Fano varieties of middle pseudoindex

Abstract

Let X be a complex smooth Fano variety of dimension n. In this paper, we give a classification of such X when the pseudoindex is equal to $\dfrac{\dim X+1}{2}$ and the Picard number greater than one.

1 Introduction

Let X be an n-dimensional smooth Fano variety, which is a complex smooth projective variety endowed with an ample anticanonical divisor $-K_X$. The index of X is defined as

$$\begin{aligned} i_X:=\max \{m\in {{\mathbb {Z}}}_{>0}\mid -K_X=mL\,\,\text{ for } \text{ some }\,\,L\in \textrm{Pic} (X)\}. \end{aligned}$$

Mukai formulated a conjecture concerning the index $i_X$ and the Picard number $\rho _X$:

Conjecture 1.1

(Mukai conjecture [15, Conjecture 4]) We have $\rho _X(i_X-1)\le n$, with equality if and only if X is isomorphic to $({{\mathbb {P}}}^{i_X-1})^{\rho _X}$.

As a specific case of this conjecture, Mukai also conjectured that if $2i_X$ is at least $n+2$, then $\rho _X$ is one unless X is isomorphic to $({{\mathbb {P}}}^{i_X-1})^2$ [15, Conjecture 4’]. To prove this, Wiśniewski [16] introduced the notion of pseudoindex $\iota _X$ of X:

$$\begin{aligned} {\iota }_X:=\min \{-K_X\cdot C\mid C\subset X~\text{ is } \text{ a } \text{ rational } \text{ curve }\}, \end{aligned}$$

and arrived at the following theorem:

Theorem 1.2

([16, 17]) For an n-dimensional smooth Fano variety X, the following statements hold:

• (i) If $2\iota _X>n+2$, then $\rho _X=1$.

• (ii) If $2i_X=n+2$ and $\rho _X>1$, then X is isomorphic to $({{\mathbb {P}}}^{i_X-1})^2$.

• (iii) If $2i_X=n+1$ and $\rho _X>1$, then X is isomorphic to one of the following:

  $$\begin{aligned} {{\mathbb {P}}}({\mathcal {O}}_{{{\mathbb {P}}}^{i_X}}(2)\oplus {\mathcal {O}}_{{{\mathbb {P}}}^{i_X}}(1)^{\oplus i_X-1}),\quad {{\mathbb {P}}}^{i_X-1}\times Q^{i_X}\quad \text{ or } \quad {{\mathbb {P}}}(T_{{{\mathbb {P}}}^{i_X}}). \end{aligned}$$

Here, $Q^m$ denotes a smooth quadric hypersurface, and $T_{{{\mathbb {P}}}^{m}}$ represents the tangent bundle of ${{\mathbb {P}}}^m$.

Theorem 1.2 (i) and (ii) provide an affirmative answer to [15, Conjecture 4’]. Following this, by substituting the index $i_X$ with the pseudoindex $\iota _X$, Bonavero-Casagrande-Debarre-Druel proposed a generalized version of Conjecture 1.1:

Conjecture 1.3

(generalized Mukai conjecture [2, Conjecture]) We have $\rho _X({\iota }_X-1)\le n$, with equality if and only if X is isomorphic to $({{\mathbb {P}}}^{{\iota }_X-1})^{\rho _X}$.

Additionally, Occhetta established in [13, Corollary 4.3] that a smooth Fano variety X is isomorphic to $({{\mathbb {P}}}^{\iota _X-1})^2$ if $2\iota _X$ equals $n+2$ and $\rho _X$ is greater than one. By combining this with Theorem 1.2 (i), a generalized version of [15, Conjecture 4’] was derived. The aim of this brief paper is to establish the following theorem as an extension of Theorem 1.2 (iii):

Theorem 1.4

Let X be a smooth Fano variety of dimension n. If $2\iota _X= n+1$ and $\rho _X>1$, then X is isomorphic to one of the following:

• (i) the blow-up of projective space ${{\mathbb {P}}}^n$ along a linear subspace ${{\mathbb {P}}}^{\iota _X-2}$, i.e., ${{\mathbb {P}}}({\mathcal {O}}_{{{\mathbb {P}}}^{\iota _X}}(2)\oplus {\mathcal {O}}_{{{\mathbb {P}}}^{\iota _X}}(1)^{\oplus \iota _X-1});$

