Proof. Suppose that $\zeta $ is big. We can write ${\mathcal {D}}_i=k_i\zeta +\Pi ^*H_i$ for some $k_i\geq 0$ and divisor $H_i$ on X. Let A be an ample divisor on X such that $A+\sum _{i=1}^{l} H_i$ is also ample for all $1\leq l\leq n$ . Then there exists the smallest integer $m>0$ such that $m\zeta -\Pi ^*A$ is effective by Kodaira’s lemma. Let ${\mathcal {E}}_0\sim m\zeta -\Pi ^*A$ be an effective divisor. For a general curve $\widetilde {\ell _1}$ in the family whose members cover the divisor ${\mathcal {D}}_1$ , we have

$$\begin{align*}{\mathcal{E}}_0.\widetilde{\ell_1} =(m\zeta-\Pi^*A).\widetilde{\ell_1} =-A.\Pi(\widetilde{\ell_1})<0. \end{align*}$$

Thus ${\mathcal {E}}_0-{\mathcal {D}}_1\geq 0$ , and we can find an effective divisor ${\mathcal {E}}_1={\mathcal {E}}_0-{\mathcal {D}}_1$ . For a general curve $\widetilde {\ell _2}$ in the family whose members cover the divisor ${\mathcal {D}}_2$ , we also have

$$\begin{align*}{\mathcal{E}}_1.\widetilde{\ell_2} =((m-k_1)\zeta-(\Pi^*A+\Pi^*H_1)).\widetilde{\ell_2} =-(A+H_1).\Pi(\widetilde{\ell_2})<0, \end{align*}$$

as $A+H_1$ is ample by the assumption. Thus, we are again able to take an effective divisor ${\mathcal {E}}_2={\mathcal {E}}_1-{\mathcal {D}}_2$ . By iterating the process, we arrive to obtain an effective divisor

$$\begin{align*}{\mathcal{E}}_n ={\mathcal{E}}_{n-1}-{\mathcal{D}}_n =\cdots ={\mathcal{E}}_0-{\mathcal{D}} =\left(m-\sum_{i=1}^n k_i\right)\zeta-\left(\Pi^*A+\sum_{i=1}^n \Pi^*H_i\right) =(m-k)\zeta-\Pi^*A-\Pi^*H \end{align*}$$

on ${\mathbb {P}}(T_X)$ . In particular, $(m-k)\zeta -\Pi ^*A$ is effective, and it contradicts the minimality of m.

Proof. Let

$$\begin{align*}[\breve{\mathcal{C}}] \sim k\zeta+b_1\Pi^*H_1+b_2\Pi^*H_2 \end{align*}$$

for some $k>0$ . Once we show that $b_i\geq 0$ for $i=1,\,2$ , the assertion follows from Lemma 2.2 because $\breve {\mathcal {C}}$ satisfies (†).

Let $x=x_1$ be the point where the first condition holds for $\ell =\ell _1$ . Let ${\mathcal {H}}={\mathbb {P}}(N_{\ell |X}|_{x})\subseteq {\mathbb {P}}(T_X|_{x})$ be the hyperplane projectively dual to the point $\tau _{\ell ,x}\in {\mathbb {P}}(\Omega _X|_{x})$ . Then, from the first condition, we have

$$\begin{align*}\emptyset\not={\mathcal{H}}\backslash\,\breve{\mathcal{C}}|_{x}\subseteq{\mathbb{P}}(T_X|_{x}). \end{align*}$$

Hence, there exists a section $\widetilde {\ell }\subseteq {\mathbb {P}}(T_X|_{\ell })\subseteq {\mathbb {P}}(T_X)$ corresponding to a quotient $T_X|_{\ell }\to {\mathcal {O}}_{{\mathbb {P}}^1}(a)$ for $a=a_{11}$ in a factor of $N_{{\ell }|X}$ passing through a point $w\in {\mathcal {H}}\backslash \breve {\mathcal {C}}|_{x}$ over x, i. e., $w=\widetilde {\ell }|_{x}=\widetilde {\ell }\cap {\mathbb {P}}(T_X|_{x})$ (see Figure 1). Thus, we have $\widetilde {\ell }|_{x}\cap \breve {\mathcal {C}}|_{x}=\emptyset $ on ${\mathbb {P}}(T_X|_{x})$ , and it implies that $\widetilde {\ell }$ is not contained in $\breve {\mathcal {C}}$ .

Figure 1

The total dual VMRT $\breve {\mathcal {C}}|_x$ at x when the VMRT ${\mathcal {C}}_x$ is finite.

From the second condition, we can observe that every section $\widetilde {\ell }\subseteq {\mathbb {P}}(T_X|_{\ell })\subseteq {\mathbb {P}}(T_X)$ which corresponds to a quotient $T_X|_{\ell }\to {\mathcal {O}}_{{\mathbb {P}}^1}(a)$ in a factor of $N_{{\ell }|X}$ satisfies $\zeta .\widetilde {\ell }=a=a_{11}\leq 0$ in ${\mathbb {P}}(T_X)$ . Therefore, for such $\widetilde {\ell _1}=\widetilde {\ell }$ , we have

$$\begin{align*}b_2(H_2.\ell_1) \geq k(\zeta.\widetilde{\ell_1})+b_1(\Pi^*H_1.\widetilde{\ell_1})+b_2(\Pi^*H_2.\widetilde{\ell_1}) =[\breve{\mathcal{C}}].\widetilde{\ell_1} \geq 0, \end{align*}$$

as $\widetilde {\ell _1}$ is not contained in $\breve {\mathcal {C}}$ . Thus, $b_2\geq 0$ . By the same argument, we can show that $b_1\geq 0$ .

