, such that every $E_{j}$ is a
$\mathbb{P}^{1}$-bundle with $E_{j}\cdot f_{j}=-1$, where $f_{j}\subset E_{j}$
is a fiber; moreover $D\cdot f_{j}>0$ and
$[f_{j}]\not\in\mathcal{N}_{1}(D,X)$. In particular $E_{j}\cap D\neq\emptyset$
and $E_{j}\neq D$.

This construction is the starting point of the proof of Theorem 8. The rest of
the proof [7, §3] is very technical and we will not enter into it.

Let us give now an application to Fano varieties with maximal Lefschetz
defect. We have seen in Remark 1 that $\delta_{X}\leq\rho_{X}-1$; the
condition $\delta_{X}=\rho_{X}-1$ is equivalent to asking that $X$ contains a
prime divisor $D$ with $\dim\mathcal{N}_{1}(D,X)=1$, for instance a prime
divisor $D$ with $\rho_{D}=1$. This implies that $\rho_{X}\leq 9$ by Theorem
8, but when $\dim X\geq 3$, the stronger bound $\rho_{X}\leq 3$ holds. This
was originally proved in [31]; we give a different proof using the previous
proposition.

######  Proposition 12.

Let $X$ be a smooth Fano variety with $\dim X\geq 3$ and
$\delta_{X}=\rho_{X}-1$. Then $\rho_{X}\leq 3$.

###### Proof.

By contradiction, suppose that $\rho_{X}\geq 4$, and let $D\subset X$ be a
prime divisor with $\dim\mathcal{N}_{1}(D,X)=1$. By Proposition 11, there
exist $s=\rho_{X}-2\geq 2$ pairwise disjoint prime divisors
$E_{1},\dotsc,E_{s}$, all intersecting $D$, and distinct from $D$.

Since $\dim X\geq 3$, we have $\dim(D\cap E_{1})\geq 1$; let $C\subset D\cap
E_{1}$ be an irreducible curve. We have $E_{2}\cdot C=0$ because $E_{1}\cap
E_{2}=\emptyset$. On the other hand $\dim\mathcal{N}_{1}(D,X)=1$, hence for
every irreducible curve $C^{\prime}\subset D$, there exists some
$\lambda\in\mathbb{Q}$ such that $[C^{\prime}]=\lambda[C]$, thus $E_{2}\cdot
C^{\prime}=0$. Finally $E_{2}$ meets $D$, $E_{2}\neq D$, and $E_{2}$ has
intersection zero with every curve in $D$, which is impossible. ∎

The boundary case where $\rho_{X}=3$ and $\delta_{X}=2$ is described by a
structure theorem [14, Th. 3.8]. In particular, Fano $4$-folds with
$\rho_{X}=3$ and $\delta_{X}=2$ are classified and studied in detail in [30];
they form 28 families, and only 3 of them are toric.

We conclude this section by considering the case of Fano $3$-folds. Example 7
and Proposition 12 immediately imply the following.

######  Lemma 13.

Let $X$ be a Fano 3-fold. Then $\delta_{X}\in\\{\rho_{X}-2,\rho_{X}-1\\}$, and
$\delta_{X}=\rho_{X}-2$ as soon as $\rho_{X}\geq 4$.

Table 1. The Lefschetz defect of Fano $3$-folds
$\begin{array}[]{||c|c||}\hline\cr\hline\cr\rho=1&\delta=0\\\
\hline\cr\rho=2&\delta\in\\{0,1\\}\\\ \hline\cr\rho=3&\delta\in\\{1,2\\}\\\
\hline\cr\rho\geq 4&\delta=\rho-2\\\ \hline\cr\hline\cr\end{array}$

######  Corollary 14.

Let $X$ be a Fano 3-fold with $\rho_{X}\geq 6$. Then $X\cong
S\times\mathbb{P}^{1}$ where $S$ is a del Pezzo surface. In particular
$\rho_{X}\leq 10$.

This corollary was originally proved by Mori and Mukai [24, Theorem 1.2], via
an explicit study of conic bundle structures on Fano 3-folds, and it was the
conclusion of their classification of Fano $3$-folds with $\rho_{X}\geq 2$.

###### Proof.

Since $\rho_{X}\geq 6$, by Lemma 13 we have $\delta_{X}=\rho_{X}-2\geq 4$, and
Theorem 8 yields the statement. ∎

In the next section (Corollary 19) we will also recover the classification of
Fano $3$-folds with $\rho_{X}=5$.

