Proof. One direction of statement (1) is clear since the pseudoeffective cone $\overline {\mathrm {Eff}}(\mathbb {P}(E))$ is closed. For the converse, we assume that $E\hspace {-0.8ex}<\hspace {-0.8ex}\delta \hspace {-0.8ex}>$ is pseudoeffective. Since D is big, by [Reference Lazarsfeld28, Chapter 2, Corollary 2.2.7], there exists an ample $\mathbb {Q}$ -Cartier $\mathbb {Q}$ -Weil divisor A and an effective $\mathbb {Q}$ -Weil divisor $N'$ , such that $D \sim _{\mathbb {Q}} A + N'$ . On the other hand, as $\Lambda +\pi ^*\delta $ is $\pi $ -ample, there exists a rational number $0<\epsilon \ll 1$ , such that the $\mathbb {Q}$ -Cartier $\mathbb {Q}$ -Weil divisor $\epsilon (\Lambda +\pi ^*\Delta )+\pi ^*A$ is ample. This implies that:

$$\begin{align*}\Lambda+\pi^*(\Delta+D)\sim_{\mathbb{Q}} \epsilon(\Lambda+\pi^*\Delta)+\pi^*A + (1-\epsilon)(\Lambda+\pi^*\Delta)+N' \end{align*}$$

is big since $\mathrm {Big}(\mathbb {P}(E))$ is the interior of $\overline {\mathrm {Eff}}(\mathbb {P}(E))$ . Then it follows again from [Reference Lazarsfeld28, Chapter 2, Corollary 2.2.7] that there exists an effective $\mathbb {Q}$ -Cartier $\mathbb {Q}$ -Weil divisor N, such that:

$$\begin{align*}N\sim_{\mathbb{Q}} \Lambda + \pi^* (\Delta + D). \end{align*}$$

One can easily obtain one implication of statement (2), since $\mathrm {Big}(\mathbb {P}(E))$ is open. Conversely, we assume that $E\hspace {-0.8ex}<\hspace {-0.8ex}\delta -\gamma \hspace {-0.8ex}>$ is pseudoeffective for some big $\mathbb {Q}$ -Cartier class $\gamma $ . Similar to the proof of statement (1), there exist a rational number $0<\epsilon \ll 1$ and an ample $\mathbb {Q}$ -Cartier $\mathbb {Q}$ -Weil divisor A and an effective $\mathbb {Q}$ -Weil divisor N such that the $\mathbb {Q}$ -Cartier divisor class $\epsilon (\Lambda +\pi ^*(\delta -\gamma ))+\pi ^*A$ is ample and:

$$\begin{align*}\Lambda+\pi^*\delta \equiv_{\mathbb{Q}} \epsilon(\Lambda+\pi^*(\delta-\gamma))+\pi^*A+(1-\epsilon)(\Lambda+\pi^*(\delta-\gamma)) + N. \end{align*}$$

Note that the $\mathbb {Q}$ -Cartier divisor class $(1-\epsilon )(\Lambda +\pi ^*(\delta -\gamma ))$ is pseudoeffective by our assumption. Then it is clear that the $\mathbb {Q}$ -Cartier divisor class $\Lambda +\pi ^*\delta $ is big.

Proof. Assume $E\langle -ac_1(E) \rangle $ is effective for some rational number $a>0$ ; that is, we have:

$$\begin{align*}H^0(X, \mathrm{Sym}^mE \otimes \det(E^*)^{\otimes (am)})\neq 0 \end{align*}$$

for some positive integer m, such that $am$ is an integer. This gives an injection:

$$\begin{align*}\det(E)^{\otimes (am)} \to \mathrm{Sym}^mE, \end{align*}$$

which yields:

$$\begin{align*}\mu_H^{\mathrm{max}} (\mathrm{Sym^m}E) \geq \mu_H (\det(E)^{\otimes (am)}) = am c_1(E) \cdot H^{n-1}. \end{align*}$$

On the other hand, as E is H-semistable, so is $\operatorname *{\mathrm {Sym}}^m E$ . Hence, we obtain:

$$\begin{align*}\mu_H^{\mathrm{max}} (\mathrm{Sym^m}E) = \mu_H (\mathrm{Sym^m} E) = \frac{m c_1(E)\cdot H^{n-1}}{r}, \end{align*}$$

which gives that $a \leq 1/r$ . In particular, it follows from Proposition 2.7 that $E\hspace {-0.8ex}<\hspace {-0.8ex}-\frac {1}{r} c_1(E)\hspace {-0.8ex}>$ is not big.

Proof. As $T_X$ is big and nef, $\mathbb {P}(T_X)$ is a weak Fano manifold, that is, $-K_{\mathbb {P}(T_X)}$ is big and nef. Then statement (1) follows from the base-point-free theorem, and statement (2) follows from [Reference Birkar, Cascini, Hacon and McKernan3, Corollary 1.3.2] since a weak Fano manifold is always log Fano.

