Proof of Theorem 1.2

Let $\varphi : X\to Y$ be a contraction of an extremal ray R as in Theorem 1.2. If $\dim Y =1$ , then it follows from Proposition 3.1 that $\varphi $ is a smooth morphism; since Y is isomorphic to $\mathbb {P}^1$ , Proposition 3.1 implies that X is isomorphic to a product of $\mathbb {P}^1$ and a variety. Thus, we assume that $\dim Y \geq 2$ . General fibers of $\varphi $ are smooth Fano varieties, so there exists a dominating family $\mathcal {M}\subset \mathrm {RatCurves}^n(X)$ such that any $\mathcal {M}$ -curve is contracted by $\varphi $ ; replacing if necessary, we may assume that the anticanonical degree of $\mathcal {M}$ is minimal among such families; then $\mathcal {M}$ is locally unsplit. We have $\deg _{(-K_X)}\mathcal {M} \leq n+1$ by Mori’s bend and break. If $\deg _{(-K_X)}\mathcal {M} \geq n$ , then Proposition 2.11 yields that $\dim \mathrm {Locus}(\mathcal {M}_x) \geq n-1$ for a general point $x\in X$ . This contradicts our assumption that $\dim Y \geq 2$ . Thus, we obtain the following inequality:

$$ \begin{align*}n-1\geq \deg_{(-K_X)}\mathcal{M}\geq \ell(R). \end{align*} $$

Then we see that $\mathcal {M}$ is unsplit; in fact, if it is not, by Lemma 2.10, there exists a degeneration of $\mathcal {M}$ -curves $Z=\sum _{i=1}^sa:i Z_i$ such that $\sum _{i=1}^sa:i>1$ . By the minimality of $\deg _{(-K_X)}\mathcal {M}$ , each $Z_i$ is not free; thus by our assumption that $\tau _X\geq n-2$ , we have

$$ \begin{align*}n-1 \geq \deg_{(-K_X)}\mathcal{M}=-K_X\cdot Z\geq 2(n-2).\end{align*} $$

This is a contradiction.

Assume that any minimal extremal rational curve of the ray R is free. Then $\mathcal {M}$ consists of these minimal extremal rational curves, and all $\mathcal {M}$ -curves are free; then Theorem 2.12 implies that $\varphi : X\to Y$ is a smooth morphism. So, assume that there exists a minimal extremal rational curve of the ray R, which is not free. Then, by our assumption, we have $\ell (R)\geq n-2$ , and we obtain the inequality

$$ \begin{align*}n-1\geq \deg_{(-K_X)}\mathcal{M}\geq \ell(R)\geq n-2. \end{align*} $$

For any $x \in X$ , again by Proposition 2.11, we have

$$ \begin{align*}\dim \mathrm{Locus}(\mathcal{M}_x)\geq \deg_{(-K_X)}\mathcal{M}-1\geq n-3. \end{align*} $$

By Theorem 2.13, $\mathbb {R}_{\geq 0}[\mathcal {M}]$ is an extremal ray, and the associated contraction is a geometric quotient $\varphi : X\to Y$ for $\mathcal {M}$ . Let us denote by B the jumping locus for fiber dimension of $\varphi $ :

$$ \begin{align*}B:=\{x \in X\mid \dim \varphi^{-1}(\varphi(x))> \dim X-\dim Y\}. \end{align*} $$

By [Reference Bonavero, Casagrande and Druel7, Proposition 1], we obtain $\dim B\leq n-2$ . Then we claim the following.

Proof of Claim 3.2

Assume that $\deg _{(-K_X)}\mathcal {M}> \ell (R)$ . Then $\deg _{(-K_X)}\mathcal {M}=n-1$ and $\ell (R)=n-2$ . Moreover, it turns out that, for any point $x\in X$ ,

$$ \begin{align*}n-2\geq \dim X-\dim Y\geq \dim \mathrm{Locus}(\mathcal{M}_x)\geq \deg_{(-K_X)}\mathcal{M}-1=n-2. \end{align*} $$

This yields $(\dim Y, \dim \mathrm {Locus}(\mathcal {M}_x))=(2, n-2)$ . Since $\dim \mathrm {Locus}(\mathcal {M}_x)=n-2=\dim X-\dim Y$ , [Reference Cho, Miyaoka and Shepherd-Barron17] implies that general fibers of $\varphi $ are isomorphic to $\mathbb {P}^{n-2}$ . Moreover, $\varphi $ is equidimensional, because $\dim B \leq n-2$ . It is flat since $\varphi $ is an equidimensional morphism between smooth varieties. Following the idea of the proof of [Reference Höring and Novelli30, Proposition 3.1], we shall prove that $\varphi $ is a projective bundle; then we get a contradiction to our assumption $\deg _{(-K_X)}\mathcal {M}> \ell (R)$ .

To prove that $\varphi $ is a projective bundle, assume the contrary. Then $\varphi $ is not smooth. We denote by $S_{\varphi }$ the locus of points of Y over which $\varphi $ is not smooth and by $R_{\varphi }$ the locus of points of Y over which the fibers of $\varphi $ are reducible. According to the rigidity of the projective space [Reference Siu57, Main Theorem] (see also [Reference Hwang and Mok31, Theorem 1’]), we remark that $\varphi |_{X\setminus \varphi ^{-1}(S_{\varphi })}: X\setminus \varphi ^{-1}(S_{\varphi })\to Y\setminus S_{\varphi }$ is a $\mathbb {P}^{n-2}$ -bundle. It follows from [Reference Araujo and Ramón-Marí5, Theorem 4] that $S_{\varphi }$ is of pure codimension one and $S_{\varphi }=\overline {R_{\varphi }}$ . Since $\dim Y=2$ , the locus $S_{\varphi }$ is a union of curves.