• (ii) the product of projective space ${{\mathbb {P}}}^{\iota _X-1}$ and a quadric hypersurface $Q^{\iota _X}$, i.e., ${{\mathbb {P}}}^{\iota _X-1}\times Q^{\iota _X};$

• (iii) the projectivization of the tangent bundle $T_{{{\mathbb {P}}}^{\iota _X}}$ of ${{\mathbb {P}}}^{\iota _X}$, i.e., ${{\mathbb {P}}}(T_{{{\mathbb {P}}}^{\iota _X}});$

• (iv) the product of projective spaces ${{\mathbb {P}}}^{\iota _X-1}$ and ${{\mathbb {P}}}^{\iota _X}$, i.e., ${{\mathbb {P}}}^{\iota _X-1}\times {{\mathbb {P}}}^{\iota _X}.$

In a manner akin to the argument presented in [17], the pivotal aspect of establishing Theorem 1.4 lies in proving that X possesses a projective bundle structure $\pi : X\rightarrow W$. Subsequently, we prove that the base variety W is either a projective space or a smooth quadric hypersurface. Applying [5, Corollary 4.7] and [2, Lemme 2.5] (detailed in Proposition 2.2 below), we thereby derive our desired conclusion.

1.1 Notation and conventions

In this paper, we work over the complex number field. Our notation is consistent with the books [6, 10, 11].

• For projective varieties X, Y and F, a smooth surjective morphism $f:X\rightarrow Y$ is called an F-bundle if any fiber of f is isomorphic to F. A surjective morphism $f:X\rightarrow Y$ with connected fibers is called an F-fibration if general fibers are isomorphic to F.

• A contraction of an extremal ray is called an elementary contraction.

• For a smooth projective variety X, we denote by $\rho _X$ the Picard number of X and by $T_X$ the tangent bundle of X.

2 Preliminaries

2.1 Fano varieties with large pseudoindex

Let us start by reviewing certain results concerning Fano varieties with large pseudoindex.

Theorem 2.1

([3, 4, 9]) Let X be a smooth Fano variety of dimension n with pseudoindex $\iota _X$. Then, the following holds.

• (i) If $\iota _X\ge n+1$, then X is isomorphic to ${{\mathbb {P}}}^n$.

• (ii) If $\iota _X= n$, then X is isomorphic to $Q^n$.

The next critical proposition contributes significantly to the proof of Theorem 1.4:

Proposition 2.2

Let X be an n-dimensional smooth Fano variety with pseudoindex $\iota _X=\dfrac{n+1}{2}$. Assume X admits either a ${{\mathbb {P}}}^{\frac{n+1}{2}}$-bundle structure $\pi : X\rightarrow W$ or a ${{\mathbb {P}}}^{\frac{n-1}{2}}$-bundle structure $\pi : X\rightarrow W$. Then X is isomorphic to one of the following:

$$\begin{aligned} {{\mathbb {P}}}({\mathcal {O}}_{{{\mathbb {P}}}^{\iota _X}}(2)\oplus {\mathcal {O}}_{{{\mathbb {P}}}^{\iota _X}}(1)^{\oplus \iota _X-1}),\quad {{\mathbb {P}}}^{\iota _X-1}\times Q^{\iota _X},\quad {{\mathbb {P}}}(T_{{{\mathbb {P}}}^{\iota _X}}), \quad {{\mathbb {P}}}^{\iota _X-1}\times {{\mathbb {P}}}^{\iota _X}. \end{aligned}$$