Proof. We first prove the following claim.

Claim. If S is a smooth del Pezzo surface of degree $d\leq 4$ , then $H^0(S,{\operatorname {Sym}}^m T_S(K_S))=0$ for all $m>0$ .

Assume that $d=4$ . Suppose that $H^0(S,{\operatorname {Sym}}^m T_S(K_S))\neq 0$ for some $m>0$ . Then we can take the smallest $m>0$ with $H^0(S,{\operatorname {Sym}}^m T_S(K_S))\neq 0$ . Let D be an effective divisor on ${\mathbb {P}}(T_S)$ such that $D\sim m\xi +\Lambda ^*K_S$ for the tautological class $\xi ={\mathcal {O}}_{{\mathbb {P}}(T_S)}(1)$ of $\Lambda :{\mathbb {P}}(T_S)\to S$ . Following the proof of [Reference Höring, Liu and Shao5, Prop. 3.5], there exist $5$ pairs $|\ell _{i1}|$ and $|\ell _{i2}|$ of pencils of conics on S such that

$$\begin{align*}[\ell_{i1}]+[\ell_{i2}]\sim -K_S \ \ \text{and}\ \ [\breve{\mathcal{C}}_{i1}]+[\breve{\mathcal{C}}_{i2}]\sim 2\xi, \end{align*}$$

where $\breve {\mathcal {C}}_{i1}$ and $\breve {\mathcal {C}}_{i2}$ are the total dual VMRTs on ${\mathbb {P}}(T_S),$ which are, respectively, associated to the pencils $|\ell _{i1}|$ and $|\ell _{i2}|$ for $i=1,\,\ldots ,\,5$ . Then, for a minimal section $\widetilde {\ell _{ij}}\subseteq {\mathbb {P}}(T_X|_{\ell _{ij}})\subseteq {\mathbb {P}}(T_X)$ of $\ell _{ij}$ , we have

$$\begin{align*}D.\widetilde{\ell_{ij}} =(m\xi+\Lambda^*K_S).\widetilde{\ell_{ij}} =K_S.\ell_{ij} <0. \end{align*}$$

So $\breve {\mathcal {C}}_{ij}$ are contained in the support of D for all $i=1,\,\ldots ,\,5$ and $j=1,\,2$ as $\widetilde {\ell _{ij}}$ cover $\breve {\mathcal {C}}_{ij}$ . Thus, there is an effective divisor

$$\begin{align*}(m-10)\xi+\Lambda^*K_S =D-\sum_{i=1}^5(\breve{\mathcal{C}}_{i1}+\breve{\mathcal{C}}_{i2}) \end{align*}$$

on ${\mathbb {P}}(T_S)$ , and it contradicts the minimality of $m>0$ . This proves the claim for $d=4$ .

Assume that $d\leq 3$ . Then, by [Reference Höring, Liu and Shao5, Th. 1.2], $T_S$ is not pseudoeffective, so $H^0(S,{\operatorname {Sym}}^m T_S)=0$ for all $m>0$ , and hence $H^0(S,{\operatorname {Sym}}^m T_S(K_S))=0$ for all $m>0$ because $-K_S$ is effective. This proves the claim for $d\leq 3$ .

Let $S=X_t$ be a general fiber of $p:X\to C$ , which is a smooth del Pezzo surface of degree $d\leq 4$ . By twisting ${\mathcal {O}}_S(K_S)$ after taking the symmetric power to the exact sequence

$$\begin{align*}0 \to T_S \to T_X|_S \to {\mathcal{O}}_S \to 0, \end{align*}$$

we obtain the following exact sequence on S.

$$\begin{align*}0 \to {\operatorname{Sym}}^m T_S(K_S) \to {\operatorname{Sym}}^m T_X|_S(K_S) \to {\operatorname{Sym}}^{m-1} T_X|_S(K_S) \to 0. \end{align*}$$

Note that $H^0(S,K_S)=0$ . Then, by induction on $m>0$ , it follows that $H^0(S,{\operatorname {Sym}}^{m}T_X|_S (K_S))=0$ for all $m>0$ due to the claim. Thus, $T_X|_S$ is not big.

Suppose that $T_X$ is big. Consider the family $\{{\mathbb {P}}(T_X|_{X_t})\}_{t\in C}$ of codimension $1$ subvarieties of ${\mathbb {P}}(T_X)$ which cover ${\mathbb {P}}(T_X)$ , i. e. ${\mathbb {P}}(T_X)=\bigcup _{t\in C}{\mathbb {P}}(T_X|_{X_t})$ . Notice that $\xi =\zeta |_{X_t}$ is the tautological class of ${\mathbb {P}}(T_X|_{X_t})$ . Since $\zeta $ is big, $|m\zeta |$ gives a birational map ${\mathbb {P}}(T_X)\dashrightarrow {\mathbb {P}}^N$ for $m\gg 1$ , so $|m\xi |$ induces a birational map ${\mathbb {P}}(T_X|_{X_t})\dashrightarrow {\mathbb {P}}^M$ , and hence $T_X|_{X_t}$ is big for general $t\in C$ , a contradiction. Therefore, $T_X$ is not big.

Proof. Let $\zeta ={\mathcal {O}}_{{\mathbb {P}}(E)}(1)$ be the tautological class of ${\mathbb {P}}(E)$ . Assume that $\zeta +\Pi ^*H$ is big for every big ${\mathbb {Q}}$ -divisor H on X. If H is a big ${\mathbb {Q}}$ -divisor on X, then $\varepsilon H$ is big for any $\varepsilon>0$ . So we can observe that $\zeta $ is given by the limit of a family of big ${\mathbb {Q}}$ -divisors $\zeta +\varepsilon \Pi ^*H$ as $\varepsilon \to 0$ . Thus, $\zeta $ is pseudoeffective.