Let $C_1$ be a minimal extremal rational curve of the ray R. Taking a family of rational curves $\mathcal {M}'\subset \mathrm {RatCurves}^n(X)$ containing $[C_1]$ , the minimality of the anticanonical degree $-K_X\cdot C_1$ and Lemma 2.10 imply that $\mathcal {M}'$ is unsplit. By Proposition 2.11, for any $x \in \mathrm { Locus}(\mathcal {M}')$ , we obtain the inequality

(1) $$ \begin{align} 2n-2=\dim X+\deg_{(-K_X)}\mathcal{M}'\leq \dim \mathrm{Locus}(\mathcal{M}')+\dim \mathrm{Locus}(\mathcal{M}_x')+1. \end{align} $$

Since any $\mathcal {M}'$ -curve is not free, $\mathrm {Locus}(\mathcal {M}')$ does not coincide with the whole space X. Moreover, for any $x \in \mathrm { Locus}(\mathcal {M}')$ , $\mathrm {Locus}(\mathcal {M}_x')$ is contained in the fiber $\varphi ^{-1}(\varphi (x))$ . Hence, the right-hand side of the inequality $ (1)$ is at most

$$ \begin{align*}(n-1)+(n-2)+1=2n-2.\end{align*} $$

As a consequence, we see that

$$ \begin{align*}\dim \mathrm{Locus}(\mathcal{M}')=n-1\,\,\,\,\,\text{and}\,\,\,\,\,\dim \mathrm{Locus}(\mathcal{M}_x')=n-2. \end{align*} $$

Since $\mathcal {M}'$ is unsplit, Mori’s bend and break tells us that the evaluation map from the universal family of $\mathcal {M}_x'$ onto $\mathrm { Locus}(\mathcal {M}_x')$ is generically finite; this yields that $\dim \mathcal {M}_x'=n-3$ for any $x \in \mathrm {Locus}(\mathcal {M}')$ . Consider the universal family $p: \mathcal {U}'\to \mathcal {M}'$ associated with $\mathcal {M}'$ and its evaluation morphism $q: \mathcal {U}'\to \mathrm {Locus}(\mathcal {M}')$ ; by dimension count, we obtain

$$ \begin{align*}\dim \mathcal{M}'= \dim \mathcal{U}'-1=\dim \mathcal{M}_x' +\dim \mathrm{Locus}(\mathcal{M}')-1=(n-3)+(n-1)-1= 2n-5. \end{align*} $$

We see that $\mathrm {Locus}(\mathcal {M}')$ is contained in $\varphi ^{-1}(S_{\varphi })$ ; since we have

$$ \begin{align*}\dim \mathrm{Locus}(\mathcal{M}')=\dim \varphi^{-1}(S_{\varphi})=n-1,\end{align*} $$

$\mathrm {Locus}(\mathcal {M}')$ is an irreducible component of $\varphi ^{-1}(S_{\varphi })$ . Moreover, $\varphi (\mathrm {Locus}(\mathcal {M}'))$ coincides with an irreducible component of $S_{\varphi }= \overline {R_{\varphi }}$ ; thus, we may find a point $x_0\in \mathrm {Locus}(\mathcal {M}')$ such that $\varphi (x_0)$ is in $R_{\varphi }$ . Set $F_0:=\varphi ^{-1}(\varphi (x_0))$ . Denoting by $D_1, \ldots , D_m~(m>1)$ the irreducible components of the fiber $(F_0)_{\mathrm {red}}$ with reduced structure, there exist positive integers $a_i$ ’s such that the cycle $[F_0]$ is algebraically equivalent to $\sum _{i=1}^ma_i[D_i]$ with $\sum _{i=1}^ma_i>1$ . Set $D_1^o:=\left (D_1\setminus \bigcup _{i\neq 1}D_i \right )$ , which is a nonempty open subset of $D_1$ . Without loss of generality, we may assume that $D_1 \subset \mathrm {Locus}(\mathcal {M}')$ . For any point $x\in D_1^o$ , any $\mathcal {M}_x'$ -curve is contained in $F_0$ and passes through the point $x \in X$ ; this implies that $\mathrm { Locus}(\mathcal {M}_x')$ is contained in $D_1$ ; since $\mathrm {Locus}(\mathcal {M}_x')$ and $D_1$ have the same dimension and $D_1$ is irreducible, we see that $\mathrm {Locus}(\mathcal {M}_x')=D_1$ . By a dimension count, we obtain

$$ \begin{align*}\dim p(q^{-1}(D_1^o))= \dim q^{-1}(D_1^o)-1=\dim \mathcal{M}_x' +\dim D_1^o-1= 2n-6. \end{align*} $$

This implies that there exists a family of rational curves $\mathcal {N}\subset \mathrm {RatCurves}^n(D_1)$ such that $\dim \mathcal {N}$ is at least $2n-6$ and $\deg _{(-K_X|_{D_1})}\mathcal {N}=\ell (R)$ . Denoting by $\nu : \tilde {D_1}\to D_1$ the normalization of $D_1$ , our family $\mathcal {N}$ lifts to $\tilde {D_1}$ . We denote by $\tilde {\mathcal {N}}$ the lifting of $\mathcal {N}$ to $\tilde {D_1}$ . Then by applying [Reference Höring and Novelli30, Theorem 2.1], we see that $D_1$ is isomorphic to $\mathbb {P}^{n-2}$ and $\tilde {\mathcal {N}}$ is the family of lines on $\mathbb {P}^{n-2}$ .