Proof

By [2, Lemme 2.5 (a)], W is a smooth Fano variety whose pseudoindex is at least $\iota _X=\dfrac{n+1}{2}$. Applying Theorem 2.1, W is isomorphic to ${{\mathbb {P}}}^{\frac{n-1}{2}}$, ${{\mathbb {P}}}^{\frac{n+1}{2}}$ or $Q^{\frac{n+1}{2}}$. By [5, Proposition 4.3], there exists a vector bundle ${{\mathcal {E}}}$ over W such that $X\cong {{\mathbb {P}}}({{\mathcal {E}}})$. When $\pi : X\rightarrow W$ is a ${{\mathbb {P}}}^{\frac{n+1}{2}}$-bundle, W is isomorphic to ${{\mathbb {P}}}^{\frac{n-1}{2}}$. In this case, [2, Lemme 2.5 (c)] and [14] tell us that X is isomorphic to ${{\mathbb {P}}}^{\iota _X-1}\times {{\mathbb {P}}}^{\iota _X}$. When $\pi : X\rightarrow W$ is a ${{\mathbb {P}}}^{\frac{n-1}{2}}$-bundle, our assertion is derived from [5, Corollary 4.7]. Thus, our assertion holds. $\square$

2.2 Extremal contractions

Extremal contractions play a pivotal role in the study of Fano varieties. Here, we gather some results concerning extremal rays and extremal contractions.

Definition 2.3

For a smooth projective variety X and its $K_X$-negative extremal ray $R\subset \overline{NE}(X)$, the length of R is defined as

$$\begin{aligned} \ell (R):=\min \{-K_X\cdot C\mid C~\text{ is } \text{ a } \text{ rational } \text{ curve } \text{ and }~[C]\in R\}. \end{aligned}$$

Theorem 2.4

(Ionescu-Wiśniewski inequality [8, Theorem 0.4], [18, Theorem 1.1]) Let X be a smooth projective variety, and let $\varphi : X \rightarrow Y$ be a contraction of a $K_X$-negative extremal ray R, with E representing its exceptional locus. Additionally, consider F as an irreducible component of a non-trivial fiber of $\varphi$. Then

$$\begin{aligned} \dim E + \dim F \ge \dim X + \ell (R)- 1. \end{aligned}$$

Theorem 2.5

([7, Theorem 1.3]) Let X be a smooth projective variety, and let $\varphi : X \rightarrow Y$ be a contraction of an extremal ray R such that any fiber has dimension d and $\ell (R)=d+1$. Then, $\varphi$ is a projective bundle.

Theorem 2.6

([1, Theorem 5.1]) For a smooth projective variety X of dimension n, the following are equivalent:

• (i) There exists an extremal ray R such that the contraction associated to R is divisorial and the fibers have dimension $\ell (R)$.

• (ii) X is the blow-up of a smooth projective variety $X'$ along a smooth subvariety of codimension $\ell (R)+1$.

Remark 2.7

For a smooth projective variety X, let $\varphi : X\rightarrow Y$ and $\psi : X\rightarrow Z$ be different elementary contractions of X. Then the fibers of $\varphi$ and $\psi$ have a finite intersection. We use this property several times in this paper.

2.3 Families of rational curves

Let X denote a smooth projective variety, and let us consider the space of rational curves $\dashrightarrow ^n(X)$ (for details, see [10, Section II.2]). A family of rational curves $\mathcal {M}$ on X refers to an irreducible component of $\dashrightarrow ^n(X)$. This family $\mathcal {M}$ is equipped with a $\mathbb {P}^1$-bundle $p: \mathcal {U} \rightarrow \mathcal {M}$ and an evaluation morphism $q: \mathcal {U} \rightarrow X$. The union of all curves parametrized by $\mathcal {M}$ is denoted by $\text {Locus}(\mathcal {M})$. For a point $x \in X$, the normalization of $p(q^{-1}(x))$ is denoted by $\mathcal {M}_x$, and $\text {Locus}(\mathcal {M}_x)$ denotes the union of all curves parametrized by $\mathcal {M}_x$.

A dominating family (resp. covering family) $\mathcal {M}$ is one where the evaluation morphism $q: \mathcal {U} \rightarrow X$ is dominant (or surjective). The family $\mathcal {M}$ is termed a minimal rational component if it is a dominating family with the minimal anticanonical degree among dominating families of rational curves on X. Additionally, $\mathcal {M}$ is called locally unsplit if for a general point $x\in \text {Locus}(\mathcal {M})$, $\mathcal {M}_x$ is proper. The family $\mathcal {M}$ is called unsplit if it is proper.