##  4\. Fano varieties with Lefschetz defect $3$

The bound $\delta_{X}\geq 4$ in Theorem 8 is sharp, as in every dimension
$\geq 3$ there are Fano varieties with Lefschetz defect $3$ that are not
products. On the other hand, the case $\delta_{X}=3$ is still very special: it
was first partially studied in [7, 15], and then completely classified in
[16]. Let us now describe this classification, which gives an explicit
geometrical construction for Fano varieties with $\delta_{X}=3$.

######  Construction 15.

We start with a smooth Fano variety $T$ with $\delta_{T}\leq 3$, and
$L_{1},L_{2},L_{3}\in\operatorname{Pic}(T)$ such that:

(1) $-K_{T}+L_{i}-L_{j}\text{ is ample on $T$ for every }i,j=1,2,3.$

We consider the $\mathbb{P}^{2}$-bundle:

$Z:=\mathbb{P}_{T}(L_{1}\oplus L_{2}\oplus
L_{3})\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}T,$

and let $S_{2},S_{3}\subset Z$ be the sections of $\varphi$ corresponding to
the projections $\oplus_{j=1}^{3}L_{j}\twoheadrightarrow L_{2}$ and
$\oplus_{j=1}^{3}L_{j}\twoheadrightarrow L_{3}$ respectively.

The construction has two variants, that we call $A$ and $B$, as follows:

  1. case $A$:

let $S_{1}\subset Z$ be the section of $\varphi$ corresponding to the
projection $\oplus_{j=1}^{3}L_{j}\twoheadrightarrow L_{1}$.

  2. case $B$:

we set $L_{2}=L_{3}=\mathcal{O}_{T}$, and assume that
$L_{1}\not\cong\mathcal{O}_{T}$. Let $H$ be the tautological line bundle of
$Z$. We assume that the complete intersection $S_{1}$ of a general element in
the linear system $|H|$ and one in $|2H|$ is smooth.222For instance, if
$L_{1}$ is globally generated, then the vector bundle
$L_{1}\oplus\mathcal{O}_{T}\oplus\mathcal{O}_{T}$ is globally generated, thus
$H$ is too, and by Bertini $S_{1}$ is smooth.

######  Lemma 16 ([16], Rem. 4.2 and 4.3).

The subvarieties $S_{1},S_{2},S_{3}$ are smooth, irreducible, of codimension
2, and pairwise disjoint.

In case B, $\varphi_{|S_{1}}\colon S_{1}\to T$ is finite of degree $2$, and
$h^{0}(T,2L_{1})>0$.

Finally let $X\to Z$ be the blow-up of $S_{1},S_{2},S_{3}$, so that $X$ is a
smooth projective variety with $\dim X=\dim T+2$ and $\rho_{X}=\rho_{T}+4$.

######  Theorem 17 ([16], Prop. 1.2 and 1.3, Th. 1.4).

The variety $X$ is Fano with $\delta_{X}=3$. Moreover every smooth Fano
variety with Lefschetz defect $3$ is obtained in this way.

It is easy to exhibit a prime divisor in $X$ which realizes $\delta_{X}=3$, by
looking at the exceptional divisors of the blow-up $X\to Z$. Recall that
$S_{2}$ is a section of $\varphi$, so that $S_{2}\cong T$. Let $E\subset X$ be
the exceptional divisor over $S_{2}$; then $E$ is a $\mathbb{P}^{1}$-bundle
over $S_{2}$ and $\rho_{E}=1+\rho_{T}=\rho_{X}-3$. One can check that the
natural map $\mathcal{N}_{1}(E)\to\mathcal{N}_{1}(X)$ is injective, hence
$\operatorname{codim}\mathcal{N}_{1}(E,X)=3$.

Given $T$, it is not difficult to classify the possible choices of $L_{i}$’s
as in cases $A$ and $B$ above; as an example, we describe the case
$T=\mathbb{P}^{1}$.

######  Example 18 (the $3$-dimensional case).

Let $X$ be a Fano $3$-fold. We assume that $\delta_{X}=3$, equivalently that
$\rho_{X}=5$ (see Table 1 on page 1).

By Theorem 17, $X$ is obtained with Construction 15 from a Fano variety $T$ of
dimension $\dim X-2=1$, so that $T=\mathbb{P}^{1}$.

We set $L_{i}:=\mathcal{O}_{\mathbb{P}^{1}}(a_{i})$ for $i=1,2,3$.

In case $A$, the $L_{i}$’s are determined up to shift and permutation, so that
we can assume $a_{3}=0$ and $a_{1}\geq a_{2}\geq 0$. Condition (1) yields
$a_{i}\leq 1$ for $i=1,2$, so that for $(a_{1},a_{2})$ we have the
possibilities $(0,0),(1,0),(1,1)$.