Let F be a general fiber of $\varphi $ . By [Reference Kollár38, Chapter I, Proposition 3.12], we obtain

$$ \begin{align*} (n-1)^{n-2} &= (-K_F)^{n-2}=[F]\cdot (-K_X)^{n-2}=[F_0]\cdot(-K_X)^{n-2} \\ &= \sum_{i=1}^m a_i\left([D_i]\cdot (-K_X)^{n-2}\right)>[D_1]\cdot (-K_X)^{n-2}=\left(\nu^{\ast}(-K_X)\right)^{n-2}. \end{align*} $$

For a positive integer b such that $\nu ^{\ast }(-K_X)=\mathcal {O}_{\mathbb {P}^{n-2}}(b)$ , the above inequality tells us that $b^{n-2}<(n-1)^{n-2}$ ; thus, we see that $b<n-1$ . On the other hand, for a line $f \subset \mathbb {P}^{n-2}$ , we have

$$ \begin{align*}n-2=\ell(R)\leq (-K_X)\cdot \nu_{\ast}(f) =\nu^{\ast}(-K_X)\cdot f =b<n-1. \end{align*} $$

Hence, we obtain $b=n-2$ .

Recall that $\mathcal {M}$ is an unsplit covering family; for any point $x \in D_1^o$ , there exists an $\mathcal {M}_x$ -curve C; we see that C is contained in $D_1$ ; it lifts to a curve $\tilde {C}$ in $\tilde {D_1}\cong \mathbb {P}^{n-2}$ . Then we obtain the following:

$$ \begin{align*}(n-2)\mathcal{O}_{\mathbb{P}^{n-2}}(1)\cdot \tilde{C}=\nu^{\ast}(-K_X)\cdot \tilde{C}=(-K_X)\cdot {C}=n-1. \end{align*} $$

This is a contradiction, because $\mathcal {O}_{\mathbb {P}^{n-2}}(1)\cdot \tilde {C}$ is an integer and $n\geq 4$ .

Let F be a smooth general fiber of $\varphi $ . As we proved, we have the following inequalities:

$$ \begin{align*} n-2 & \geq \dim F \geq n-3, \\ \quad n-1\geq \deg_{(-K_X)}\mathcal{M}&=\ell(R)\geq n-2. \end{align*} $$

By [Reference Cho, Miyaoka and Shepherd-Barron17] and [Reference Dedieu and Höring20] (see also [Reference Miyaoka47]), F is isomorphic to $\mathbb {P}^{n-2}, Q^{n-2}$ , or $\mathbb {P}^{n-3}$ . Assume that F is isomorphic to $\mathbb {P}^{n-2}$ or $Q^{n-2}$ ; then $\varphi $ is equidimensional, because $\dim B \leq n-2$ . Thus, if F is isomorphic to $\mathbb {P}^{n-2}$ , [Reference Höring and Novelli30] yields that $\varphi $ is a $\mathbb {P}^{n-2}$ -bundle; this contradicts our assumption that $\varphi $ is not smooth. Consequently, we see that $\varphi $ is an equidimensional $Q^{n-2}$ -fibration. Finally, we assume that F is isomorphic to $\mathbb {P}^{n-3}$ . If $\varphi $ is equidimensional, then again by [Reference Höring and Novelli30] $\varphi $ should be a $\mathbb {P}^{n-3}$ -bundle; it is a contradiction because $\varphi $ is not smooth. As a consequence, $\varphi $ is a $\mathbb {P}^{n-3}$ -fibration which is not equidimensional.

Proof. Remark that $\mathbb {P}(\mathcal {O}_{\mathbb {P}^{n-1}}\oplus \mathcal {O}_{\mathbb {P}^{n-1}}(1))$ is the blow up of $\mathbb {P}^n$ at a point. In the following, we denote by E its exceptional divisor.

$ \mathrm{(i)}\Rightarrow \mathrm{(ii)}$ Assume that X is isomorphic to $\mathbb {P}(\mathcal {O}_{\mathbb {P}^{n-1}}\oplus \mathcal {O}_{\mathbb {P}^{n-1}}(1))$ . We denote by $\mathcal {O}_X(1)$ the tautological line bundle of $\mathbb {P}(\mathcal {O}_{\mathbb {P}^{n-1}}\oplus \mathcal {O}_{\mathbb {P}^{n-1}}(1))$ and by $\pi : X\to \mathbb {P}^{n-1}$ the natural projection. Then we have the following exact sequence:

$$ \begin{align*}0\to \mathcal{O}_X(2)\otimes \pi^{\ast}(T_{\mathbb{P}^{n-1} }(-1))\to \wedge^2 T_X \to \wedge^2\pi^{\ast}T_{\mathbb{P}^{n-1}}\to 0.\end{align*} $$

Since the bundles $\mathcal {O}_X(2), \pi ^{\ast }(T_{\mathbb {P}^{n-1} }(-1))$ , and $\pi ^{\ast }T_{\mathbb {P}^{n-1}}$ are nef, so is $\wedge ^2 T_X$ . Moreover, Theorem 2.12 tells us that $T_X$ is not nef.

$ \mathrm{(ii)}\Rightarrow \mathrm{(iii)}$ This follows from Lemma 2.7.

$ \mathrm{(iii)}\Rightarrow \mathrm{(i)}$ , $ (iv)\Rightarrow (i)$ These follow from the same argument as in [Reference Yasutake74] or [Reference Watanabe66, Theorem 1.7].

$ \mathrm{(i)}\Rightarrow \mathrm{(iv)}$ If X is $\mathbb {P}(\mathcal {O}_{\mathbb {P}^{n-1}}\oplus \mathcal {O}_{\mathbb {P}^{n-1}}(1))$ , the above arguments imply that $\tau _X\geq n-1$ . For a line $\ell $ in the exceptional divisor $E\cong \mathbb {P}^{n-1}$ of X, we have $-K_X\cdot \ell =n-1$ . Since $\ell $ is not free, our assertion holds.

Proof. Combining $\nu _X=3$ with Lemma 2.7 and Theorem 4.1, we see that $\tau _X=n-2$ .

Proof. By definition, $\iota _X$ is a multiple of $i_X$ ; then our assertion follows from Theorem 2.4.