Theorem 2.8

([3, 9]) Let X be an n-dimensional smooth Fano variety and ${{\mathcal {M}}}$ a locally unsplit dominating family of rational curves on X. If the anticanonical degree of ${{\mathcal {M}}}$ is at least $n+1$, then X is isomorphic to ${{\mathbb {P}}}^n$.

Proposition 2.9

([10, IV Corollary 2.6]) Let X be a smooth projective variety and ${{\mathcal {M}}}$ a locally unsplit family of rational curves on X. For a general point $x \in \textrm{Locus}({{\mathcal {M}}})$,

Moreover, if ${{\mathcal {M}}}$ is unsplit, this inequality holds for any point $x \in \textrm{Locus}({{\mathcal {M}}})$.

3 Proof of the main theorem

3.1 The case when X admits a birational elementary contraction

In this subsection, we aim to establish the following proposition:

Proposition 3.1

Let X be a smooth Fano variety with $\iota _X=\dfrac{n+1}{2}$ and $\rho _X>1$. Assume there exists a birational contraction $\varphi : X\rightarrow Y$ of an extremal ray R. Then X is isomorphic to ${{\mathbb {P}}}({\mathcal {O}}_{{{\mathbb {P}}}^{\iota _X}}(2)\oplus {\mathcal {O}}_{{{\mathbb {P}}}^{\iota _X}}(1)^{\oplus \iota _X-1})$.

To prove this proposition, throughout this subsection, let X be a smooth Fano variety with $\iota _X=\dfrac{n+1}{2}$ and $\rho _X>1$. Assume there exists a birational contraction $\varphi : X\rightarrow Y$ of an extremal ray R. We denote by E the exceptional locus of $\varphi$ and by F an irreducible component of a nontrivial fiber of $\varphi$.

Claim 3.2

The exceptional locus E forms a divisor, meaning that $\varphi : X\rightarrow Y$ is a divisorial contraction.

Proof

Let us consider a minimal rational component $\mathcal {M}$ on X. According to Theorem 2.8, the anticanonical degree of $\mathcal {M}$ is at most n. Combining with our assumption that $\iota _X=\frac{n+1}{2}$ and [10, II, Proposition 2.2], it follows that $\mathcal {M}$ is an unsplit covering family. For any $x \in F$, Proposition 2.9 implies $\dim \textrm{Locus}(\mathcal {M}_x)\ge \frac{n-1}{2}$. To establish our assertion, let us assume the contrary, namely $\text {codim}_XE\ge 2$. Then, by Theorem 2.4, it follows that $\dim F\ge \frac{n+3}{2}$. Consequently, $\dim (\textrm{Locus}(\mathcal {M}_x)\cap F)\ge 1$. By [10, II, Corollary 4.21], this leads to a contradiction. $\square$

Utilizing Theorem 2.4, we infer $\dim F\ge \iota _X=\dfrac{n+1}{2}$. Since the Kleiman-Mori cone $\overline{NE}(X)$ of a Fano variety X is polyhedral and each extremal ray is generated by a rational curve, we can identify an extremal ray $R'$ and a rational curve $C'$ such that $R'={{\mathbb {R}}}_{\ge 0}[C']$, $\ell (R')=-K_X\cdot C'$ and $E\cdot C'>0$. We denote by $\psi : X\rightarrow Z$ the contraction of an extremal ray $R'$.

Claim 3.3

$\psi : X\rightarrow Z$ is of fiber type.

Proof

Assuming the contrary, that is, $\psi$ is of birational type, let $E'$ be the exceptional locus and $F'$ an irreducible component of a nontrivial fiber of $\psi$. By Theorem 2.4, we have $\dim F'\ge \iota _X=\dfrac{n+1}{2}$. Since $E\cdot C'>0$, we have $E\cap E'\ne \emptyset$. By replacing the fibers F and $F'$ if necessary, we may assume that $F\cap F'\ne \emptyset$. Then we obtain

$$\begin{aligned} \dim F+\dim F'-\dim X \ge \dfrac{n+1}{2} \times 2 -n =1. \end{aligned}$$