Proof. Claims (1) and (2) follow from [Reference Muñoz, Occhetta, Solá Conde, Watanabe and Wiśniewski37, Proposition 5.10] and [Reference Wierzba52, Theorem 1.3]. To prove (3), we follow the argument of [Reference Wierzba52, Proposition 2.13]. Let:

$$\begin{align*}0\rightarrow E\rightarrow T_{{\mathcal X}}\rightarrow {\mathcal O}_{{\mathcal X}}(1)\rightarrow 0 \end{align*}$$

be the natural contact structure on ${\mathcal X}$ . By our assumption, we have ${\mathcal O}_{{\mathcal X}}(1)|_F\cong {\mathcal O}_F$ . Then the contact structure induces a natural isomorphism $E|_F\cong E^*|_F$ . Let:

$$\begin{align*}E|_F\cong \bigoplus_{i=1}^{2n-2} {\mathcal O}_{\mathbb{P}^1}(a_i) \end{align*}$$

be the decomposition with $a_1\geq \cdots \geq a_{2n-2}$ . Since the problem is local in ${\mathcal Y}$ , after removing a subvariety of codimension at least $4$ of ${\mathcal Y}$ , we may assume that all fibres of $\varepsilon $ are at most one dimensional. Then the argument of [Reference Wierzba52, Proposition 2.13] applies verbatim to our situation to obtain $h^1(F,\Omega _X|_F)=1$ and the short exact sequence below:

(3.3) $$ \begin{align} 0\longrightarrow {\mathcal O}_F \longrightarrow \Omega_X|_F \longrightarrow E^*|_F=E|_F\longrightarrow 0 \end{align} $$

implies $h^1(F,E|_F)=1$ . This implies that $a_{2n-2}=-2$ and $a_{2n-3}\geq -1$ . The isomorphism $E|_F\cong E^*|_F$ shows that $E|_F$ must be of the form:

$$\begin{align*}{\mathcal O}_{\mathbb{P}^1}(2)\oplus {\mathcal O}_{\mathbb{P}^1}(1)^{\oplus a} \oplus {\mathcal O}_{\mathbb{P}^1}^{\oplus 2n-2a-4} \oplus {\mathcal O}_{\mathbb{P}^1}(-1)^{\oplus a} \oplus {\mathcal O}_{\mathbb{P}^1}(-2). \end{align*}$$

Then it follows from (3.3) and the fact $h^1(E|_F)=h^1(E^*|_F)=1$ that $T_X|_F$ is either of the form $E\oplus {\mathcal O}_{\mathbb {P}^1}$ or of the form:

$$\begin{align*}{\mathcal O}_{\mathbb{P}^1}(2)\oplus {\mathcal O}_{\mathbb{P}^1}(1)^{\oplus a} \oplus {\mathcal O}_{\mathbb{P}^1}^{\oplus 2n-2a-4} \oplus {\mathcal O}_{\mathbb{P}^1}(-1)^{\oplus a+2}. \end{align*}$$

It is clear that the $\mathrm {Chow}({\mathcal X})$ has dimension $\geq 2n-3$ at $[F]$ as $\Gamma $ has dimension $2n-2$ and the deformation of F dominates $\Gamma $ . Hence, we have $h^0(F,T_{{\mathcal X}}|_F)\geq 2n$ and $T_X|_F$ is of the form $E\oplus {\mathcal O}_{\mathbb {P}^1}$ .

Proof. Recall that ${\mathcal X}$ is a Mori dream space by Lemma 3.7. By [Reference Hu and Keel17, Proposition 1.11], there exists a SQM , such that:

$$\begin{align*}[g^*(\Lambda-\beta_X\pi^*H)]\in \mathrm{Nef}({\mathcal X}'). \end{align*}$$

Moreover, as ${\mathcal X}'$ is again a Mori dream space, it follows that $\mathrm {Nef}({\mathcal X}')$ is generated by semiample $\mathbb {Q}$ -Cartier divisors. Hence, $\beta _X$ is a rational number.