Conversely, assume that $\zeta $ is pseudoeffective, and let H be an arbitrary big ${\mathbb {Q}}$ -divisor on X. Then $H\sim A+N$ for some ample divisor A and effective divisor N. Because $\zeta +m\Pi ^*A$ is ample on ${\mathbb {P}}(T_X)$ for sufficiently large $m\gg 0$ , we can write

$$\begin{align*}m(\zeta+\Pi^*H) =(\zeta+m\Pi^*A) + m\Pi^*N + (m-1)\zeta, \end{align*}$$

as the sum of a big divisor $(\zeta +m\Pi ^*A)+m\Pi ^*N$ and a pseudoeffective divisor $(m-1)\zeta $ on ${\mathbb {P}}(T_X)$ . Thus, $\zeta +\Pi ^*H$ is big.

Proof. We continue to use the notation above. Note that $\alpha $ , $\beta $ , and $\widetilde {f}$ are birational morphisms in (3.1). Thus, we have the following implication.

$$\begin{align*}\zeta \text{ is big} \ \ \Leftrightarrow\ \ \alpha^*\zeta \text{ is big} \ \ \Rightarrow\ \ \alpha^*\zeta+R \sim (\widetilde{f}\circ \beta)^*\eta \text{ is big} \ \ \Leftrightarrow\ \ \eta \text{ is big.} \end{align*}$$

Now, assume that $T_X$ is pseudoeffective and let $H'$ be a big ${\mathbb {Q}}$ -divisor on Z. By Lemma 3.1, $T_X\otimes {\mathcal {O}}_X(f^*H')$ is big as $f^*H'$ is big on X. Then we can show that $T_Z\otimes {\mathcal {O}}_Z(H')$ is big on Z using the same argument as before. Since $H'$ was arbitrary, by Lemma 3.1 again, $T_Z$ is pseudoeffective.

Proof. Let ${\mathcal {M}}$ be the family of lines ${l}$ on Q, which is known to be ${\mathcal {M}}\cong {\mathbb {P}}^3$ . Then $N_{{l}|Q}\cong {\mathcal {O}}_{{\mathbb {P}}^1}(1)\oplus {\mathcal {O}}_{{\mathbb {P}}^1}$ , and ${\mathcal {M}}$ is a family of unbendable rational curves on Q. Let $\breve {\mathcal {D}}$ be the total dual VMRT on ${\mathbb {P}}(T_Q)$ associated with ${\mathcal {M}}$ . As Q is homogeneous and the VMRT ${\mathcal {D}}_{z}$ of ${\mathcal {M}}$ at $z\in Q$ is irreducible and reduced, so is $\breve {\mathcal {D}}|_z$ for all $z\in Q$ , and hence $\breve {\mathcal {D}}$ is irreducible and reduced.

Let $H'={\mathcal {O}}_Q(1)$ and $\eta ={\mathcal {O}}_{{\mathbb {P}}(T_Q)}(1)$ be the tautological class of $\Phi : {\mathbb {P}}(T_Q)\to Q$ . It is known that

$$\begin{align*}[\breve{\mathcal{D}}]\sim 2\eta-2\Phi^*H' \end{align*}$$

(see [Reference Solá Conde and Wiśniewski19, Proof of Prop. 3.1] for instance). Therefore, we can write

$$\begin{align*}[\widetilde{f}^*\breve{\mathcal{D}}]=2\widetilde{f}^*\eta-2\widetilde{\Phi}^*H, \end{align*}$$

where $\widetilde {f}: {\mathbb {P}}(f^*T_Q)\to {\mathbb {P}}(T_Q)$ and $\widetilde {\Phi }:\mathbb P(f^*T_Q)\rightarrow X$ are the natural morphisms induced by $f: X\to Q$ and $\Phi : {\mathbb {P}}(T_Q)\to Q$ .

Let ${\mathcal {K}}$ be the family of rational curves on X containing the strict transform $[\ell ]$ of $[{l}]$ for general $[{l}]\in {\mathcal {M}}$ . Since a general member $[{l}]\in {\mathcal {M}}$ does not intersect with the blow-up center $\Gamma $ , ${\mathcal {K}}$ is a family of unbendable rational curves on X.

Let $\breve {\mathcal {C}}$ be the total dual VMRT on ${\mathbb {P}}(T_X)$ associated to ${\mathcal {K}}$ . Because the total dual VMRT $\breve {\mathcal {D}}$ is given by the union of minimal sections of rational curves of ${\mathcal {M}}$ , $\breve {\mathcal {C}}|_U$ and $\breve {\mathcal {D}}|_V$ coincide under the isomorphism ${\mathbb {P}}(T_X|_{U})\cong {\mathbb {P}}(T_Q|_{Z})$ over $U=X\backslash D$ and $V=Q\backslash \Gamma $ . As the closure is uniquely determined by a subset out of codimension $1$ , $\breve {\mathcal {C}}$ is the strict transform of $\breve {\mathcal {D}}$ from ${\mathbb {P}}(T_Q)$ to ${\mathbb {P}}(T_X)$ . Moreover, $\breve {\mathcal {C}}$ is the strict transform of $\widetilde {f}^*\breve {\mathcal {D}}$ from ${\mathbb {P}}(f^*T_Q)$ to ${\mathbb {P}}(T_X)$ . As $\breve {\mathcal {D}}$ is irreducible and reduced, so is its strict transform $\breve {\mathcal {C}}$ .