On the other hand, in case $A$ we also have a commutative diagram:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{P}_{T}((-L_{1})\oplus(-L_{2})\oplus(-L_{3}))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z=\mathbb{P}_{T}(L_{1}\oplus
L_{2}\oplus
L_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T}$

(see [16, Lemma 6.3]), which fiberwise over $T$ is just the standard Cremona
transformation in $\mathbb{P}^{2}$. This means that the choice
$L_{1}=L_{2}=\mathcal{O}_{\mathbb{P}^{1}}(1),L_{3}=\mathcal{O}_{\mathbb{P}^{1}}$
yields the same Fano $3$-fold $X$ as
$L_{1}=L_{2}=\mathcal{O}_{\mathbb{P}^{1}}(-1),L_{3}=\mathcal{O}_{\mathbb{P}^{1}}$
and hence as
$L_{1}=\mathcal{O}_{\mathbb{P}^{1}}(1),L_{2}=L_{3}=\mathcal{O}_{\mathbb{P}^{1}}$.

In conclusion, case $A$ yields two distinct Fano $3$-folds. The trivial choice
$L_{1}=L_{2}=L_{3}=\mathcal{O}_{\mathbb{P}^{1}}$ yields
$Z=\mathbb{P}^{2}\times\mathbb{P}^{1}$ and
$X=(\operatorname{Bl}_{3}\mathbb{P}^{2})\times\mathbb{P}^{1}$.

The choice
$L_{1}=\mathcal{O}_{\mathbb{P}^{1}}(1),L_{2}=L_{3}=\mathcal{O}_{\mathbb{P}^{1}}$
yields
$Z=\mathbb{P}_{\mathbb{P}^{1}}(\mathcal{O}(1)\oplus\mathcal{O}\oplus\mathcal{O})$,
the blow-up of $\mathbb{P}^{3}$ along a line, and $X$ is the blow-up of $Z$
along three curves $S_{1},S_{2},S_{3}$, where $S_{1}$ is the transform of a
general line in $\mathbb{P}^{3}$, while $S_{2}$ and $S_{3}$ are non-trivial
fibers of the blow-up $Z\to\mathbb{P}^{3}$.

In case $B$ we have $a_{2}=a_{3}=0$, $|a_{1}|\leq 1$ by condition (1),
$a_{1}\neq 0$ because $L_{1}\not\cong\mathcal{O}_{T}$ by assumption, and
finally $a_{1}>0$ because $2L_{1}$ is effective by Lemma 16; thus the only
possibility is $a_{1}=1$, and again
$Z=\mathbb{P}_{\mathbb{P}^{1}}(\mathcal{O}(1)\oplus\mathcal{O}\oplus\mathcal{O})$.
We note that $L_{1}=\mathcal{O}_{\mathbb{P}^{1}}(1)$ is globally generated, so
that the additional assumption for case $B$ is fulfilled. Then $X$ is the
blow-up of $Z$ along two curves $S_{2},S_{3}$, which as before are non-trivial
fibers of the blow-up $Z\to\mathbb{P}^{3}$, and the transform of a general
conic in $\mathbb{P}^{3}$.

We have recovered the classification of Fano $3$-folds with $\rho_{X}=5$ [19,
Table 12.6], as follows.

######  Corollary 19.

There are three Fano $3$-folds with $\rho_{X}=5$, they have $\delta_{X}=3$ and
are the following:

  1. 1)

$X=(\operatorname{Bl}_{3}\mathbb{P}^{2})\times\mathbb{P}^{1}$.

This is [19, Table 12.6, No. 3], and it is obtained with Construction 15,
$T=\mathbb{P}^{1}$, case $A$, $L_{1}=L_{2}=L_{3}=\mathcal{O}$.

  2. 2)

$X$ is the blow-up of $\mathbb{P}^{3}$ along two skew lines and along two non-
trivial fibers of the blow-up of one of the lines.

This is [19, Table 12.6, No. 2], and is obtained with Construction 15,
$T=\mathbb{P}^{1}$, case $A$, $L_{1}=\mathcal{O}(1)$,
$L_{2}=L_{3}=\mathcal{O}$.

  3. 3)

$X$ is the blow-up of $\mathbb{P}^{3}$ along a line and a conic (disjoint from
each other), and along two non-trivial fibers of the blow-up of the line.

This is [19, Table 12.6, No. 1], and is obtained with Construction 15,
$T=\mathbb{P}^{1}$, case $B$, $L_{1}=\mathcal{O}(1)$.