Now assume that $\alpha _X\not =\beta _X$ . Then $\Lambda -\alpha _X\pi ^*H$ is not movable. Moreover, as X has Picard number $1$ , it is clear that $R=\mathbb {R}_{>0}[\Lambda -\alpha _X\pi ^*H]$ is an extremal ray of $\overline {\mathrm {Eff}}({\mathcal X})$ . Then, by Lemma 2.4, there exists a unique prime divisor $\Gamma \subset {\mathcal X}$ , such that $[\Gamma ]\in \mathbb {R}_{>0}[\Lambda -\alpha _X \pi ^*H]$ . In particular, we have:

$$\begin{align*}\alpha_X=\frac{b(\Gamma)}{a(\Gamma),} \end{align*}$$

and, hence, $\alpha _X$ is again a rational number. Denote by $\Gamma '$ the divisor $g^*\Gamma $ . Note that the pseudoeffective cones and movables are preserved by $g^*$ . In particular, by Lemma 2.2, we obtain:

$$\begin{align*}{\mathbb B}_+^1(g^*(\Lambda - \beta_X\pi^*H))\subset {\mathbb B}_{-}^1(\Gamma')\subset \Gamma'. \end{align*}$$

Since $\Lambda -\beta _X \pi ^*H$ is not contained in the interior of $\overline {\mathrm {Mov}}({\mathcal X})$ , so is the pull-back $g^*(\Lambda -\beta _X\pi ^*H)$ . In particular, by Lemma 2.3, we have:

$$\begin{align*}{\mathbb B}_+^1(g^*(\Lambda - \beta_X \pi^*H))=\Gamma'. \end{align*}$$

On the other hand, as $g^*(\Lambda -\beta _X\pi ^*H)$ is nef, by [Reference Birkar2, Theorem 1.4], $\Gamma '$ is contained in the null locus of $g^*(\Lambda -\beta _X\pi ^*H)$ . In particular, we obtain:

$$\begin{align*}g^*\Gamma \cdot \left(g^*(\Lambda-\beta_X\pi^*H)\right)^{2n-2} = \Gamma'\cdot \left(g^*(\Lambda-\beta_X\pi^*H)\right)^{2n-2} = 0. \end{align*}$$

This completes the proof.

Proof. Since $\Lambda $ is big and nef, it follows from [Reference Birkar2, Theorem 1.4] that ${\mathbb B}_{+}(\Lambda )$ coincides with the exceptional locus $\Gamma $ of $\varepsilon $ . In particular, $\Gamma \cdot \Lambda ^{2n-2}=(\Lambda |_{\Gamma })^{2n-2} = 0$ . Moreover, according to Lemma 2.3, $[\Lambda ]$ is not contained in the interior of $\overline {\mathrm {Mov}}({\mathcal X})$ . This implies $\beta _X=0$ . Then it follows from Corollary 2.5 that $[\Gamma ]\in \mathbb {R}_{>0}[\Lambda -\alpha _X\pi ^*H]$ . Combining Proposition 3.10 with the same argument as in the proof of Theorem 3.4(3) shows that $b(\Gamma )\pi ^*H\cdot F\leq 2$ with equality if and only if $\Gamma $ is smooth along F. Then we obtain that $b(\Gamma )\leq 2$ with equality if and only if $\Gamma $ is smooth along F and $\pi ^*H\cdot F=1$ . Now the result follows from the following two claims.

Proof. Let H be the ample generator of $\operatorname *{\mathrm {Pic}}(X)$ and denote by $\alpha _X:=\alpha (X,H)$ the pseudoeffective threshold of X with respect to H. Let $i_X$ be the index of X, that is, $-K_X=i_XH$ . Then the normalised tangent bundle of X is pseudoeffective if and only if the following inequality holds:

$$\begin{align*}\alpha_X\geq \frac{i_X}{\dim(X)}. \end{align*}$$

On the other hand, since ${\mathcal C}_x$ is not dual defective, the total dual VMRT $\check {{\mathcal C}}\subset \mathbb {P}(T_X)$ is a prime divisor. Write $[\check {{\mathcal C}}]\equiv a \Lambda - b\pi ^*H$ . Then, by Theorem 3.4, we obtain:

$$\begin{align*}a=\operatorname*{\mathrm{codeg}}({\mathcal C}_x),\quad 0<bH\cdot {\mathcal K}\leq 2 \quad \text{and} \quad \alpha_X=\frac{b}{a}. \end{align*}$$

Therefore we get:

$$\begin{align*}\frac{2}{\operatorname*{\mathrm{codeg}}({\mathcal C}_x)}\geq \alpha_X H\cdot {\mathcal K} \geq \frac{i_X H\cdot {\mathcal K}}{\dim(X)}=\frac{\dim({\mathcal C}_x)+2}{\dim(X)}, \end{align*}$$

and the result follows. Here, we use the fact that $\dim ({\mathcal C}_x)=-K_X\cdot {\mathcal K}-2=i_XH\cdot {\mathcal K}-2$ .