Let $m\geq 0$ be the order of vanishing of $\widetilde {f}^*\breve {\mathcal {D}}$ on $S\cong {\mathbb {P}}_{D}(T_{D/\Gamma }(D))\subseteq {\mathbb {P}}(f^*T_Q)$ . Then the strict transform $\breve {\mathcal {C}}$ of $\widetilde {f}^*\breve {\mathcal {D}}$ from ${\mathbb {P}}(f^*T_Q)$ to ${\mathbb {P}}(T_X)$ satisfies

$$\begin{align*}[\breve{\mathcal{C}}] \sim \alpha_*\beta^*(2\widetilde{\eta}-2\widetilde{\Phi}^*H)-m\Pi^*D =2\zeta-2\Pi^*H+(2-m)\Pi^*D. \end{align*}$$

We will complete the proof by showing that $m=0$ , which is equivalent to showing that $S\not \subseteq \widetilde {f}^*\breve {\mathcal {D}}$ .

Let $z\in \Gamma $ . Then ${\mathbb {P}}(f^*T_Q|_{f^{-1}(z)})\cong {\mathbb {P}}(T_Q|_z)\times f^{-1}(z)$ . Moreover, ${\mathbb {P}}(T_{D/\Gamma }(D)|_{f^{-1}(z)})$ is a subvariety of ${\mathbb {P}}(f^*T_Q|_{f^{-1}(z)})$ whose image is $\widetilde {f}({\mathbb {P}}(T_{D/\Gamma }(D)|_{f^{-1}(z)}))={\mathbb {P}}(N_{\Gamma |Q}|_z)$ in ${\mathbb {P}}(T_Q|_z)$ . Since the line ${\mathbb {P}}(N_{\Gamma |Q}|_z)$ is not contained in the conic $\breve {\mathcal {D}}|_z\subseteq {\mathbb {P}}(T_Q|_z)$ , ${\widetilde {f}}^*\breve {\mathcal {D}}|_z$ does not contain ${\mathbb {P}}(T_{D/\Gamma }(D)|_{f^{-1}(z)})=S|_{f^{-1}(z)}$ (see Figure 2). Thus, we can conclude that $S\not \subseteq \widetilde {f}^*\breve {\mathcal {D}}$ .

Figure 2

The restriction $\widetilde {f}^*\breve {\mathcal {D}}|_x\subseteq {\mathbb {P}}(f^*T_Q|_x)$ of $\widetilde {f}^*\breve {\mathcal {D}}\subseteq {\mathbb {P}}(f^*T_Q)$ over $x\in D$ .

Proof. Let $\breve {\mathcal {D}}$ be the total dual VMRT on ${\mathbb {P}}(T_{V_5})$ associated to the family of lines on $V_5$ . Then, according to [Reference Höring, Liu and Shao5, Th. 5.4],

$$\begin{align*}[\breve{\mathcal{D}}] \sim 3\eta-\Phi^*H' \end{align*}$$

for $H'={\mathcal {O}}_{V_5}(1)$ and the tautological class $\eta ={\mathcal {O}}_{{\mathbb {P}}(T_{V_5})}(1)$ of $\Phi : {\mathbb {P}}(T_{V_5})\to {V_5}$ .

Let $\breve {\mathcal {C}}$ be the total dual VMRT on ${\mathbb {P}}(T_X)$ associated to the family of the strict transforms of lines on $V_5$ . Then $\breve {\mathcal {C}}$ is the strict transform of $\breve {\mathcal {D}}$ from ${\mathbb {P}}(T_{V_5})$ to ${\mathbb {P}}(T_X)$ . By the same argument in the proof of Proposition 3.6, the proof is completed once we show that $S\not \subseteq \widetilde {f}^*\breve {\mathcal {D}}$ .

The VMRT of the family of lines on $V_5$ at a point $z\in V_5$ is the union of three points on ${\mathbb {P}}(\Omega _{V_5}|_z)$ with multiplicity [Reference Furushima and Nakayama3, Lem. 2.3(1)]. So $\breve {\mathcal {D}}|_z$ is the union of three lines on ${\mathbb {P}}(T_{V_5}|_z)$ . If there is a point $z\in \Gamma $ such that every line passing through z is not tangent to $\Gamma $ at z, then ${\mathbb {P}}(N_{\Gamma |V_5}|_z)\not \subseteq \breve {\mathcal {D}}|_z$ on ${\mathbb {P}}(T_{V_5}|_z)$ , and it implies that $S\not \subseteq \widetilde {f}^*\breve {\mathcal {D}}$ . Thus, it suffices to show that there are only finitely many lines on $V_5$ which is tangent to $\Gamma $ .

We consider $V_5$ as the subvariety of ${\mathbb {P}}^6$ and fix a $4$ -dimensional linear subspace L of ${\mathbb {P}}^6$ containing $\Gamma $ . Then $L\cap V_5$ is a union of finitely many curves on $V_5$ . If a line l on ${\mathbb {P}}^6$ is tangent to $\Gamma $ , then $l\subseteq L$ , and if l is also contained in $V_5$ , then $l\subseteq L\cap V_5$ . Therefore, we can conclude that there are only finitely many lines l on $V_5$ which is tangent to $\Gamma $ .