Proof. We may take $\tilde {\ell _i}$ in a fiber of p for every i and take $\tilde {\ell }$ to be a line in a fiber of q. Let $\Gamma $ be a projective curve on X. Then there exist nonnegative real numbers $c_1,\ldots , c_r$ such that $q_{\ast }[\Gamma ]=\sum _{i=1}^rc_i[\ell _i]$ in $\overline { NE}(Y)$ . Since $[\Gamma ]-\sum _{i=1}^rc_i[\tilde {\ell _i}]$ is contained in $\mathrm {Ker}(q_{\ast })=\langle [\tilde {\ell }] \rangle _{\mathbb {R}}$ , there exists $a\in \mathbb {R}$ such that $[\Gamma ]=a[\tilde {\ell }]+\sum _{i=1}^rc_i[\tilde {\ell _i}]$ . Then we have $a=p^{\ast }(\mathcal {O}_{\mathbb {P}^m}(1))\cdot \Gamma \geq 0$ .

Assumption 5.1. Let X be a smooth Fano variety of dimension $n \geq 4$ and $\tau _X\geq n-2$ . Moreover, assume that X admits a birational contraction $\varphi : X \to Y$ of an extremal ray R. We denote by E the exceptional locus of $\varphi $ and by F a fiber of $\varphi |_{E}: E\to W:=\varphi (E)$ .

Proof. Since any rational curve C contracted by $\varphi $ is not free, we have $-K_X\cdot C \geq n-2$ . Then Proposition 2.3 yields that

$$ \begin{align*}2(n-1)\geq \dim E +\dim F \geq n+\ell(R)-1\geq n+(n-2)-1=2n-3. \end{align*} $$

As a consequence, we obtain our assertion.

Proof. Take a rational curve $C_0 \subset X$ such that $E\cdot C_0>0$ . According to the Cone theorem, there exist finitely many extremal rational curves $C_i$ ’s and positive real numbers $a_i$ ’s such that $[C_0]$ is numerically equivalent to $\sum a_i [C_i]$ ; since we have

$$ \begin{align*}\sum a_i (E\cdot C_i)=E\cdot C_0>0, \end{align*} $$

there exists an extremal rational curve $C_{i_0}$ such that $E\cdot C_{i_0}>0$ as desired.

The latter part of our assertion follows from a standard dimension count.

Proof. We claim that $\psi (E)$ coincides with $Y'$ . If not, a curve D exists in the exceptional divisor E, which is contracted by $\psi $ ; this is a contradiction because the curve $[D]$ lives in the rays R and $R'$ . Then it turns out that $\psi ^{-1}(y')\cap E$ is a nonempty finite set for any point $y' \in Y'$ . This yields an inequality

$$ \begin{align*}0=\dim \psi^{-1}(y')\cap E\geq \dim \psi^{-1}(y')+\dim E -n=\dim \psi^{-1}(y')-1. \end{align*} $$

This tells us that $\dim \psi ^{-1}(y') \leq 1$ for any point $y' \in Y'$ ; since $\psi $ is of fiber type, it is a conic bundle. By our assumption $\tau _X\geq n-2$ , the anticanonical degree of any rational curve is at least two; this implies that $\psi $ is a $\mathbb {P}^1$ -bundle. Applying [Reference Fujita23, Proposition 2.2] and its proof, $Y'$ is an $(n-1)$ -dimensional smooth Fano variety of $\rho _{Y'}=1$ whose Fano index $i_{Y'}$ satisfies $i_{Y'}>s>0$ ; moreover, denoting by $\mathcal {O}_{Y'}(1)$ the ample generator of $\mathrm {Pic}(Y')$ , X is isomorphic to $\mathbb {P}(\mathcal {O}_{Y'}\oplus \mathcal {O}_{Y'}(s))$ and E gives a section of $\psi : X \to Y'$ .

Let f be a minimal extremal rational curve of the ray R; by our assumption $\tau _X\geq n-2$ and [Reference Cho, Miyaoka and Shepherd-Barron17], the curve f satisfies $-K_X\cdot f \geq n-2$ and $n \geq -K_E\cdot f$ . We have $K_X=\varphi ^{\ast }K_Y+\alpha E$ for some $\alpha \in \mathbb {Q}$ ; then $-\alpha E=-K_X+ \varphi ^{\ast }K_Y$ is $\varphi $ -nef. Since $-\alpha E$ is $\varphi $ -exceptional, the negativity lemma [Reference Kollár and Mori40, Lemma 3.39] yields $-\alpha E\leq 0$ , and hence $\alpha>0$ . Since $K_X\cdot f=\alpha E\cdot f<0$ , it follows that $E\cdot f<0$ . Thus, we have

$$ \begin{align*}n \geq -K_E\cdot f=-K_X\cdot f-E\cdot f\geq (n-2)+1=n-1. \end{align*} $$

This implies that $(-K_E\cdot f, -K_X\cdot f, -E\cdot f)=(n, n-1, 1), (n-1, n-2, 1)$ , or $(n, n-2,2)$ .

Let us assume that $(-K_E\cdot f, -K_X\cdot f, -E\cdot f)=(n, n-1, 1)$ . Since $Y'$ is an $(n-1)$ -dimensional smooth Fano variety of $\rho _{Y'}=1$ , we see that $i_{Y'}=n$ and $s=1$ ; by Kobayashi–Ochiai theorem [Reference Kobayashi and Ochiai37], we obtain $E\cong Y'\cong \mathbb {P}^{n-1}$ and $X\cong \mathbb {P}(\mathcal {O}_{\mathbb {P}^{n-1}}\oplus \mathcal {O}_{\mathbb {P}^{n-1}}(-1))$ . If $(-K_E\cdot f, -K_X\cdot f, -E\cdot f)=(n-1, n-2, 1)$ , the same argument implies that $X\cong \mathbb {P}(\mathcal {O}_{Q^{n-1}}\oplus \mathcal {O}_{Q^{n-1}}(-1))$ . Thus, assume that $(-K_E\cdot f, -K_X\cdot f, -E\cdot f)=(n, n-2,2)$ . Applying [Reference Cho, Miyaoka and Shepherd-Barron17], we see that $E\cong Y' \cong \mathbb {P}^{n-1}$ and $X\cong \mathbb {P}(\mathcal {O}_{\mathbb {P}^{n-1}}\oplus \mathcal {O}_{\mathbb {P}^{n-1}}(-2))$ .