Proof. Denote by $\bar {L}$ the restriction $L|_S$ . As $g=q$ , by [Reference Zak53] (see also [Reference Sommese49, Corollary 1.5.2]), we know that either S is a geometrically ruled surface with smooth $C\in |\bar {L}|$ as sections or the pair $(S,\bar {L})$ is isomorphic to one of the following:

$$ \begin{align*} (\mathbb{P}^2,{\mathcal O}_{\mathbb{P}^2}(1))\quad \text{or} \quad (\mathbb{P}^2,{\mathcal O}_{\mathbb{P}^2}(2)). \end{align*} $$

First, we note that the case $(S,\bar {L})=(\mathbb {P}^2,{\mathcal O}_{\mathbb {P}^2}(2))$ does not happen. This was already proved by Gaetano Scorza. Indeed, by Bott’s formula, we have $H^1(\mathbb {P}^2,T_{\mathbb {P}^2}(-2))=0$ . Then we can apply Zak’s inextendibility theorem (see [Reference Zak54]) to conclude that Z is a cone over S. In particular, Z is singular, which is a contradiction.

Next, we assume that the pair $(S,\bar {L})$ is isomorphic to $(\mathbb {P}^2,{\mathcal O}_{\mathbb {P}^2}(1))$ . Then, we have $L^3=\bar {L}^2=1$ . In particular, $(Z,L)$ itself is isomorphic to $(\mathbb {P}^3,{\mathcal O}_{\mathbb {P}^3}(1))$ , which contradicts our assumption.

Finally, we assume that S is a geometrically ruled surface over a smooth curve. According to [Reference Liu30, Theorem 1.3], the pair $(Z,L)$ is one of the following varieties:

1. (1) $(\mathbb {Q}^3,{\mathcal O}_{\mathbb {Q}^3}(1))$ ;

2. (2) $(\mathbb {P}^3,{\mathcal O}_{\mathbb {P}^3}(2))$ or

3. (3) there exists a vector bundle E of rank $3$ over B, such that $Z=\mathbb {P}(E)$ and S is an element in the linear system $|{\mathcal O}_{\mathbb {P}(E)}(1)|$ .

In Case (1), it is clear that $Z\subset \mathbb {P}^N$ is linearly normal, and, hence, it is a quadric hypersurface of $\mathbb {P}^4$ . In Case (2), one can easily obtain that $g(S)=1$ , while $q(S)=0$ , which does not satisfy our assumption. In Case (3), we note that $Z\subset \mathbb {P}^N$ is actually a three-dimensional scroll, such that all the fibres of $\mathbb {P}(E)\rightarrow B$ are linearly embedded. As B is a curve, it is well known that the dual defect of Z is equal to $1$ in this case (see, for instance, [Reference Tevelev50, Theorem 7.21]) and the fact $\operatorname *{\mathrm {codeg}}(Z)=\deg (Z)=c_1(E)$ follows from Lemma 4.6 below.

Proof. Let H be a hyperplane section of Z and denote by $\pi :\mathbb {P}(E)\rightarrow B$ the natural projection. By [Reference Beltrametti, Fania and Sommese1] (see also [Reference Tevelev50, Theorem 6.1]), we have:

$$\begin{align*}\operatorname*{\mathrm{codeg}}(Z)= c_2({\mathcal J}(H))\cdot H=c_1(\Omega_Z\otimes H)\cdot H^2 + c_2(\Omega_Z\otimes H)\cdot H, \end{align*}$$

where ${\mathcal J}(H)$ is the first jet bundle. By a straightforward computation, we get:

$$\begin{align*}c_1(\Omega_Z)=\pi^*c_1(E)+\pi^*K_C-3H, \end{align*}$$

and:

$$\begin{align*}c_2(\Omega_Z)=-3\pi^*K_C\cdot H-2\pi^*c_1(E)\cdot H + 3H^2. \end{align*}$$

As a consequence, we obtain $\operatorname *{\mathrm {codeg}}(Z)=c_1(E)\cdot H^2=H^3=\deg (Z)$ .

Proof of Proposition 1.8

Denote by L the restriction of ${\mathcal O}_{\mathbb {P}^N}(1)|_Z$ , and let $S\subset Z$ be a general hyperplane section. According to Lemma 4.6, we shall assume that Z is not dual defective (cf. [Reference Tevelev50, Example 7.6]). By the codegree formula (cf. [Reference Lanteri and Turrini27, Proposition 1.1]), we have:

(4.3) $$ \begin{align} d^*=(b_3(Z)-b_1(Z))+2(b_2(S)-b_2(Z))+2(g(S)-q(S)). \end{align} $$