Proof. Let ${\mathcal {K}}$ be the family of unbendable rational curves on X containing the strict transforms of general secant lines of $\Gamma $ on ${\mathbb {P}}^3$ . Note that $N_{\ell |X}\cong {\mathcal {O}}_{{\mathbb {P}}^1}\oplus {\mathcal {O}}_{{\mathbb {P}}^1}$ for general $[\ell ]\in {\mathcal {K}}$ . We can observe that $\overline {{\mathcal {K}}}\cong S^2\Gamma $ . Indeed, $S^2\Gamma $ is smooth, and there exists a finite morphism from $S^2\Gamma $ to the locus of secant lines of $\Gamma $ in the Grassmannian $\text {Gr}(2,4)$ of lines on ${\mathbb {P}}^3$ . Let ${\overline {q}}_0:\overline {{\mathcal {U}}_0}\to \overline {{\mathcal {K}}}$ be the universal family of the secant lines of $\Gamma $ on ${\mathbb {P}}^3$ and ${{\overline {e}}_0}:\overline {{\mathcal {U}}_0}\to {\mathbb {P}}^3$ be its evaluation morphism. Note that ${\overline {q}}_0:\overline {{\mathcal {U}}_0}\to \overline {{\mathcal {K}}}$ is a ${\mathbb {P}}^1$ -fibration and $\overline {{\mathcal {U}}_0}\cong {\mathbb {P}}_{\overline {\mathcal {K}}}(V)$ for some vector bundle V of rank $2$ on $\overline {\mathcal {K}}$ , where V is given by the pull-back of the universal bundle via the induced morphism $\overline {{\mathcal {K}}}\to \mathrm {Gr}(2,4)$ .

As a general hyperplane meets a general secant line at $1$ point on ${\mathbb {P}}^3$ , we take ${V={{\overline {q}}_0}_*{{\overline {e}}_0}^*{\mathcal {O}}_{{\mathbb {P}}^3}(1)}$ so that ${\mathcal {O}}_{{\mathbb {P}}_{\overline {\mathcal {K}}}(V)}(1)={{\overline {e}}_0}^*{\mathcal {O}}_{{\mathbb {P}}^2}(1)$ .

We first prove the case where $\Gamma $ has a trisecant line. Let ${\overline {q}}:\overline {\mathcal {U}}\to \overline {\mathcal {K}}$ be the normalization of the universal family of curves on X with the evaluation morphism ${\overline {e}}: \overline {\mathcal {U}}\to X$ . Then ${\overline {q}}:\overline {\mathcal {U}}\to \overline {\mathcal {K}}$ is a conic bundle over $\overline {\mathcal {K}}$ whose discriminant locus is the locus of trisecant lines of $\Gamma $ on ${\mathbb {P}}^3$ . Moreover, $\overline {\mathcal {U}}$ is obtained by the blow-up $\sigma :\overline {{\mathcal {U}}}\to \overline {{\mathcal {U}}_0}$ along some curve in $\overline {{\mathcal {U}}_0}$ over the locus in $\overline {\mathcal {K}}$ of trisecant lines.

Let $\xi =\sigma ^*{\mathcal {O}}_{{\mathbb {P}}_{\overline {\mathcal {K}}}(V)}(1)$ and E be the exceptional divisor of $\sigma :\overline {{\mathcal {U}}}\to \overline {{\mathcal {U}}_0}$ . Then we can write

$$\begin{align*}K_{\overline{\mathcal{U}}/\overline{\mathcal{K}}} =\sigma^*K_{{\mathbb{P}}_{\overline{\mathcal{K}}}(V)/\overline{\mathcal{K}}}+E =-2\xi+{\overline{q}}^*c_1(V)+E. \end{align*}$$

Let $\breve {\mathcal {C}}$ be the total dual VMRT on ${\mathbb {P}}(T_X)$ associated to ${\mathcal {K}}$ . Then $\breve {\mathcal {C}}$ is irreducible by its construction, and it is reduced as a general VMRT ${\mathcal {C}}_x$ of $\overline {\mathcal {K}}$ is reduced for general $x\in X$ . Moreover, as ${\mathcal {K}}$ is locally unsplit, we can apply Proposition 2.3 to obtain the linear equivalence

$$\begin{align*}[\breve{\mathcal{C}}] \sim k\zeta+\Pi^*{\overline{e}}_*(K_{\overline{\mathcal{U}}/\overline{\mathcal{K}}}) \end{align*}$$

on ${\mathbb {P}}(T_X)$ for $k=\deg ({\overline {e}})$ . From the commutativity, we have $\xi =\sigma ^*{{\overline {e}}_0}^*{\mathcal {O}}_{{\mathbb {P}}^3}(1)={\overline {e}}^*f^*{\mathcal {O}}_{{\mathbb {P}}^3}(1)={\overline {e}}^*H$ , and

$$\begin{align*}{\overline{e}}_*(K_{\overline{{\mathcal{U}}}/\overline{{\mathcal{K}}}}) =-2{\overline{e}}_*\xi +{\overline{e}}_*{\overline{q}}^*c_1(V)+ {\overline{e}}_*E =-2kH+{\overline{e}}_*{\overline{q}}^*c_1(V)+ {\overline{e}}_*E. \end{align*}$$

First, let

$$\begin{align*}{\overline{e}}_*{\overline{q}}^*c_1(V)=aH+bD \end{align*}$$

for some $a,\,b\in {\mathbb {Z}}$ . We know from blow-up formulas (e. g. see [Reference Mori and Mukai14, (4.3)]) that

$$\begin{align*}H^3=1,\ \ H^2.D=0,\ \ H.D^2=-d. \end{align*}$$

Thus,

$$\begin{align*}aH^3 ={\overline{e}}_*{\overline{q}}^*c_1(V).H^2 ={\overline{q}}^*c_1(V).\xi^2 ={\overline{q}}^*c_1(V).({\overline{q}}^*c_1(V)\xi-{\overline{q}}^*c_2(V)) ={c_1(V)}^2, \end{align*}$$

and ${c_1(V)}^2=kH^3+c_2(V)$ due to

$$ \begin{align*} kH^3 &=\xi^3 ={\overline{q}}^*c_1(V)\xi^2-{\overline{q}}^*c_2(V)\xi ={\overline{q}}^*c_1(V)({\overline{q}}^*c_1(V)\xi-{\overline{q}}^*c_2(V))-{\overline{q}}^*c_2(V)\xi\\ &={c_1(V)}^2-c_2(V). \end{align*} $$

Then the number $r:=c_2(V)$ can be calculated by the number of points of the degeneracy locus of a general member of $|{\mathcal {O}}_{{\mathbb {P}}_{\overline {\mathcal {K}}}(V)}(1)|$ . That is, $r=\#\{[\ell ]\in \overline {\mathcal {K}}\,|\,\ell \subseteq P\}$ for general $P\in |H|$ .