Proof. By Lemma 5.2, we have $\ell (R)=n-2$ ; thus, [Reference Andreatta and Occhetta3, Theorem 5.1] implies that $\varphi : X \to Y$ is the blow-up of Y along a smooth curve $W\subset Y$ ; by our assumption $\tau _X\geq n-2$ and [Reference Kollár38, Chapter II, Lemma 3.13], we have $\iota _X\geq 2$ . Since any fiber $F\cong \mathbb {P}^{n-2}$ of $E=\mathbb {P}(N_{W/Y}^{\vee }) \to W$ is not contracted by $\psi $ , $\psi |_{F}: F\cong \mathbb {P}^{n-2}\to \psi (F)$ is a finite morphism. Thus, it turns out that $n-2 =\dim \psi (F)\leq \dim Y'<n$ . Then our assertion follows from [Reference Tsukioka64, Lemma 2 and Propositions 3 and 4] and Examples 4.7 and 4.10. Remark that the pseudoindices of the Fano varieties $3, 4$ , and $5$ in [Reference Tsukioka64, Proposition 4] are all one. Thus, our assertion holds.

Proof. First assume that there exists an extremal ray $R"$ such that $E\cdot R">0$ and the contraction associated with $R"$ is of fiber type. Then Propositions 5.4 and 5.5 imply that there is no birational contraction of an extremal ray other than $\varphi $ . Hence, our assertion holds. Consequently, we may assume that $\psi $ is of birational type. We denote by $E'$ the exceptional divisor of $\psi $ and by $F'$ a fiber of $\psi |_{E'}: E'\to W':=\psi (E')$ . By the same argument as in Lemma 5.2, we see that $(\dim E', \dim W')=(n-1, 0)$ or $(n-1,1)$ . By our assumption, $E\cdot R'>0$ ; this implies that $E\cap E'\neq \emptyset $ . Then we may find a fiber $F_0$ of $\varphi |_{E}: E\to W$ and a fiber $F_0'$ of $\psi |_{E'}: E'\to W' $ such that $F_0\cap F_0'\neq \emptyset $ . Then we have

$$ \begin{align*}\dim F_0\cap F_0' \geq \dim F_0+\dim F_0'-\dim X. \end{align*} $$

Remark that $\dim F_0$ and $\dim F_0'$ are $n-1$ or $n-2$ . If $\dim F_0$ or $\dim F_0'$ is $n-1$ , then the above inequality tells us that $\dim F_0 \cap F_0'\geq 1$ , because we assume that $n\geq 4$ . This contradicts $\varphi \neq \psi $ . This yields that $\dim F_0=\dim F_0'=n-2$ . The same argument shows that the only possible case is $(n, \dim F_0, \dim F_0')=(4, 2, 2)$ . We assume this from now on. By Lemma 5.2, we have $\ell (R')=2$ ; thus, [Reference Andreatta and Occhetta3, Theorem 5.1] implies that $\psi : X \to Y'$ is the blow-up of $Y'$ along a smooth curve $W'\subset Y'$ . Then, by [Reference Tsukioka63, Theorem 1], X is isomorphic to the blow-up of $\mathbb {P}^4$ along a smooth, complete intersection of a hyperplane and two quadrics. Then, another contraction of X is the blow-up of a singular variety (see [Reference Tsukioka63, 2 An example]). This contradicts the fact that Y and $Y'$ are smooth.

Proof. The “if” part is trivial, and the “only if” part is immediate from [Reference Kollár38, Chapter II, Lemma 3.13].

Assumption 6.4. Let X be a smooth Fano fourfold of $\tau _X\geq 2$ and $\rho _X\geq 2$ . Assume that any elementary contraction of X is of fiber type and X does not admit a contraction onto $\mathbb {P}^1$ .

Proof. Let us first assume that a smooth elementary contraction $\varphi : X \to Y$ exists. Let F be a fiber of $\varphi $ ; by [Reference Casagrande12, Example 3.8], we see that the Picard number $\rho _F$ is one; thus, F is a Fano variety of $\rho _F=1$ . If $\dim F=1$ or $2$ , then $\varphi : X \to Y$ is a $\mathbb {P}^d$ -bundle ( $d=1, 2$ ). So we assume that $\dim F=3$ ; in this case, Y is isomorphic to $\mathbb {P}^1$ ; this is a contradiction. So, we may assume that any elementary contraction is not smooth. Then, our assertion follows from Theorem 1.2.

Proof. Let us first claim that $\rho _X=2$ . To prove this, assume the contrary, that is, $\rho _X\geq 3$ . Then it follows from [Reference Kollár, Miyaoka and Mori39, Corollary 2.9] that Y is a smooth Fano threefold of $\rho _Y\geq 2$ . Applying [Reference Druel22, Lemmas 4.2 and 4.4 and Theorem 6.3] (remark that $\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^1$ is missing in the list of [Reference Druel22, Theorem 6.3]), we see that Y is one of the following:

1. (1) A double cover of $W=\mathbb {P}(T_{\mathbb {P}^2})$ whose branch locus is a divisor in $|{-}K_W|$ .

2. (2) A divisor on $\mathbb {P}^2\times \mathbb {P}^2$ of bidegree $(1, 1)$ , which is isomorphic to $\mathbb {P}(T_{\mathbb {P}^2})$ .