Set $A=b_3(Z)-b_1(Z)$ , $B=b_2(S)-b_2(Z)$ and $C=2(g(S)-q(S))$ . Then both A and C are even nonnegative integers and B is a positive integer since Z is not dual defective by our assumption (see [Reference Lanteri and Turrini27, Propositions 1.2 and 1.4]). If $C=0$ , then we can conclude by Lemma 4.5 that Z satisfies $d^*=d$ . Hence, we may assume also that $C>0$ in the sequel. On the other hand, if $p_g(S)\not =0$ , then it follows from [Reference Lanteri and Turrini27, Proposition 2.5] that we have $g(S)-q(S)\geq N-1$ . In particular, we obtain:

$$\begin{align*}d^*\geq 2B+2(g(S)-q(S))\geq 2+2(N-1)=2N. \end{align*}$$

From now on, we shall assume that $B>0$ , $C>0$ and $p_g(S)=0$ . In particular, it follows [Reference Sommese49, Theorem 1.5] that $K_S+\bar {L}$ is globally generated and [Reference Sommese49, Proposition 2.1] implies that we have:

$$\begin{align*}d=\bar{L}^2\leq K_S^2+4g(S)-4, \end{align*}$$

where $\bar {L}$ is the restriction $L|_S$ and the equality holds if and only if $\Phi _{|K_S+\bar {L}|}$ is not generically finite, that is, $\dim (\Phi _{|K_S+\bar {L}|}(S))\leq 1$ . In particular, S is uniruled and so is Z. On the other hand, by Noether’s formula, we have:

$$\begin{align*}K_S^2=12\chi({\mathcal O}_S)-\chi_{\mathrm{top}}(S)=10-8q(S)-h^{1,1}(S). \end{align*}$$

This implies:

$$\begin{align*}d\leq 6+4g(S)-8q(S)-h^{1,1}(S). \end{align*}$$

Applying the codegree formula (4.3), we get:

$$ \begin{align*} d & \leq 6+2(d^*-2B-A)-4q(S)-h^{1,1}(S) \\ & \leq 2d^*+6-4(B+q(S))-2A-h^{1,1}(S). \end{align*} $$

Note that $q(S)$ and A are nonnegative integers. Thus, since B and $h^{1,1}(S)=b_2(S)$ are positive integers, it follows that we have $d\leq 2d^*$ unless the following condition happens:

$$\begin{align*}B=h^{1,1}(S)=b_2(S)=1\quad \mathrm{and}\quad A=q(S)=0. \end{align*}$$

This is impossible since we have $b_2(S)>b_2(Z)\geq 1$ by our assumption. Moreover, an easy similar argument shows that if the equality $d=2d^*$ holds, then we must have:

$$\begin{align*}B=1,\quad h^{1,1}(S)=b_2(S)=2\quad \mathrm{and}\quad A=q(S)=0. \end{align*}$$

This implies that $\rho (Z)=b_2(Z)=1$ . In particular, as Z is uniruled, it follows that Z is a Fano threefold of Picard number $1$ . Note that $(K_Z+2L)|_S=K_S+\bar {L}$ is globally generated but not big. This implies that $-K_Z=2L$ . In particular, the pair $(Z,L)$ is isomorphic to either $(\mathbb {P}^3,{\mathcal O}_{\mathbb {P}^3}(2))$ or a del Pezzo threefold. If Z is a del Pezzo threefold, then S is a del Pezzo surface with $b_2(S)=2$ and $-K_S=\bar {L}$ . However, according to the classification of del Pezzo threefolds of Picard number $1$ , we must have $d=\bar {L}^2=K_S^2\leq 5$ . This implies that $b_2(S)\geq 4$ , which is a contradiction. Hence, $2d^*=d$ if and only if $(Z,L)$ is isomorphic to $(\mathbb {P}^3,{\mathcal O}_{\mathbb {P}^3}(2))$ ; that is, the projective variety $Z\subset \mathbb {P}^N$ is projectively equivalent to either the second Veronese variety $\nu _2(\mathbb {P}^3)\subset \mathbb {P}^9$ or its isomorphic projection in $\mathbb {P}^8$ .

Finally, we assume that Z satisfies the equality (1.2). Then we have $5d^*=2(N+1)$ . In particular, by our results above, we must have:

$$\begin{align*}\frac{4(N+1)}{5}=2d^*\geq d\geq N-2. \end{align*}$$

This implies $N\leq 14$ and $d^*\leq 6$ . Then, by the classification of smooth projective threefolds of codegree at most $6$ given in [Reference Lanteri and Turrini27], one can easily check that the only possibilities are the quadric threefold $\mathbb {Q}^3\subset \mathbb {P}^4$ (with $d^*=2$ ) and the Veronese variety $\nu _2(\mathbb {P}^3)\subset \mathbb {P}^9$ (with $d^*=4$ ).

Proof. Note that $T_S$ is always semistable with respect to $-K_S$ by [Reference Fahlaoui9], and its normalisation is not nef by Theorem 1.1. On the other hand, since S is simply connected, there does not exist nontrivial unramified coverings of S. In particular, applying [Reference Nakayama38, IV, Theorem 4.8] to S and $T_S$ , we see that only Case (A) and Case (C) of [Reference Nakayama38, IV, Theorem 4.8] may happen in our situation. In other words, either $T_S$ splits as a direct sum $L_1\oplus L_2$ as in Case (A) or $-K_S\equiv 2L$ for some line bundle L on S as in Case (C). In the latter case, it is easy to see that S is isomorphic to the quadric surface from the classification of del Pezzo surfaces. In the former case, the surface S is isomorphic to a product of curves (see, for instance, [Reference Höring13, Theorem 1.4]). This implies immediately that S is isomorphic to the product $\mathbb {P}^1\times \mathbb {P}^1$ as S is rationally connected.