On the other hand,

$$ \begin{align*} bH.D^2 &={\overline{e}}_*{\overline{q}}^*c_1(V).H.D ={\overline{q}}^*c_1(V).\xi.{\overline{e}}^*D =(\xi^2+{\overline{q}}^*c_2(V)).{\overline{e}}^*D =kH^2.D+{\overline{q}}^*c_2(V).{\overline{e}}^*D\\ &=2c_2(V), \end{align*} $$

as a general secant line meets $\Gamma $ at $2$ points on ${\mathbb {P}}^3$ so that ${\overline {e}}^*D\equiv 2\xi $ in $\text {N}^1(\overline {\mathcal {U}}/\overline {\mathcal {K}})$ .

Moreover, we can deduce that

$$\begin{align*}{\overline{e}}_*E=mD \end{align*}$$

for some $m>0$ from the observation that the image ${\overline {e}}(E)$ lies on the exceptional locus of $f:X\to {\mathbb {P}}^3$ , and the multiplicity m is given by the number of trisecant lines of $\Gamma $ on ${\mathbb {P}}^3$ passing through a general point of $\Gamma $ .

By the projection $\pi _x$ from a general point x in $\mathbb P^3$ , it gives a one-to-one correspondence between the number of secant lines of $\Gamma $ through x and the number of nodes of the curve $\pi _x(\Gamma )$ in ${\mathbb {P}}^2$ , which is k. Also, by the projection $\pi _y$ from a general point $y\in \Gamma $ , it gives a one-to-one correspondence between the number of trisecant lines of $\Gamma $ through y and the number of nodes of the curve $\pi _y(\Gamma )$ in ${\mathbb {P}}^2$ , which is m. Then the genus-degree formula for plane curves gives k and m. The number r can also be computed (cf. [Reference Höring, Liu and Shao5, Lem. 5.3]). That is,

$$ \begin{align*} k &=(\text{the number of nodes in a general hyperplane projection} \Gamma\to{\mathbb{P}}^2)\\ &=\frac{1}{2}(d-1)(d-2)-g,\\ r &=(\text{the number of choices of two points among } \Gamma\cap{\mathbb{P}}^2 \text{ for a general hyperplane } {\mathbb{P}}^2\subseteq {\mathbb{P}}^3)\\ &=\genfrac(){0pt}{}{d}{2},\\ m &=(\text{the number of nodes in the projection } \Gamma\to{\mathbb{P}}^2 \text{ from a general point of } \Gamma)\\ &=\frac{1}{2}(d-2)(d-3)-g. \end{align*} $$

Then we obtain the formula in the statement after plugging in the numbers to the equation

$$ \begin{align*} k\zeta+\Pi^*(-2kH+{\overline{e}}_*{\overline{q}}^*c_1(V)+ {\overline{e}}_*E) &=k\zeta-2k\Pi^*H+\left\{(r+k)\Pi^*H-\frac{2r}{d}\Pi^*D\right\}+m\Pi^*D \\&=k\zeta+(r-k)\Pi^*H-\left(\frac{2r}{d}-m\right)\Pi^*D. \end{align*} $$

The proof for the second case is similar to the first case except $\sigma =\text {Id}_{\overline {{\mathcal {U}}_0}}$ , and hence $m=0$ .

Proof. Let ${\mathcal {K}}$ be the family of unbendable rational curves on X containing the strict transforms of general lines on ${\mathbb {P}}^3$ meeting $\Gamma $ . Note that $N_{\ell |X}\cong {\mathcal {O}}_{{\mathbb {P}}^1}(1)\oplus {\mathcal {O}}_{{\mathbb {P}}^1}$ for general $[\ell ]\in {\mathcal {K}}$ . We denote by ${\mathcal {C}}_x$ the VMRT of ${\mathcal {K}}$ at $x\in X$ and $\breve {\mathcal {C}}$ the total dual VMRT on ${\mathbb {P}}(T_X)$ associated to ${\mathcal {K}}$ .

Let $x\in X\backslash D$ and $z=f(x)$ . We define $C_z\subseteq {\mathbb {P}}(\Omega _{{\mathbb {P}}^3}|_z)$ to be the plane curve parametrizing the tangent directions of lines on ${\mathbb {P}}^3$ passing through z and a point of $\Gamma $ , and denote by $\breve C_z\subseteq {\mathbb {P}}(T_{{\mathbb {P}}^3}|_z)$ the projectively dual curve of $C_z$ . Notice that $C_z$ is isomorphic to the plane curve which is the image of $\Gamma $ under the projection ${\mathbb {P}}^3\dashrightarrow {\mathbb {P}}^2$ from z. By the natural isomorphism ${\mathbb {P}}(\Omega _X|_x)\cong {\mathbb {P}}(\Omega _{{\mathbb {P}}^3}|_z)$ , we have the identifications ${\mathcal {C}}_x\cong C_z$ and $\breve {\mathcal {C}}|_x\cong \breve C_z$ for general $x\in X\backslash D$ . As ${\mathcal {C}}_x$ is irreducible and reduced, so is $\breve {\mathcal {C}}|_x$ for general $x\in X$ , and hence $\breve {\mathcal {C}}$ is irreducible and reduced.

Let

$$\begin{align*}[\breve{\mathcal{C}}]\sim k\zeta+a\Pi^*H+b\Pi^*D \end{align*}$$

for some $k>0$ and $a,\,b\in {\mathbb {Z}}$ . Then k is equal to the degree of $\breve C_z$ .

Let $[\ell ]$ be a general member of ${\mathcal {K}}$ and $\widetilde {\ell }\subseteq {\mathbb {P}}(T_X|_{\ell })\subseteq {\mathbb {P}}(T_X)$ be its minimal section. Since the members of the family $\widetilde {{\mathcal {K}}}$ of rational curves on ${\mathbb {P}}(T_X)$ containing $[\widetilde {\ell }]$ dominates $\breve {\mathcal {C}}$ , we have

$$\begin{align*}\dim\widetilde{{\mathcal{K}}} =3 \geq -K_{\breve{\mathcal{C}}}.\widetilde{\ell}+\dim\breve{\mathcal{C}}-|{\operatorname{Aut}}({\mathbb{P}}^1)\,| =(3\zeta-(k\zeta+a\Pi^*H+b\Pi^*D)).\widetilde{\ell}+1. \end{align*}$$

We note that the dimension of ${\mathcal {K}}$ is 3 and $\widetilde \ell $ is uniquely determined by $\ell $ , and hence we have $\dim \widetilde {\mathcal {K}}=3$ . Therefore, $b\geq -a-2$ .

Let ${l}_0$ be a general line on ${\mathbb {P}}^3$ and $\ell _0$ be its strict transform on X. Note that $N_{\ell _0|X}\cong N_{{l}_0|{\mathbb {P}}^3} \cong {\mathcal {O}}_{{\mathbb {P}}^1}(1)\oplus {\mathcal {O}}_{{\mathbb {P}}^1}(1)$ . Let $\widetilde {\ell _0}\subseteq {\mathbb {P}}(T_X|_{\ell _0})\subseteq {\mathbb {P}}(T_X)$ be the section of $\ell _0$ corresponding to a quotient $T_X|_{\ell _0}\to N_{{\ell _0}|X}\to {\mathcal {O}}_{{\mathbb {P}}^1}(1)$ . We will complete the proof by showing that

$$\begin{align*}\breve{\mathcal{C}}.\widetilde{\ell_0} =k+(aH+bD).\ell_0 =k+a =0, \end{align*}$$

and it is sufficient to show that $\breve {\mathcal {C}}\cap \widetilde {\ell _0}\neq \emptyset $ if and only if $\widetilde {\ell _0}\subseteq \breve {\mathcal {C}}$ .

Assume that $\breve {\mathcal {C}}\cap \widetilde {\ell _0}\neq \emptyset $ . That is, $\widetilde {\ell _0}|_x\in \breve {\mathcal {C}}|_x$ for some $x\in \ell _0$ . Then it is enough to show that $\widetilde {\ell _0}|_x\in \breve {\mathcal {C}}|_x$ for all $x\in \ell _0$ . Let $\widetilde {{l}_0}\subseteq {\mathbb {P}}(T_{{\mathbb {P}}^3}|_{{l}_0})\subseteq {\mathbb {P}}(T_{{\mathbb {P}}^3})$ be the image of $\widetilde {\ell _0}$ under the isomorphism ${\mathbb {P}}(T_X|_{\ell _0})\xrightarrow {\simeq }{\mathbb {P}}(T_{{\mathbb {P}}^3}|_{{l}_0})$ . So it is reduced to prove the following claim.

Notice that $\widetilde {{l}_0}$ is a section of ${l}_0$ corresponding to a quotient $T_{{\mathbb {P}}^3}|_{{l}_0}\to N_{{{l}_0}|{{\mathbb {P}}^3}}\to {\mathcal {O}}_{{\mathbb {P}}^1}(1)$ . Let P be a plane containing ${l}_0$ on ${\mathbb {P}}^3$ . Then it determines a quotient

(4.1) $$ \begin{align} T_{{\mathbb{P}}^3}|_{{l}_0}\to N_{{l}_0|{\mathbb{P}}^3}\to N_{P|{\mathbb{P}}^3}|_{{l}_0}\cong {\mathcal{O}}_{{\mathbb{P}}^1}(1), \end{align} $$

and vice versa. That is, given ${l}_0$ on ${\mathbb {P}}^3$ , there is a $1$ -to- $1$ correspondence between a plane P containing ${l}_0$ and a section $\widetilde {{l}_0}$ of ${l}_0$ corresponding to a quotient $T_{{\mathbb {P}}^3}|_{{l}_0}\to N_{{l}_0|{\mathbb {P}}^3}\to {\mathcal {O}}_{{\mathbb {P}}^1}(1)$ , both are parametrized by $\mathbb P^2$ .

Let $z\in l_0$ . As the kernel of the quotient (4.1) is $T_P|_{{l}_0}$ , the line $L_z:={\mathbb {P}}(\Omega _P|_z) \subseteq {\mathbb {P}}(\Omega _{{\mathbb {P}}^3}|_z)$ parametrizes the tangent directions of P at z. Thus, the line $L_z$ is tangent to $C_z\subseteq {\mathbb {P}}(\Omega _{{\mathbb {P}}^3}|_z)$ if and only if the point ${\mathbb {P}}(N_{P|{\mathbb {P}}^3}|_z)$ is contained in $\breve C_z\subseteq {\mathbb {P}}(T_{{\mathbb {P}}^3}|_z)$ as $L_z={\mathbb {P}}(\Omega _P|_z)$ and ${\mathbb {P}}(N_{P|{\mathbb {P}}^3}|_z)$ are projectively dual. This is equivalent to say that P is tangent to $\Gamma $ on ${\mathbb {P}}^3$ because $C_z$ and $L_z$ are linear projections of $\Gamma $ and P, respectively, after identifying ${\mathbb {P}}^3\dashrightarrow {\mathbb {P}}(\Omega _{{\mathbb {P}}^3}|_z)$ to the projection from z. If we choose a plane P that contains $l_0$ and tangents to $\Gamma $ , then any point $z\in l_0$ satisfies the above. Therefore, we have the claim.