This leads to $\varphi =\psi$; this is a contradiction. Therefore, $\psi : X\rightarrow Z$ is of fiber type. $\square$

Let $F_\textrm{gen}'$ denote any fiber of $\psi$ whose dimension is equal to $\dim X-\dim Z$. Applying Theorem 2.4, we have

$$\begin{aligned} \dim F'_\textrm{gen}\ge \ell (R')-1\ge \dfrac{n-1}{2}. \end{aligned}$$

(1)

According to $E\cdot C'>0$, $\psi |_E: E\rightarrow Z$ is surjective. Since $\psi |_F: F \rightarrow Z$ is finite, we have

$$\begin{aligned} \dim Z \ge \dim F\ge \dfrac{n+1}{2}. \end{aligned}$$

(2)

Now we have $n=\dim X=\dim F'_\textrm{gen}+\dim Z \ge \dfrac{n-1}{2}+\dfrac{n+1}{2}=n$. This yields

$$\begin{aligned} \left( \dim F'_\textrm{gen}, \dim Z \right) = \left( \dfrac{n-1}{2},\dfrac{n+1}{2}\right) \end{aligned}$$

Moreover (1) and (2) imply that $\ell (R')=\dim F =\dfrac{n+1}{2}$. Assume there exists a jumping fiber $F'_\textrm{sp}$ of $\psi$, meaning $\dim F'_\textrm{sp}>\dim F'_\textrm{gen}=\dfrac{n-1}{2}$. Taking an irreducible component F of a nontrivial fiber of $\varphi$ such that $F'_\textrm{sp}\cap F\ne \emptyset$, we have

$$\begin{aligned} \dim F'_\textrm{sp}+\dim F-\dim X>\dfrac{n-1}{2}+\dfrac{n+1}{2}-n=0. \end{aligned}$$

This is a contradiction. As a consequence, $\psi$ is equidimensional. Since $\ell (R')=\dfrac{n+1}{2}=\dim F'_\textrm{gen}+1$, Theorem 2.5 tells us that $\psi$ is a ${{\mathbb {P}}}^{\frac{n-1}{2}}$-bundle.

By Theorem 2.4, we have

$$\begin{aligned} n-1 +\dfrac{n+1}{2} =\dim E+\dim F \ge n+\ell (R)-1\ge n-1+\dfrac{n+1}{2}. \end{aligned}$$

This yields that, for any nontrivial fiber F of $\varphi$, we have $\dim F=\dfrac{n+1}{2}=\ell (R)$. Applying Theorem 2.6, we see that $\varphi : X\rightarrow Y$ is the blow-up of a smooth variety Y along a smooth subvariety $\varphi (E)$ of codimension $\ell (R)+1=\dfrac{n+3}{2}$. Hence any nontrivial fiber F of $\varphi$ is isomorphic to ${{\mathbb {P}}}^{\frac{n+1}{2}}$. Since we have a finite morphism $\psi |_F: F\cong {{\mathbb {P}}}^{\frac{n+1}{2}} \rightarrow Z$ between smooth projective varieties of dimension $\dfrac{n+1}{2}$, $\psi |_F: F\rightarrow Z$ is a finite surjective morphism. By [12, Theorem 4.1], Z is isomorphic to ${{\mathbb {P}}}^{\frac{n+1}{2}}$. Since $\psi : X\rightarrow {{\mathbb {P}}}^{\frac{n+1}{2}}$ is a ${{\mathbb {P}}}^{\frac{n-1}{2}}$-bundle, Proposition 2.2 implies Proposition 3.1.

3.2 The case when any elementary contraction of X is of fiber type

In this subsection, we aim to establish the following proposition:

Proposition 3.4

Let X be a smooth Fano variety with $\iota _X=\dfrac{n+1}{2}$ and $\rho _X>1$. Assuming that any elementary contraction of X is of fiber type, then X is isomorphic to ${{\mathbb {P}}}^{\iota _X-1}\times Q^{\iota _X}$, ${{\mathbb {P}}}(T_{{{\mathbb {P}}}^{\iota _X}})$, or ${{\mathbb {P}}}^{\iota _X-1}\times {{\mathbb {P}}}^{\iota _X}$.