Proof. By the biduality theorem, the dual variety $\check {{\mathcal C}}_x$ does not contain hyperplanes as irreducible components since ${\mathcal C}_x$ is purely dimensional, and, hence, $\operatorname *{\mathrm {codeg}}({\mathcal C}_x)\geq 2$ . Let us denote by $\mathbb {P}^m=\mathbb {P}(W)\subset \mathbb {P}(\Omega _{X,x})$ the linear span of ${\mathcal C}_x$ .

First, we assume that $\operatorname *{\mathrm {codeg}}({\mathcal C}_x)=2$ , that is, the dual $\check {{\mathcal C}}_x\subset \mathbb {P}(T_{X,x})$ is an irreducible quadric hypersurface of $\mathbb {P}(T_{X,x})$ . Then the VMRT ${\mathcal C}_x$ itself is irreducible. On the other hand, if $\check {{\mathcal C}}_x$ is not smooth, then it is an irreducible quadric cone. According to the biduality theorem, since ${\mathcal C}_x$ is not dual defective, the VMRT ${\mathcal C}_x$ is a smooth quadric hypersurface in $\mathbb {P}^m\subset \mathbb {P}(\Omega _{X,x})$ . Then it follows from [Reference Hwang18, Propositions 2.4 and 2.6] that we must have $\mathbb {P}^m=\mathbb {P}(\Omega _{X,x})$ . Therefore, by Theorem 1.5, the variety X is isomorphic to a quadric hypersurface.

Next, we assume that ${\mathcal C}_x$ is smooth with $\operatorname *{\mathrm {codeg}}({\mathcal C}_x)=3$ and the normalised tangent bundle of X is pseudoeffective. Then, $\check {{\mathcal C}}_x$ is an irreducible hypersurface of degree $3$ , and, hence, ${\mathcal C}_x$ is irreducible and smooth. By Zak’s classification of linearly nondegenerate smooth varieties with codegree $3$ [Reference Zak55, Theorem 5.2], we obtain that $\dim ({\mathcal C}_x)\geq 2$ and:

$$\begin{align*}\dim({\mathcal C}_x)>\frac{m-1}{2}, \end{align*}$$

unless ${\mathcal C}_x\subset \mathbb {P}^m$ is $\nu _2(\mathbb {P}^2)\subset \mathbb {P}^5$ . On the other hand, it can be directly checked that the tangential variety of $\nu _2(\mathbb {P}^2)\subset \mathbb {P}^5$ is linearly nondegenerate. Therefore, it follows from [Reference Hwang18, Proposition 2.6] that the tangential variety of ${\mathcal C}_x$ is linearly nondegenerate. Then we can apply [Reference Hwang18, Proposition 2.4] to obtain that $\mathbb {P}^m=\mathbb {P}(\Omega _{X,x})$ . In particular, as ${\mathcal C}_x$ is assumed to be not dual defective, it follows from Proposition 4.3 that ${\mathcal C}_{x}\subset \mathbb {P}(\Omega _{X,x})$ is projectively equivalent to one of the four Severi varieties. Then one can apply Theorem 1.5 to conclude that X is isomorphic to one of the four varieties in the theorem.

Proof of Theorem 1.9

For statement (1), we assume that the VMRT ${\mathcal C}_x\subset \mathbb {P}(\Omega _{X,x})$ is irreducible, linearly nondegenerate and not dual defective. Let $\check {{\mathcal C}}\subset \mathbb {P}(T_X)$ be the total dual VMRT. Write $[\check {{\mathcal C}}]\equiv a\Lambda - b\pi ^*H$ , where $\Lambda $ is the tautological divisor of the projectivised tangent bundle $\pi :\mathbb {P}(T_X)\rightarrow X$ and H is the ample generator of $\operatorname *{\mathrm {Pic}}(X)$ . Let $i_X$ be the index of X. Without loss of generality, we may assume that $\alpha (X,-K_X)>0$ , or equivalently, $T_X$ is big. In particular, by Theorem 3.4, we have $b>0$ and:

$$\begin{align*}\alpha(X,-K_X) = \frac{b}{a i_X} =\frac{b}{i_X \operatorname*{\mathrm{codeg}}({\mathcal C}_x)}\leq \frac{b}{i_X}\cdot \frac{\dim({\mathcal C}_x)+2}{2\dim(X)}. \end{align*}$$