Proof. Let d and g be the degree and genus of $\Gamma $ . If $d=3$ , then $T_X$ is big by Remark 3.5. Otherwise, if $d\geq 4$ , then let $\breve {\mathcal {C}}_1$ and $\breve {\mathcal {C}}_2$ be the total dual VMRTs on ${\mathbb {P}}(T_X)$ , respectively given by Propositions 4.2 and 4.1. Then, for general, $x\in X$ , $\breve {\mathcal {C}}_1|_x$ is projectively dual to a plane curve of degree d with $\delta =\frac {1}{2}(d-1)(d-2)-g$ nodes, hence the degree of $\breve {\mathcal {C}}_1|_x$ is given by

$$\begin{align*}d(d-1)-2\delta =d(d-1)-2\cdot\left(\frac{1}{2}(d-1)(d-2)-g\right) =2d+2g-2. \end{align*}$$

So $\breve {\mathcal {C}}_1$ and $\breve {\mathcal {C}}_2$ satisfy

$$ \begin{align*} [\breve{\mathcal{C}}_1] &\sim (2d+2g-2)\zeta-(2d+2g-2)\Pi^*H+\left((2d+2g-4)+c\right)\Pi^*D,\\ [\breve{\mathcal{C}}_2] &\sim \left(\frac{1}{2}(d-1)(d-2)-g\right)\zeta+(d+g-1)\Pi^*H-\left((d-1)-\frac{1}{2}(d-2)(d-3)+g\right)\Pi^*D \end{align*} $$

for some $c\geq 0$ where $H=f^*{\mathcal {O}}_{{\mathbb {P}}^3}(1)$ and D is the exceptional divisor of $f: X\to {\mathbb {P}}^3$ (the second formula is valid even in the case $d=4$ and $g=1$ where $\Gamma $ has no trisecant line on ${\mathbb {P}}^3$ ). Then they yield an effective divisor

$$\begin{align*}[\breve{\mathcal{C}}_1]+2[\breve{\mathcal{C}}_2] =k\zeta+\left( (d-1)(d-4)+c \right)\Pi^*D \end{align*}$$

on ${\mathbb {P}}(T_X)$ satisfying (†) for some $k>0$ and $c\geq 0$ . Thus by Lemma 2.2, $\zeta $ is not big on ${\mathbb {P}}(T_X)$ .

Proof. We continue to use the notation in the proof of Proposition 4.1. The proof is similar to the proposition. Let ${\mathcal {K}}$ be the family of unbendable rational curves on X containing the strict transforms of lines meeting $\Gamma $ on Q and $\breve {\mathcal {C}}$ be the total dual VMRT on ${\mathbb {P}}(T_X)$ associated to ${\mathcal {K}}$ . From the diagram

we have the formula

$$\begin{align*}[\breve{\mathcal{C}}] \sim k\zeta+\Pi^*{\overline{e}}_*(K_{\overline{\mathcal{U}}/\overline{\mathcal{K}}}) =k\zeta -2\Pi^*{\overline{e}}_*\xi +\Pi^*{\overline{e}}_*{\overline{q}}^*c_1(V) +\Pi^*{\overline{e}}_*E \end{align*}$$

for $k=\deg ({\overline {e}})$ .

As a general hyperplane section meets a general line at $1$ point on Q, we take $V={{\overline {q}}}_*{{\overline {e}}}^*H$ so that $\xi ={{\overline {e}}}^*H$ for the tautological class $\xi ={\mathcal {O}}_{{\mathbb {P}}_{\overline {{\mathcal {K}}}}}(1)$ . Then ${\overline {e}}_*\xi =kH$ . Let

$$\begin{align*}{\overline{e}}_*{\overline{q}}^*c_1(V)=aH+bD\ \ \text{and}\ \ {\overline{e}}_*E=mD \end{align*}$$

for some $a,\,b\in {\mathbb {Z}}$ and $m\geq 0$ . In this case, $H^3=2$ , $H^2.D=0$ , and $H.D^2=-d$ . Thus from the same calculation in the proof of Proposition 4.1,

$$\begin{align*}a =\frac{1}{2}{c_1(V)}^2 =k+\frac{1}{2}r,\ \ b =-\frac{1}{2}{\overline{q}}^*c_2(V).{\overline{e}}^*D =-\frac{1}{d}r \end{align*}$$

for $r:=c_2(V)$ as a general line is chosen to meet $\Gamma $ at 1 point on Q so that ${\overline {e}}^*D\equiv \xi $ in $\text {N}^1(\overline {{\mathcal {U}}}/\overline {{\mathcal {K}}})$ .

Thus, we obtain the formula in the statement using the following correspondences.

$$ \begin{align*} k &=(\text{the number of lines on } Q \text{ joining } \Gamma \text{ and a general point of } Q)\\ &=d,\\ r &=(\text{the number of lines on } Q \text{ meeting } \Gamma \text{ and contained in a general hyperplane section of } Q)\\ &=2d,\\ m &=(\text{the number of secant lines of } \Gamma \text{ on } Q \text{ passing through a general point of } \Gamma).\\[-42pt] \end{align*} $$