In a similar way, Theorem 17 yields the classification of Fano $4$-folds with
$\delta_{X}=3$. Now $T$ is a del Pezzo surface with $\delta_{T}\leq 3$; on the
other hand $\rho_{T}=\delta_{T}+1$ (see Remark 2), so that $1\leq\rho_{T}\leq
4$ and $5\leq\rho_{X}\leq 8$. Here we sum up the outcome.

######  Proposition 20 ([15] and [16], Prop. 1.5).

Let $X$ be a Fano $4$-fold with $\delta_{X}=3$. Then $5\leq\rho_{X}\leq 8$ and
there are 19 families for $X$, among which 14 are toric. More precisely:

  1. –

if $\rho_{X}=8$, then
$X\cong\operatorname{Bl}_{3}\mathbb{P}^{2}\times\operatorname{Bl}_{3}\mathbb{P}^{2}$;

  2. –

if $\rho_{X}=7$, then
$X\cong\operatorname{Bl}_{3}\mathbb{P}^{2}\times\operatorname{Bl}_{2}\mathbb{P}^{2}$;

  3. –

if $\rho_{X}=6$, there are 11 families for $X$, among which 8 are toric, and
one is $\mathbb{P}^{1}\times Y$, $Y$ the non-toric Fano $3$-fold with
$\rho_{Y}=5$;

  4. –

if $\rho_{X}=5$, there are 6 families for $X$, among which 4 are toric.

More generally, given the classification of $(n-2)$-dimensional Fano
varieties, Theorem 17 allows – in principle – to completely classify
$n$-dimensional Fano varieties with Lefschetz defect $3$.

##  5\. Fano varieties with Lefschetz defect $2$

After Theorems 8 and 17, the structure of Fano varieties with $\delta_{X}\geq
3$ is well understood. We consider in this section the next case,
$\delta_{X}=2$, studied in [10]. The following result shows that $X$ still has
some interesting properties.

######  Theorem 21 ([10], Th. 1.2).

Let $X$ be a smooth Fano variety with $\delta_{X}=2$. Then one of the
following holds.

  1. $(i)$

There exist a small modification $X\dasharrow X^{\prime}$ and a conic bundle
$X^{\prime}\to Y$ where $X^{\prime}$ and $Y$ are smooth, and
$\rho_{X}-\rho_{Y}=2$.

  2. $(ii)$

There is an equidimensional fibration in del Pezzo surfaces $\psi\colon X\to
T$, where $T$ has locally factorial, canonical singularities,
$\operatorname{codim}\operatorname{Sing}(T)\geq 3$, and $\rho_{X}-\rho_{T}=3$.

It is easy to find examples (among toric Fano varieties) of $X$ as in Theorem
21$(i)$ where a birational modification is necessary, namely $X$ itself does
not have a conic bundle structure, nor a fibration in del Pezzo surfaces; for
instance the two toric Fano $4$-folds of combinatorial type $J$ in Batyrev’s
classification of toric Fano $4$-folds [2, 3.2.7].

We do not know whether the above result is optimal; indeed, based on examples,
it may very well be that $(i)$ holds for every Fano variety with
$\delta_{X}=2$. The author believes that the case $\delta_{X}=2$ has yet to be
fully studied and understood.

######  Example 22 (Fano $3$-folds with $\delta_{X}=2$).

Let $X$ be a Fano $3$-fold with $\delta_{X}=2$; we have
$\rho_{X}\in\\{3,4\\}$, see Table 1 on page 1.

If $\rho_{X}=3$, then $\delta_{X}=\rho_{X}-1$ is maximal, and the structure
theorem [14, Theorem 3.8] yields all the possibilities for $X$; in particular
there exists a conic bundle $X\to\mathbb{P}^{2}$. These are 6 among the 31
families of Fano $3$-folds with $\rho_{X}=3$ (see [19, Table 12.4]), the
remaining ones have $\delta_{X}=1$.

Again by Table 1, all Fano $3$-folds with $\rho_{X}=4$ have $\delta_{X}=2$;
there are 13 families [19, Table 12.5], [25]. By [19, Theorem on p. 141,
Proposition 7.1.11], there is a conic bundle $X\to Y$ where $Y$ is a smooth
del Pezzo surface. It is not difficult to see that $Y\not\cong\mathbb{P}^{2}$,
because the discriminant curve should have at least two connected components,
so that $\rho_{Y}\in\\{2,3\\}$. If $\rho_{Y}=3$, then
$Y\cong\text{Bl}_{2}\mathbb{P}^{2}$ and
$X\cong\mathbb{P}^{1}\times\operatorname{Bl}_{2}\mathbb{P}^{2}$ by [19,
Theorem 7.1.15], so that we can replace $X\to Y$ with the conic bundle
$X\to\mathbb{P}^{1}\times\mathbb{F}_{1}\to\mathbb{P}^{1}\times\mathbb{P}^{1}$.
In any case we get a conic bundle $X\to Y$ with $\rho_{Y}=2$.

We summarize this example in the following table.

Table 2. Fano $3$-folds with $\delta=2$
$\begin{array}[]{||c|c|c||}\hline\cr\hline\cr\rho=3&\text{6
families}&\exists\text{ conic bundle }X\to\mathbb{P}^{2}\\\
\hline\cr\rho=4&\text{13 families}&\exists\text{ conic bundle
}X\to\mathbb{P}^{1}\times\mathbb{P}^{1}\text{ or }X\to\mathbb{F}_{1}\\\
\hline\cr\hline\cr\end{array}$

We conclude that all Fano $3$-folds with $\delta_{X}=2$ have a conic bundle
$X\to Y$ with $\rho_{X}-\rho_{Y}=2$, hence satisfy $(i)$ of Theorem 21.

In the $4$-dimensional case, we have a more refined version of Theorem 21.

######  Theorem 23.

Let $X$ be a Fano $4$-fold with $\delta_{X}=2$. Then $\rho_{X}\leq 12$.

If moreover $\rho_{X}\geq 7$, then there exist a small modification
$X\dasharrow X^{\prime}$ and a conic bundle $f\colon X^{\prime}\to Y$ where
$X^{\prime}$ and $Y$ are smooth, $\rho_{X}-\rho_{Y}=2$, and $Y$ is _weak Fano_
, i.e. $-K_{Y}$ is nef and big.

The bound $\rho_{X}\leq 12$ is obtained by studying the geometry of the weak
Fano $3$-fold $Y$ and proving that $\rho_{Y}\leq 10$, see [8, Theorem 4.4].

###### Proof.

By [9, Theorem 1.2] we have $\rho_{X}\leq 12$, and if $\rho_{X}\geq 7$, there
exists a diagram:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{{X^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{{X_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{{X_{1}^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{Y}$

where all varieties are smooth, $X_{1}$ is Fano, $g$ is the blow-up of a
smooth surface, $h$ is a small modification, and $f_{1}$ is an elementary
conic bundle.

We show that $Y$ is weak Fano. It follows from [27, Lemma 2.8] that $Y$ is of
Fano type, in particular $-K_{Y}$ is big.

Let $R$ be an extremal ray of $\operatorname{NE}(Y)$, and $\varphi\colon Y\to
Y^{\prime}$ the associated elementary contraction. If $\varphi$ is of fiber
type, then $-K_{Y}\cdot R>0$, because of bigness. If $\varphi$ is birational
and small, then $-K_{Y}\cdot R=0$ by [8, Lemma 4.5].

Finally suppose that $\varphi$ is birational divisorial. We show that the
composition $f_{1}\circ h\colon X_{1}\dasharrow Y$ cannot be regular, so that
$-K_{Y}\cdot R>0$ by [8, Lemma 4.6]. Indeed let $E\subset Y$ be the
exceptional divisor of $\varphi$. Since $\dim\varphi(E)\leq 1$, as in Example
7 we see that $\dim\mathcal{N}_{1}(E,Y)\leq 2$. If $f_{1}\circ h$ were
regular, let $E_{X}\subset X$ be a prime divisor whose image in $Y$ is $E$.
Then
$\dim\mathcal{N}_{1}(E_{X},X)\leq\rho_{X}-\rho_{Y}+\dim\mathcal{N}_{1}(E,Y)\leq
4$, which contradicts the assumptions $\delta_{X}=2$ and $\rho_{X}\geq 7$.

In conclusion $-K_{Y}\cdot R\geq 0$ for every extremal ray $R$ of
$\operatorname{NE}(Y)$, hence $-K_{Y}$ is nef. ∎

We can deduce from the previous results some general properties of Fano
varieties with $\delta_{X}\geq 2$.

######  Lemma 24.

Let $X$ be a smooth Fano variety with $\delta_{X}\geq 2$. Then the following
hold:

  1. $(a)$

$X$ is covered by a family of rational curves having intersection $2$ with
$-K_{X}$;

  2. $(b)$

$X$ contains a smooth prime divisor $E$, with a $\mathbb{P}^{1}$-bundle
structure, such that $E\cdot f=-1$, where $f\subset E$ is a fiber of the
$\mathbb{P}^{1}$-bundle.

If instead $X$ has $\delta_{X}=1$, then at least one of $(a)$ or $(b)$ holds.

###### Proof.

Statement $(b)$ follows from [7, Proposition 2.5]. When $\delta_{X}\geq 4$
(respectively $\delta_{X}=3$), $(a)$ is a straightforward conseguence of
Theorem 8 (respectively Theorem 17).

We show $(a)$ when $\delta_{X}=2$. We apply [10, Theorem 5.2], which is a more
refined version of Theorem 21. By this result, in case $(i)$ of Th. 21, the
general fiber of the conic bundle $X^{\prime}\to Y$ is contained in the open
subset where $X\dasharrow X^{\prime}$ is an isomorphism, so it yields the
desidered family in $X$. Moreover, in case $(ii)$ of Th. 21, the fibration
$\psi\colon X\to T$ is “quasi-elementary”, which means that for a general
fiber $F$ we have $\dim\mathcal{N}_{1}(F,X)=\rho_{X}-\rho_{T}$. Then $F$ is a
del Pezzo surface with $\rho_{F}\geq\rho_{X}-\rho_{T}=3$, in particular it has
a covering family of rational curves of anticanonical degree $2$, and again
this yields the desired family in $X$.

Finally the statement for the case $\delta_{X}=1$ follows from [10,
Proposition 6.1]. ∎

##  6\. An overview of known Fano $4$-folds with $\rho\geq 6$

In this section we describe the known examples of Fano $4$-folds with
$\rho\geq 6$. In fact, they are very few: we have products, toric examples,
two additional families with $\delta=3$ (see Proposition 20), one $4$-fold
contained in a product of grassmannians [23], and four families constructed
from the blow-ups of $\mathbb{P}^{4}$ in points. Let us start with these last
ones.

######  Example 25 (Fano models of the blow-ups of $\mathbb{P}^{4}$ in up to 8
points).

Let $\operatorname{Bl}_{m}\mathbb{P}^{4}$ be the blow-up of $\mathbb{P}^{4}$
in $m\geq 2$ general points. This $4$-fold is not Fano, as the transform
$\ell$ of a line through 2 blown-up points has $-K\cdot\ell=-1$. On the other
hand, for $m\leq 8$ there exists a small modification
$\operatorname{Bl}_{m}\mathbb{P}^{4}\dasharrow X_{m}$ such that $X_{m}$ is
smooth and Fano.

Let us describe this birational map. Consider in
$\operatorname{Bl}_{m}\mathbb{P}^{4}$ the transforms of the $\binom{m}{2}$
lines through 2 blown-up points, and, for $m\in\\{7,8\\}$, of the
$\binom{m}{7}$ rational normal quartics through 7 blown-up points. These
curves in $\operatorname{Bl}_{m}\mathbb{P}^{4}$ are pairwise disjoint, and are
smooth rational curves with normal bundle
$\mathcal{O}_{\mathbb{P}^{1}}(-1)^{\oplus 3}$.

Let $Y_{m}\to\operatorname{Bl}_{m}\mathbb{P}^{4}$ be the blow-up of all these
curves. The exceptional divisors are isomorphic to
$\mathbb{P}^{1}\times\mathbb{P}^{2}$ with normal bundle $\mathcal{O}(-1,-1)$.
Moreover $Y_{m}$ is also obtained by blowing-up a smooth $4$-fold $X_{m}$
along some pairwise disjoint smooth rational surfaces, with the same
exceptional divisors.

The $4$-fold $X_{m}$ is Fano, has $\rho_{X_{m}}=m+1\leq 9$, and it is toric if
and only if $m\leq 5$. For $m=7$, $X_{7}$ is isomorphic to one of Küchle’s
Fano $4$-folds [22], more precisely (b9), see [11]. For $m=8$, this example
has been constructed in [13], and is studied in detail in [33].

It follows from the description of the generators of the cone of effective
divisors of $X_{8}$ [13, §6.5] that $X_{m}$ cannot contain a prime divisor $E$
with a $\mathbb{P}^{1}$-bundle structure such that $E\cdot f=-1$, where
$f\subset E$ is a fiber of the $\mathbb{P}^{1}$-bundle. Therefore
$\delta_{X_{m}}\leq 1$ by Lemma 24.

On the other hand, for $m\leq 5$, $X_{m}$ is toric and it is easy to see that
it contains pairs of disjoint prime divisors, so that $\delta_{X_{m}}=1$ in
this case, by Example 6.

Note that $\operatorname{Bl}_{m}\mathbb{P}^{4}$ is not a Mori dream space for
$m\geq 9$ [26], therefore there cannot be a small modification to a smooth
Fano $4$-fold, and this construction works only for $m\leq 8$.

Let us consider now the toric case. Toric Fano $4$-folds are classified [2,
28], they are 124, and have $\rho\leq 8$; we list the ones with $6\leq\rho\leq
8$ in Table 3, where $X_{5}$ is from Example 25, while the combinatorial type
$U$ refers to Batyrev’s notation in [2].

Table 3. Toric Fano $4$-folds with $\rho\geq 6$
$\begin{array}[]{||c|c|c||}\hline\cr\hline\cr\rho=8&\operatorname{Bl}_{3}\mathbb{P}^{2}\times\operatorname{Bl}_{3}\mathbb{P}^{2}&\delta=3\\\
\hline\cr\rho=7&\operatorname{Bl}_{3}\mathbb{P}^{2}\times\operatorname{Bl}_{2}\mathbb{P}^{2}&\delta=3\\\
\hline\cr\rho=6&\operatorname{Bl}_{2}\mathbb{P}^{2}\times\operatorname{Bl}_{2}\mathbb{P}^{2}&\delta=2\\\
\cline{2-3}\cr&\text{combinatorial type $U$ (8 varieties)}&\delta=3\\\
\cline{2-3}\cr&X_{5}&\delta=1\\\ \hline\cr\hline\cr\end{array}$

In the recent paper [23], Manivel constructs a Fano $4$-fold $X_{L}$ with
$\rho_{X_{L}}=6$, contained in the product
$\text{Gr}(1,\mathbb{P}^{3})\times\text{Gr}(2,\mathbb{P}^{4})$. Its numerical
invariants are given in [23, Prop. 4.1], and using these one can check that
this $4$-fold is different from the other ones mentioned in this section.

Finally, excluding products and toric ones, to our knowledge the known
examples of Fano $4$-folds with $\rho\geq 6$ are the $4$-folds
$X_{6},X_{7},X_{8}$ of Example 25, the $4$-fold $X_{L}$, and the two new Fano
$4$-folds $X_{B_{1}}$ and $X_{B_{2}}$ with $\delta_{X}=3$ and $\rho_{X}=6$
constructed in [16, §7] (see Proposition 20), as shown in Table 4.

Table 4. Known Fano $4$-folds with $\rho\geq 6$, excluding toric ones and
products
$\begin{array}[]{||c|c|c||}\hline\cr\hline\cr\rho=7,8,9&X_{\rho-1}&\delta\in\\{0,1\\}\\\
\hline\cr\rho=6&X_{B_{1}}&\delta=3\\\ \cline{2-3}\cr&X_{B_{2}}&\delta=3\\\
\cline{2-3}\cr&X_{L}&\delta\in\\{0,1,2\\}\\\ \hline\cr\hline\cr\end{array}$

In the references [2, 15, 16, 23] one can find the main numerical invariants
of the $4$-folds in Tables 3 and 4, such as Hodge numbers, $K_{X}^{4}$, and
$h^{0}(X,-K_{X})$. Note that all these varieties are rational.

In particular we see that all known Fano $4$-folds with $\delta_{X}=2$ have
$\rho_{X}\leq 6$, quite far from the bound $\rho_{X}\leq 12$ of Theorem 23. It
would be very interesting to construct new examples, maybe starting from the
geometric description of the case $\rho_{X}\geq 7$ given by Theorem 23.

Finally, it is still an open question to determine a good bound on $\rho_{X}$
when $\delta_{X}\in\\{0,1\\}$, see [12] for related results. It is expected
that $\rho_{X}\leq 18$ for every Fano $4$-fold, and that $X$ is a product of
surfaces when $\rho_{X}$ is close to $18$.

##  7\. Special Fano’s: index $>1$ and the toric case

We recall that the _index_ $r_{X}$ of a Fano variety is the divisibility of
$-K_{X}$ in $\operatorname{Pic}(X)$, namely

$r_{X}=\max\\{m\in\mathbb{Z}_{>0}\,|\,-K_{X}\sim mH\text{ for
}H\in\operatorname{Pic}(X)\\}.$

A related invariant is the _pseudo-index_ , defined as

$\iota_{X}=\min\\{-K_{X}\cdot C\,|\,C\text{ a rational curve in }X\\},$

which is a multiple of $r_{X}$.

One expects “most” Fano varieties to have $r_{X}=\iota_{X}=1$, and that Fano
varieties with $\iota_{X}>1$ or $r_{X}>1$ should be simpler. For instance,
Fano 4-folds with $r_{X}>1$ are classified and have $\rho_{X}\leq 4$, see [19,
Cor. 3.1.15, Table 12.1, §5.2, and Table 12.7], so that the missing step in
the classification of Fano $4$-folds is the case of index $r_{X}=1$.