The last inequality follows from the Segre inequality (1.1). In particular, note that we have $\dim ({\mathcal C}_x)+2=-K_X\cdot {\mathcal K} = i_X H \cdot {\mathcal K}$ and $bH\cdot {\mathcal K}\leq 2$ , thus, we get:

$$\begin{align*}\alpha(X,-K_X)\leq \frac{1}{\dim(X)} \end{align*}$$

with equality only if the VMRT ${\mathcal C}_x\subset \mathbb {P}(\Omega _{X,x})$ satisfies the equality (1.2). Hence, if the normalised tangent bundle of X is pseudoeffective and Conjecture 1.6 holds, then the VMRT ${\mathcal C}_x\subset \mathbb {P}(\Omega _{X,x})$ is projectively equivalent to one of the varieties listed in Conjecture 1.6 and we then conclude by Theorem 1.5 and Table 1.

For statement (2), assume that the VMRT ${\mathcal C}_x\subset \mathbb {P}(\Omega _{X,x})$ is not dual defective and the normalised tangent bundle of X is pseudoeffective. By Proposition 4.3, if the condition (2.1) (respectively, condition (2.2)) holds, then we get $\operatorname *{\mathrm {codeg}}({\mathcal C}_x)<3$ (respectively, $\operatorname *{\mathrm {codeg}}({\mathcal C}_x)<4$ ) and the results follow from Theorem 4.9 above. If the condition (2.3) holds, then it is clear that $\dim ({\mathcal C}_x)\geq 1$ as the VMRT can not be a single point. Then the result follows from statement (1.2), Proposition 1.8 and Theorem 4.4.

Proof of Corollary 1.10

By assumption, the VMRT ${\mathcal C}_x\subset \mathbb {P}(\Omega _{X,x})$ is either a nonlinear smooth curve, a nonlinear smooth surface or a nonlinear smooth hypersurface $(n=5)$ . In particular, the VMRT ${\mathcal C}_x$ is not dual defective by [Reference Tevelev50, Examples 1.19 and 7.5 and Theorem 4.25]. Then the result follows directly from Theorem 1.9 (2.2).

Proof of Corollary 1.12

The result follows from Theorem 1.9 (2.2) and (2.3).

Proof. For classical cases, this follows from the proof of [Reference Namikawa40, Proposition 3.1]. For exceptional cases, assume G is of exceptional type. In most cases, $\mathcal {O}$ is an even orbit or an orbit with trivial fundamental group, which implies that $\widehat {s}$ is birational. For the remaining cases, the degree is computed in [Reference Fu10, Appendix].

Proof. Since $G/P$ is not isomorphic to projective spaces, the birational contraction $\varepsilon : \mathbf {P} (T^*_{G/P}) \to \mathbf {P}(\widetilde {{\mathcal O}}) $ is not an isomorphism and $\mathrm {Nef}(\mathbf {P} (T^*_{G/P}) )=\langle [\Lambda ],[\pi ^*H]\rangle $ . Let $\alpha (G/P, H)$ be the pseudoeffective threshold of $G/P$ with respect to H, then $[\Lambda -\alpha (G/P, H)\pi ^*H]$ generates an extremal ray of $\overline {\mathrm {Eff}}(\mathbf {P} (T^*_{G/P}) )$ . Statement (2) follows directly from Proposition 3.13. Thus, it remains to prove statement (1).

By our assumption, the birational contraction $\varepsilon $ is small. By Propositions 5.2 and 5.3, there exists a flop $\mu : \mathbf {P} (T^*_{G/P}) \dashrightarrow \mathbf {P} (T^*_{G/Q})$ , with $G/P \simeq G/Q$ as projective varieties. It follows that the pull-back $\mu ^*\mathrm {Nef}( \mathbf {P} (T^*_{G/Q}))$ is contained in $\overline {\mathrm {Mov}}( \mathbf {P} (T^*_{G/P}))$ (cf. Lemma 3.7). Moreover, it is clear that we have $\mu ^*\Lambda =\Lambda $ since $-K_{\mathbf {P} (T^*_{G/P})}=n\Lambda $ and $\mu $ is a SQM. This implies that the pull-back $\mu ^*\mathrm {Nef}(\mathbf {P} (T^*_{G/P}))$ is contained in the cone $\langle [\Lambda -\alpha (G/P, H)\pi _1^*H], [\Lambda ]\rangle $ . Nevertheless, as $\pi _2^*H$ is not big, it follows that $[\mu ^*\pi _2^*H]$ is not contained in the interior of $\overline {\mathrm {Eff}}(\mathbf {P} (T^*_{G/P}))$ . So we get:

$$\begin{align*}\overline{\mathrm{Eff}}(\mathbf{P} (T^*_{G/P}))=\overline{\mathrm{Mov}}(\mathbf{P} (T^*_{G/P}))=\langle [\mu^*\pi_2^*H], [\pi_1^*H]\rangle. \end{align*}$$

On the other hand, as $\pi _2^*H$ is globally generated and $\mu $ is a SQM, the stable base locus ${\mathbb B}(\mu ^*\pi _2 H)$ has codimension at least $2$ . Hence, we obtain:

$$\begin{align*}\overline{\mathrm{Eff}}(\mathbf{P} (T^*_{G/P}))=\mathrm{Eff}(\mathbf{P} (T^*_{G/P}))=\overline{\mathrm{Mov}}(\mathbf{P} (T^*_{G/P}))=\mathrm{ Mov}(\mathbf{P} (T^*_{G/P})). \end{align*}$$

This finishes the proof.