3. (3) A divisor on $\mathbb {P}^2\times \mathbb {P}^2$ of bidegree $(1, 2)$ .

4. (4) A divisor on $\mathbb {P}^2\times \mathbb {P}^2$ of bidegree $(2, 2)$ .

On the other hand, it follows from [Reference Bonavero, Casagrande, Debarre and Druel6, Lemma 2.5] that $\iota _Y \geq \iota _X \geq 2$ . Since the pseudoindices of the Fano varieties appearing in $(1)$ , $(3)$ , and $(4)$ are one (for the variety in $(1)$ , see [Reference Matsuki46, p. 53, ${n}^o$ 6 (6.B)]), we are in case $(2)$ , that is, Y is isomorphic to $\mathbb {P}(T_{\mathbb {P}^2})$ . Let Y be $\mathbb {P}(T_{\mathbb {P}^2})$ . Since Y is rational, its Brauer group vanishes. This yields that there exists a rank two vector bundle $\mathcal {E}$ on Y such that $X=\mathbb {P}(\mathcal {E})$ and $\varphi $ is the natural projection $X=\mathbb {P}(\mathcal {E})\to Y$ . Remark that $\iota _X = \iota _Y =2$ . Since $\mathbb {P}(T_{\mathbb {P}^2})$ is isomorphic to a divisor on $\mathbb {P}^2\times \mathbb {P}^2$ of bidegree $(1, 1)$ , there exist two projections $p_i: \mathbb {P}(T_{\mathbb {P}^2}) \to \mathbb {P}^2$ for $i=1, 2$ as follows:

In this diagram, the projective plane $\mathbb {P}^2$ on the left can be seen as a unique family of lines on the projective plane $\mathbb {P}^2$ on the right; then, this diagram is nothing but the universal family of lines. For any fiber $\ell _i$ of $p_i$ , [Reference Bonavero, Casagrande, Debarre and Druel6, Lemma 2.5] tells us that $\mathcal {E}|_{\ell _i}\cong \mathcal {O}_{\mathbb {P}^1}(a_i)^{\oplus 2}$ for some $a_i \in \mathbb {Z}\, (i=1,2)$ . Tensoring a line bundle, we may assume that $\mathcal {E}|_{\ell _i}\cong \mathcal {O}_{\mathbb {P}^1}^{\oplus 2}$ for any fiber $\ell _i$ of $p_i$ . Then it follows from Grauert’s theorem and Nakayama’s lemma that ${p_2}_{\ast }\mathcal {E}$ is a rank $2$ vector bundle on $\mathbb {P}^2$ and ${p_2}^{\ast }({p_2}_{\ast }\mathcal {E})\cong \mathcal {E}$ . Moreover, we have

$$ \begin{align*}({p_2}_{\ast}\mathcal{E})|_{p_2(\ell_1)}\cong {p_2}^{\ast}({p_2}_{\ast}\mathcal{E})|_{\ell_1}\cong \mathcal{E}|_{\ell_1}\cong \mathcal{O}_{\mathbb{P}^1}^{\oplus 2}. \end{align*} $$

This means that ${p_2}_{\ast }\mathcal {E}$ is a uniform vector bundle on $\mathbb {P}^2$ ; then [Reference Van de Ven65] implies that ${p_2}_{\ast }\mathcal {E}\cong \mathcal {O}_{\mathbb {P}^2}^{\oplus 2}$ . As a consequence, X is isomorphic to $\mathbb {P}^1\times \mathbb {P}(T_{\mathbb {P}^2})$ , which contradicts our assumption. Thus, we see that $\rho _X=2$ .

Since Y is a Fano threefold of $\rho _Y=1$ and $\iota _Y\geq 2$ , [Reference Novelli and Occhetta50, Lemma 2.16] implies that Y is $\mathbb {P}^3$ or $Q^3$ . Since Y is rational, there exists a rank two vector bundle $\mathcal {E}$ on Y such that $X=\mathbb {P}(\mathcal {E})$ . From the classification of rank two Fano bundles over $\mathbb {P}^3$ [Reference Szurek and Wiśniewski61, Theorem 2.1] and that of $Q^3$ [Reference Sols, Szurek and Wiśniewski60, Theorem], X is isomorphic to one of the following:

1. (i) $\mathbb {P}(\mathcal {N})$ , where $\mathcal {N}$ is the null correlation bundle over $\mathbb {P}^3$ ;

2. (ii) $\mathbb {P}(\mathcal {S})$ , where $\mathcal {S}$ is the spinor bundle over $Q^3$ ;

3. (iii) $\mathbb {P}({\pi }^{\ast }\mathcal {N})$ , where $\mathcal {N}$ is the null correlation bundle over $\mathbb {P}^3$ and $\pi : Q^3\to \mathbb {P}^3$ is a double cover.

By Examples 6.6 and 6.7, X is isomorphic to $\mathbb {P}(\mathcal {N})\cong \mathbb {P}(\mathcal {S})$ .

Proof. By [Reference Kollár, Miyaoka and Mori39, Corollary 2.9] and [Reference Druel22, Lemmas 4.2 and 4.4 and Proposition 6.1], we see that Y is isomorphic to $\mathbb {P}^2$ or $\mathbb {P}^1\times \mathbb {P}^1$ . In the latter case, X admits a contraction onto $\mathbb {P}^1$ , which contradicts our assumption. Thus, Y is isomorphic to $\mathbb {P}^2$ ; our assertion follows from [Reference Szurek and Wiśniewski62].

Proof. In both cases, there exists a $\mathbb {P}^{1}$ -fibration whose special fibers are two-dimensional, which is denoted by $\varphi : X\to Y$ . Remark that the structure of $\varphi $ is described in [Reference Kachi33]. By [Reference Kachi33, Lemma 5.2], there exist only finitely many two-dimensional fibers of $\varphi $ , which are denoted by $F_1=\varphi ^{-1}(y_1), \ldots , F_m=\varphi ^{-1}(y_m)$ . By [Reference Kachi33, Theorem 0.6] (see also [Reference Andreatta and Wiśniewski4, Lemma 5.9.4]), Y is smooth and we have $F_i\cong \mathbb {P}^2$ and $N_{F_i/X}\cong \Omega _{\mathbb {P}^2}(1)$ . Let $\pi : \tilde {X}\to X$ be the blow-up of X along $F_1 \sqcup \cdots \sqcup F_m$ and $\tilde {Y}$ the blow-up of Y along $\{y_1, \ldots , y_m\}$ . Then the exceptional divisor E of $\tilde {X}\to X$ is

$$ \begin{align*}\mathbb{P}(N_{F_1/X}^{\vee}) \sqcup \cdots \sqcup \mathbb{P}(N_{F_m/X}^{\vee})\cong\mathbb{P}(T_{\mathbb{P}^2}(-1)) \sqcup \cdots \sqcup \mathbb{P}(T_{\mathbb{P}^2}(-1)). \end{align*} $$

By the universal property of blow-up (see, for instance, [Reference Hartshorne28, Proposition 7.14 and Corollary 7.15]), there exists a morphism $\tilde {\varphi }: \tilde {X}\to \tilde {Y}$

making a commutative diagram as shown. Remark that $\tilde {\varphi }: \tilde {X}\to \tilde {Y}$ is a $\mathbb {P}^1$ -bundle (cf. [Reference Andreatta and Wiśniewski2, Remark 4.13]). Each irreducible component $\mathbb {P}(N_{F_i/X}^{\vee })\cong \mathbb {P}(T_{\mathbb {P}^2}(-1))$ of the exceptional divisor E is isomorphic to a hyperplane section of $\mathbb {P}^2\times \mathbb {P}^2$ ; the restrictions $\tilde {\varphi }|_{\mathbb {P}(N_{F_i/X}^{\vee })}$ and $\pi |_{\mathbb {P}(N_{F_i/X}^{\vee })}$ give two natural projections of $\mathbb {P}(T_{\mathbb {P}^2}(-1)) \subset \mathbb {P}^2\times \mathbb {P}^2$ onto $\mathbb {P}^2$ .

By our assumption, we have another elementary contraction $\psi : X \to Z$ which is

• • a $\mathbb {P}^{1}$ -fibration whose special fibers are two-dimensional; or

• • an equidimensional $Q^{2}$ -fibration.

In each case, Z is a smooth variety of dimension at most three (see [Reference Andreatta, Ballico and Wiśniewski1, Proposition 1.4.1] and [Reference Kachi33, Theorem 0.6]). We proceed as in the proof of [Reference Casagrande12, Theorem 4.1]. Now, we have the following diagram:

This yields a morphism $\tilde {Y}\to \mathrm {Chow}(Z)$ ; we denote by $Y'$ the image of this morphism. Let us consider a geometric quotient of Z for $Y'$ ; by applying [Reference Bonavero, Casagrande and Druel7, Corollary 1], there exists an elementary contraction $f: Z\to W$ which contracts a rational curve $\psi \circ \pi (\tilde {\varphi }^{-1}(\tilde {y}))$ for any closed point $\tilde {y} \in \tilde {Y}$ . This yields that the surface $\psi (F_i)$ is contracted by f. Since f is constructed as a geometric quotient for $Y'$ , it is of fiber type; if the relative dimension of f is one, then f is a conic bundle, which contradicts the fact that f contracts a surface. This yields that f has relative dimension at least two. However, if the relative dimension of f is two, then W is isomorphic to $\mathbb {P}^1$ ; it turns out that X admits a contraction $f \circ \psi : X \to \mathbb {P}^1$ ; this contradicts our assumption. Consequently, we see that W is a point and $\rho _Z=1$ . This implies that $\rho _X=2$ . Since Z is a uniruled variety of $\rho _Z=1$ , it is a Fano variety. Since $\dim Z=2$ or $3$ , Z is $\mathbb {P}^2$ or a Fano threefold of $\rho _Z =1$ . Then there exists a rational curve $\ell \subset Z$ such that $-K_Z\cdot \ell $ is equal to the Fano index of Z (see, for instance, [Reference Šokurov58]).

If $\psi $ is a $\mathbb {P}^{1}$ -fibration which is not equidimensional, then it is a smooth morphism away from finitely many fibers (see [Reference Kachi33, Lemma 5.2]). This means that $\psi |_{\psi ^{-1}({\ell })}: \psi ^{-1}({\ell }) \to \ell $ is smooth over a nonempty open subset of $\ell $ . Thus, the existence of the desired rational curve $\tilde {\ell } \subset X$ follows from the same argument as in [Reference Watanabe66, Section 4.3]. So assume that $\psi : X\to Z$ is an equidimensional $Q^2$ -fibration. In this case, Z is isomorphic to $\mathbb {P}^2$ ; then, by replacing $\ell $ if necessary, we may assume that $\psi |_{\psi ^{-1}({\ell })}: \psi ^{-1}({\ell }) \to \ell $ is smooth over a nonempty open subset of $\ell $ . Thus, we are done.

By this claim, we may take a family of minimal sections $\mathcal {M} \subset \mathrm {RatCurves}^n(X)$ as in [Reference Watanabe66, Section 4.3], which is unsplit. Moreover, we claim the following.