With respect to the Lefschetz defect, we have the following.

######  Theorem 26 ([7], Th. 1.2).

Let $X$ be a smooth Fano variety with $r_{X}>1$, or more generally with
$\iota_{X}>1$. Then one of the following holds:

  1. $(i)$

$\delta_{X}=0$;

  2. $(ii)$

$\delta_{X}=1$, $\iota_{X}=2$, $r_{X}\in\\{1,2\\}$, and there exists a smooth
morphism $X\to Y$ with fiber $\mathbb{P}^{1}$, where $Y$ is smooth, Fano, and
$\iota_{Y}>1$.

Let us consider now the case of toric Fano varieties; we have the following.

######  Proposition 27.

Every smooth toric Fano variety $X$ has $\delta_{X}\leq 3$.

Moreover, if $\delta_{X}=3$, then there exist a toric Fano variety $T$ and
$L_{1},L_{2},L_{3}\in\operatorname{Pic}(T)$ with $-K_{T}+L_{i}-L_{j}$ ample
for every $i,j=1,2,3$, such that $X$ is obtained by blowing-up
$Z:=\mathbb{P}_{T}(L_{1}\oplus L_{2}\oplus L_{3})$ along the three torus
invariant sections of the $\mathbb{P}^{2}$-bundle $Z\to T$.

###### Proof.

If $X$ is a Fano variety with $\delta_{X}\geq 4$, then by Theorem 8 we have
$X\cong S\times T$ where $S$ is a del Pezzo surface with
$\rho_{S}=\delta_{X}+1\geq 5$. In particular $S$ is not toric, as toric del
Pezzo surfaces have $\rho\leq 4$, therefore $X$ is not toric either.

Now if $X$ is a toric Fano variety with $\delta_{X}=3$, then we can apply
Theorem 17; moreover, since $X$ is toric, only case $A$ can occur, and $T$
must be toric too, see [16, Remark 6.1]. ∎

It is interesting to note that Proposition 27 is very much related to [5,
Theorem 3.4], proved much earlier with completely combinatorial methods. This
last result says that for every torus invariant prime divisor $D\subset X$,
one has $\operatorname{codim}\mathcal{N}_{1}(D,X)\leq 3$, and if there is a
$D$ with $\operatorname{codim}\mathcal{N}_{1}(D,X)=3$, then $X$ is a toric
$(\operatorname{Bl}_{3}\mathbb{P}^{2})$-bundle onto a toric Fano variety $T$.
If instead $\operatorname{codim}\mathcal{N}_{1}(D,X)=2$, then one gets a
statement similar to Theorem 21.

##  8\. Beyond smooth Fano’s

The notion of Lefschetz defect can be introduced for any projective variety;
anyway, the good properties of $\delta_{X}$ strongly depend on the Fano
assumption. Indeed, it seems that already when $X$ is weak Fano there is no
analog of Theorem 8, as the following example shows.

######  Example 28 (a toric weak Fano $3$-fold).

After [21] we know that $\rho_{X}\leq 35$ for every smooth, toric weak Fano
$3$-fold $X$, and the bound is sharp. One example with $\rho_{X}=35$ is
described in detail in [29, §7], together with the extremal rays of
$\operatorname{NE}(X)$. In particular, this $X$ contains a divisor isomorphic
to $\mathbb{P}^{2}$, so that $\delta_{X}=34$.

For singular Fano varieties, we have the following.

######  Theorem 29 ([17]).

Let $X$ be a $\mathbb{Q}$-factorial Gorenstein Fano variety, with canonical
singularities, and at most finitely many non-terminal points. Then
$\delta_{X}\leq 8$.

Moreover, if $\delta_{X}\geq 4$, then there is a finite morphism $X\to S\times
T$, where $S$ and $T$ are normal with rational singularities, and $\dim S=2$.

The main point where the technique used in the proof of Theorem 8 breaks, when
$X$ is too singular or not Fano, is the following. As explained in §3, the
properties of elementary birational contraction with fibers of dimension $\leq
1$ play a crucial role in the construction which is the starting point for the
proof of the theorem. In particular, if $X$ is smooth and Fano, such a
contraction is always divisorial. Instead, when $X$ is weak Fano, it can have
small (crepant) elementary contractions having fibers of dimension $\leq 1$.
Similarly, if $X$ is a $\mathbb{Q}$-factorial Gorenstein Fano variety with
canonical singularities, again it can have small elementary contractions
having fibers of dimension $\leq 1$, see [17, Example 2.11].

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