To prove this proposition, throughout this subsection, we stay within the confines of the present subsection, maintaining the setting where X is a smooth Fano variety with $\iota _X=\dfrac{n+1}{2}$ and $\rho _X>1$. Assume that any elementary contraction of X is of fiber type. For different extremal rays R and $R'$ of $\overline{NE}(X)$, consider the elementary contractions $\varphi :X\rightarrow Y$ and $\psi : X\rightarrow Z$ associated to R and $R'$ respectively. We denote by F (resp. $F'$) any fiber of $\varphi$ (resp. $\psi$). Using Theorem 2.4, we infer

$$\begin{aligned} \dim F\ge \ell (R)-1\ge \dfrac{n-1}{2}\quad \text{ and }\quad \dim F'\ge \ell (R')-1\ge \dfrac{n-1}{2}. \end{aligned}$$

(3)

Since we have

$$\begin{aligned} \dim F+\dim F'-n\le 0, \end{aligned}$$

(4)

it turns out that $\dim F$ and $\dim F'$ are at most $\dfrac{n+1}{2}$. Thus, denoting by $F_\textrm{gen}$ (resp. $F_\textrm{gen}'$) any fiber of $\varphi$ (resp. $\psi$) whose dimension is equal to $\dim X-\dim Y$ (resp. $\dim X- \dim Z$), $(\dim F_\textrm{gen }, \dim Y)$ and $(\dim F_\textrm{gen}', \dim Z)$ are either:

$$\begin{aligned} \left( \dfrac{n+1}{2}, \dfrac{n-1}{2}\right) \quad \text{ or }\quad \left( \dfrac{n-1}{2}, \dfrac{n+1}{2}\right) . \end{aligned}$$

We now claim:

Claim 3.5

$\varphi$ and $\psi$ are one of the following:

• (i) a ${{\mathbb {P}}}^{\frac{n+1}{2}}$-bundle;

• (ii) a $Q^{\frac{n+1}{2}}$-fibration;

• (iii) a ${{\mathbb {P}}}^{\frac{n-1}{2}}$-fibration.

Proof

It is enough to consider the structure of $\varphi$. Assume $\dim F_\textrm{gen }=\dfrac{n+1}{2}$. By inequality (4), $\varphi$ is equidimensional. By inequality (3), we see that $\ell (R)=\dfrac{n+3}{2}$ or $\dfrac{n+1}{2}$. In the former case, it follows from Theorem 2.5 that $\varphi$ is a ${{\mathbb {P}}}^{\frac{n+1}{2}}$-bundle. In the latter case, following Theorem 2.1, $\varphi$ is a $Q^{\frac{n+1}{2}}$-fibration. On the other hand, if $\dim F_\textrm{gen }=\dfrac{n-1}{2}$, then Theorem 2.1 yields that $\varphi$ is a ${{\mathbb {P}}}^{\frac{n-1}{2}}$-fibration. $\square$

Without loss of generality, we may assume that $\dim F_\textrm{gen } \ge \dim F_\textrm{gen }'$. Then the pair of $\varphi$ and $\psi$ is one of the following:

• (A) $\varphi$ is a ${{\mathbb {P}}}^{\frac{n+1}{2}}$-bundle and $\psi$ is a ${{\mathbb {P}}}^{\frac{n-1}{2}}$-fibration;

• (B) $\varphi$ is a $Q^{\frac{n+1}{2}}$-fibration and $\psi$ is a ${{\mathbb {P}}}^{\frac{n-1}{2}}$-fibration;

• (C) $\varphi$ and $\psi$ are ${{\mathbb {P}}}^{\frac{n-1}{2}}$-fibrations.

By inequality (4) and Theorem 2.5, in case (B), $\psi$ is a ${{\mathbb {P}}}^{\frac{n-1}{2}}$-bundle. In case (C), either $\varphi$ or $\psi$ turns into a ${{\mathbb {P}}}^{\frac{n-1}{2}}$-bundle. Consequently, Proposition 2.2 infers Proposition 3.4.

4 Conclusion

By Proposition 3.1 and Proposition 3.4, we obtain Theorem 1.4.