To prove this, assume the contrary; that is, there exists an $\mathcal {M}$ -curve which is not contracted by $\varphi $ . Then, any $\mathcal {M}$ -curve is not contracted by $\varphi $ . If $\mathcal {M}$ is a covering family, it follows from [Reference Bonavero, Casagrande and Druel7, Corollary 1] that $\varphi : X\to Y$ is the geometric quotient for $\mathcal {M}$ . This contradicts our assumption. Thus, $\operatorname {\mathrm {codim}}_X\mathrm {Locus}(\mathcal {M})\geq 1$ ; then by Proposition 2.11, for any point $x \in \mathrm {Locus}(\mathcal {M})$ , we have

$$ \begin{align*}\dim \mathrm{Locus}(\mathcal{M}_x) \geq \deg_{(-K_X)}\mathcal{M}+\operatorname{\mathrm{codim}}_X\mathrm{Locus}(\mathcal{M}) -1. \end{align*} $$

By our assumption, we have $\deg _{(-K_X)}\mathcal {M} \geq 2$ ; combining this with the above inequality, we see that $\dim \mathrm {Locus}(\mathcal {M}_x)\geq 2$ for any point $x \in \mathrm {Locus}(\mathcal {M})$ and $\operatorname {\mathrm {codim}}_X\mathrm {Locus}(\mathcal {M})=1$ . Let us put $D:=\mathrm {Locus}(\mathcal {M})$ . Here, we prove that there exists a curve $\ell _{\varphi } \subset D$ (resp. $\ell _{\psi }\subset D$ ) which is contracted by $\varphi $ (resp. $\psi $ ). If $\dim D>\dim \varphi (D)$ and $\dim D>\dim \psi (D)$ , then we may find such curves. So assume $\dim D=\dim \varphi (D)=\dim \psi (D)$ . Since $\varphi $ and $\psi $ have two-dimensional fibers and D is a divisor on X, we may find the desired curves by a dimension count.

We claim that the curve $\ell _{\varphi }$ is not contained in $\mathrm {Locus}(\mathcal {M}_x)$ for any $x\in D$ . In fact, if $\ell _{\varphi }$ is contained in $\mathrm {Locus}(\mathcal {M}_{x_0})$ for some $x_0\in D$ , it follows from [Reference Kollár38, Chapter II, Corollary 4.21] that any $\mathcal {M}$ -curve is contracted by $\varphi $ ; this is a contradiction. Thus, we see that $\dim \mathrm {Locus}(\mathcal {M}_x)=2$ for any $x\in D$ and we obtain

$$ \begin{align*}D=\bigcup_{x\in \ell_{\varphi}} \mathrm{Locus}(\mathcal{M}_x). \end{align*} $$

By [Reference Occhetta51, Lemma 3.2 and Remark 3.3], there exist rational numbers $\lambda \geq 0, \mu $ , and $[C]\in \mathcal {M}$ such that $\ell _{\psi }=\lambda \ell _{\varphi } +\mu C\in N_1(X)$ . For any ample divisor A on Z, we have

$$ \begin{align*}0=\psi^{\ast}(A)\cdot \ell_{\psi}=\lambda \psi^{\ast}(A)\cdot\ell_{\varphi} +\mu \psi^{\ast}(A)\cdot C. \end{align*} $$

This implies that $\mu \leq 0$ . Therefore, we obtain

$$ \begin{align*}\ell_{\psi} +(-\mu) C=\lambda \ell_{\varphi}\in \mathbb{R}_{\geq 0}[\ell_{\varphi}]. \end{align*} $$

Since $\mathbb {R}_{\geq 0}[\ell _{\varphi }]$ is an extremal ray, this yields $\ell _{\psi } \in \mathbb {R}_{\geq 0}[\ell _{\varphi }]$ ; this is a contradiction. As a consequence, Claim 6.12 holds.

On the other hand, since Y is a Fano threefold of $\rho _Y=1$ , we may find a rational curve $\ell ' \subset Y$ such that $-K_Y\cdot \ell '=i_Y$ . By the same argument as in Claims 6.11 and 6.12, we may find a rational curve $\tilde {\ell '}\subset X$ , which is contracted by $\psi $ .

Let $H_Y$ and $H_Z$ be the ample generators of $ Pic(Y)$ and $ Pic(Z)$ , respectively. Since $\rho _X=2$ , $\{\varphi ^{\ast }H_Y, \psi ^{\ast }H_Z\}$ is a basis of $N^1(X)$ . Thus, there exist rational numbers $a, b$ such that $-K_X=a\varphi ^{\ast }H_Y+b\psi ^{\ast }H_Z$ . Denoting by $C_{\varphi }$ and $C_{\psi }$ the minimal extremal rational curves of the rays associated with $\varphi $ and $\psi $ , respectively, we have

$$ \begin{align*}2=-K_X\cdot C_{\psi}=aH_Y\cdot \varphi_{\ast}(C_{\psi}). \end{align*} $$

On the other hand, we also have

$$ \begin{align*}2\leq -K_X\cdot \tilde{\ell'}=aH_Y\cdot \varphi_{\ast}(\tilde{\ell'})=a. \end{align*} $$

These imply that $H_Y\cdot \varphi _{\ast }(C_{\psi })=1$ and $a=2$ . In the same way, we also obtain $H_Z\cdot \psi _{\ast }(C_{\varphi })=1$ and $b=2$ . As a consequence, $-K_X=2(\varphi ^{\ast }H_Y+\psi ^{\ast }H_Z)$ ; this yields that $i_X=2$ .

Proof. This can be proved by the same argument as in Proposition 6.10.

Proof. Let X be a smooth projective variety of dimension n and $\rho _X>1$ . Assume that $\wedge ^kT_X$ is strictly nef for $1\leq k<n$ . Then [Reference Li, Ou and Yang45, Theorem 1.2] yields that X is rationally connected. Applying [Reference Watanabe67, Proposition 1.4], X is a Fano variety. Then we see that $\iota _X\geq n+2-k$ (see, for instance, [Reference Gachet27, Lemma 2.1]). Thus, our assertion follows from [Reference Watanabe68, Theorem 1.4] and Theorem 1.4.