Proof. Since $\mu $ is a SQM and $\ell _i$ ’s are general, we may assume that both $\mu $ and $\mu ^{-1}$ are isomorphisms in a neighborhood of $\ell _i$ . In particular, we have:

$$\begin{align*}a(H) = (a(H)\Lambda - b(H) \pi^*_1 H)\cdot \ell_1 = \mu^*\pi_2^*H\cdot \ell_1 = \pi_2^*H\cdot \mu_*(\ell_1) \end{align*}$$

and:

$$\begin{align*}0=\pi_2^*H\cdot \ell_2 = \mu^*\pi_2^*H \cdot \mu^{-1}_*(\ell_2) = a(H) - b(H)\pi_1^*H\cdot \mu^{-1}_*(\ell_2). \end{align*}$$

Here, we note that $\mu ^*\Lambda =\Lambda $ and $\Lambda \cdot \ell _i=1$ . Now it remains to show that the morphism $\mu (\ell _1)\rightarrow C:=\pi _2(\mu (\ell _1))$ is birational. Let $f:\mathbb {P}^1\rightarrow C$ be the normalisation. As $T_{G/Q}$ is nef, there exist integers $a_1\geq \dots \geq a_k>a_{k+1}=\dots =a_n=0$ (with $k\leq n$ ), such that:

$$\begin{align*}f^*T_{G/Q}\cong \bigoplus_{i=1}^k{\mathcal O}_{\mathbb{P}^1}(a_i)\oplus {\mathcal O}_{\mathbb{P}^1}^{\oplus (n-k)}. \end{align*}$$

Denote by d the degree of $\mu (\ell _1)\rightarrow C$ . Then we have:

$$\begin{align*}\pi_2^*T_{G/Q}|_{\mu(\ell_1)} \cong \bigoplus_{i=1}^k{\mathcal O}_{\mathbb{P}^1}(d a_i) \oplus {\mathcal O}_{\mathbb{P}^1}^{\oplus (n-k)}. \end{align*}$$

As $\Lambda \cdot \mu (\ell _1)=1$ , if $d\geq 2$ , then $\mu (\ell _1)$ is contained in:

$$\begin{align*}\mathbb{P}({\mathcal O}_{\mathbb{P}^1}^{\oplus (n-k)}) \subset \mathbb{P}(f^*T_{G/P}). \end{align*}$$

On the other hand, as $\mathbb {P}({\mathcal O}_{\mathbb {P}^1}^{\oplus (n-k)})$ is dominated by curves with $\Lambda $ -degree $0$ , thus, $\mathbb {P}({\mathcal O}_{\mathbb {P}^1}^{\oplus (n-k)})$ is contained in the exceptional locus of $\varepsilon $ and so is $\mu (\ell _1)$ , which is absurd.

Proof. First, note that the morphism $\varepsilon $ is a Mori extremal contraction with respect to a klt pair $(\mathbf {P}(T^*_{G/P}),\Delta )$ (see [Reference Muñoz, Occhetta, Solá Conde, Watanabe and Wiśniewski37, Proposition 5.5]). Thus, as $\rho (\mathbf {P}(T^*_{G/P}))=2$ , $\varepsilon $ is divisorial and $\mathbf {P}(T^*_{G/P})$ is rationally connected, it follows that $\mathbf {P}(\widetilde {\mathcal {O}})$ is a $\mathbb {Q}$ -factorial Fano variety of Picard number $1$ . Moreover, note that we have $\Gamma \cdot \Lambda ^{2n-2}=0$ . This implies immediately:

$$\begin{align*}\frac{b(\Gamma)}{a(\Gamma)}=\frac{\Lambda^{2n-1}}{\Lambda^{2n-2}\cdot \pi^*H}. \end{align*}$$

Let $\widetilde {H}$ be a general member in $|\pi ^*H|$ , and set $H'=\varepsilon _*\widetilde {H}$ . As $\mathbf {P}(\widetilde {\mathcal {O}})$ is $\mathbb {Q}$ -factorial, there exists a rational number r, such that $H'\equiv r \Lambda '$ . Moreover, by the negativity lemma, there exists a nonnegative rational number $\alpha $ , such that: