1. Introduction
Throughout the paper, all the varieties are defined over the field 
$\mathbb{C}$ of complex numbers.
Let S be a minimal resolution of a projective surface Sʹ obtained as a complete intersection of a hyperquadric and a hypercubic of 
$\mathbb{P}^4$. Assume that Sʹ has at most rational double points as the singularities. Then S is a K3 surface, and a general member of the complete linear system of the hyperplane sections is a nonhyperelliptic curve of genus 4.
Denote by Λ the complete linear system of the hyperplane sections of S, and let 
$\mathcal{P}\subset\Lambda$ be a subpencil. Assume that a general member of 
$\mathcal{P}$ is smooth. Let 
$\nu:\widetilde{S}\to S$ be a blow-up such that the complete linear system of the proper transform of the member of 
$\mathcal{P}$ is base point free. We assume that ν is the shortest among the blow-up with the above property. Then there exists a surjective morphism 
$f:\widetilde{S}\to\mathbb{P}^1$ whose general fibre is a nonhyperelliptic curve of genus 4. Furthermore, f is relatively minimal. For each fibre 
$\mathcal{F}$ of f, the invariant 
${\rm Ind}(\mathcal{F})$ named Horikawa index (H-index) is defined as we mention in the next section. 
${\rm Ind}(\mathcal{F})$ is a non-negative rational number, and we have 
${\rm Ind}(\mathcal{F})=0$ except for a finite number of fibres of f.
Nonhyperelliptic curve C of genus 4 is obtained as the complete intersection of a hyperquadric 
$\frak{Q}_0$ and a hypercubic 
$\frak{Y}$ in 
$\mathbb{P}^3$. Since the defining equation of 
$\frak{Q}_0$ is given as a quadratic form of the homogeneous coordinates of 
$\mathbb{P}^3$, the rank 
${\rm rk}(\frak{Q}_0)$ is defined, which is equal to 3 or 4. 
$\frak{Q}_0$ is expressed as follows:
(I) If
${\rm rk}(\frak{Q}_0)=4$, we have $\frak{Q}_0\cong\mathbb{P}^1\times\mathbb{P}^1$, and the hyperplane section of $\frak{Q}_0$ is a diagonal divisor. If we consider C as a divisor of $\frak{Q}_0$, then C is linearly equivalent to the triple of the diagonal. According to [Reference Ashikaga and Yoshikawa1], in this case, C is called as Eisenbud–Harris general (EH-general).(II) If
${\rm rk}(\frak{Q}_0)=3$, $\frak{Q}_0$ is a cone over a smooth conic. If we denote by Δ0 the tautological divisor of the Hirzebruch surface $\Sigma_2:=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(2))$, then $\frak{Q}_0$ is the image of Σ2 by the morphism $\Phi_{|\Delta_0|}$ defined by the complete linear system $|\Delta_0|$. If Γ is a fibre of the ruling $\mu:\Sigma_2\to\mathbb{P}^1$ and if $\Delta_{\infty}$ is a section of µ with $\Delta_{\infty}\sim\Delta_0-2\Gamma$, then $\Delta_{\infty}$ is contracted to the vertex of $\frak{Q}_0$. Since C does not go through the vertex, we may consider as $C\subset\Sigma_2$. If we consider that C is a divisor of Σ2, then we have $C\sim 3\Delta_0$. In this case, let us call C Eisenbud–Harris special (EH-special).
If a general fibre of f above is EH-general (respectively, EH-special), f is called the EH-general fibration (respectively, the EH-special fibration), 
$\mathcal{P}$ the EH-general subpencil (respectively, the EH-special subpencil). The definition of H-index depends on whether f is EH-general or EH-special. The sum of H-indices for the fibration we investigate in the present paper is as follows (see § 2 for details):
• When f is EH-general, then the sum is equal to
$1/2$.• When f is EH-special, then the sum is equal to
$6/7$.
Each case is divided into several cases as follows:
(I) The case where f is EH-general.
- (I-i) There are two fibres with H-index
$1/4$.- (I-ii-a) There is a fibre with H-index
$1/2$, and the rank of the hyperquadric containing the fibre is 3.- (I-ii-b) There is a fibre with H-index
$1/2$, and the rank of the hyperquadric containing the fibre is 2.
(II) The case where f is EH-special.
- (II-i-a) There are three fibres with H-index
$2/7$.- (II-i-b) There is a fibre with H-index
$2/7$ and a fibre with H-index $4/7$.- (II-i-c) There is a fibre with H-index
$6/7$, and the rank of the hyperquadric containing the fibre is 3.- (II-ii-a) There are two fibres with H-index
$3/7$.- (II-ii-b) There is a fibre with H-index
$6/7$, and the rank of the hyperquadric containing the fibre is 2.- (II-ii-c) There is a fibre with H-index
$6/7$, and the rank of the hyperquadric containing the fibre is 1.
Let 
$Q\subset\mathbb{P}^4$ be the hyperquadric containing Sʹ. As in the case of 
$\frak{Q}_0$, the rank 
${\rm rk}(Q)$ is also defined, and we have that 
${\rm rk}(Q)$ is one of 3, 4 and 5. As we will see in § 4, if 
${\rm rk}(Q)=3$, then Q has a singular curve 
$\ell$ as the compound rational double point of type A 1, and Sʹ has rational double points on 
$\ell$. We restrict our arguments to the generic case as follows:
Assumption 1. When 
${\rm rk}(Q)=5$, we assume 
$S^{\prime}=S$. When 
${\rm rk}(Q)=4$, we assume Sʹ does not go through the vertex of Q and that 
$S^{\prime}=S$ holds. When 
${\rm rk}(Q)=3$, we assume that Sʹ does not have singularities except for the intersection points with 
$\ell$.
The main result of the present paper is as follows:
Theorem 1. Let the notation and the conditions be as above. The classification of subpencils of Λ without fixed component is as follows:
(1) The case where
${\rm rk}(Q)=3$
(2) The case where
${\rm rk}(Q)=4$
(3) The case where
${\rm rk}(Q)=5$
We set the notation as follows:
Notation 1.
∼ means the linear equivalence of two divisors. For a non-negative integer d, denote by 
$\mu:\Sigma_d:=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(d))\to\mathbb{P}^1$ the Hirzebruch surface. Let Δ0 be the tautological divisor of 
$\Sigma_d$, and 
$\Delta_{\infty}$ the section of µ with 
$\Delta_{\infty}^2=-d$ and 
$\Delta_0\Delta_{\infty}=0$. For a linear system Ω of divisors for some variety, denote by 
$\Phi_{\Omega}$ the rational map defined by Ω. For a global section s of an invertible sheaf, denote by (s) the divisor defined by s = 0. For a linear system Ω, denote by 
${\rm Bs}\Omega$ the base locus of Ω.
2. Horikawa index
In this section, we review the H-index for nonhyperelliptic fibrations of genus 4 for both EH-general case and EH-special case. Let 
$f:S\to B$ be a nonhyperelliptic fibration of genus 4 over a smooth projective curve B. Assume that f is relatively minimal and that f is not isotrivial. Let 
$K_{S/B}=K_S-f^*K_B$ be the relative canonical divisor and 
$\omega_{S/B}
:=\mathcal{O}_S(K_{S/B})$ the relative dualizing sheaf. Then 
$E:=f_*\omega_{S/B}$ is a locally free sheaf of rank 4 over B. If we put 
$\chi_f:=\deg E$, then it is known that the inequalities 
$K_{S/B}^2 \gt 0$ and 
$\chi_f \gt 0$ both hold. Furthermore, the following inequalities are known:
Theorem 2. ([Reference Chen4], [Reference Konno11], [Reference Konno12])
Under the above conditions, we have
\begin{equation*}K_{S/B}^2\geq{\displaystyle\frac{24}{7}}\chi_f.\end{equation*}Moreover, if we assume that a general fibre of f is EH-general, then
\begin{equation*}K_{S/B}^2\geq{\displaystyle\frac{7}{2}}\chi_f\end{equation*}holds.
Remark 1. Similar results to Theorem 2 are obtained as follows:
(i) ([Reference Xiao17], [Reference Konno11]) If a general fibre of f is a hyperelliptic curve of genus
$g\,(\geq 2)$, then we have
\begin{equation*}K_{S/B}^2\geq{\displaystyle\frac{4(g-1)}{g}}\chi_f.\end{equation*}(ii) ([Reference Horikawa9], [Reference Konno11], [Reference Reid13]) If a general fibre of f is a nonhyperelliptic curve of genus 3, then we have
\begin{equation*}K_{S/B}^2\geq 3\chi_f.\end{equation*}
Let us return to the case of genus 4 fibrations. For each fibre 
$\mathcal{F}$ of f, denote by 
$\omega_{\mathcal{F}}$ the dualizing sheaf. The multiplication map
\begin{equation*}{\rm Sym}^2H^0(\omega_{\mathcal{F}})\to H^0(\omega_{\mathcal{F}}^{\otimes 2})\end{equation*}defines the multiplication map
\begin{equation*}\varphi:{\rm Sym}^2E\to f_*\omega_{S/B}^{\otimes2}.\end{equation*}By our assumption and Max–Noether’s theorem, φ is generically surjective. We obtain the following exact sequence:
\begin{equation*}0\to L\to{\rm Sym}^2E\to f_*\omega_{S/B}^{\otimes2}\to\mathcal{T}\to0,\end{equation*} where L is a line bundle and 
$\mathcal{T}$ is a sheaf supported over finitely many points of B. Denote by 
$\pi:\frak{W}:=\mathbb{P}(E)\to B$ the 
$\mathbb{P}^3$-bundle defined by E and by T the tautological divisor of 
$\frak{W}$. The natural morphism 
$f^*E\to\omega_{S/B}$ defines the rational map 
$\psi:S\cdots\!\! \gt \frak{W}$ over B. ψ is called the relative canonical map, and the image 
$\psi(S)\subset\frak{W}$ is called the relative canonical image of S by ψ.
Lemma 1. ([Reference Konno, Ashikaga and Yoshikawa10])
Let the notation and conditions be as above. Then there exists an irreducible relative hyperquadric 
$\frak{Q}\in|2T-\pi^*L|$ containing 
$\psi(S)$.
Notation 2.
Denote by 
$\pi_{\frak{Q}}$ the restriction of π to 
$\frak{Q}$.
2.1. H-index for EH-general case
Assume that f is the EH-general fibration. Let 
$q\in H^0(\mathcal{O}_W(2T-\pi^*L))$ be the global section defining 
$\frak{Q}$. Since q can be considered as an element of 
$H^0(B,({\rm Sym}^2E)\otimes L^{-1})$, q defines the morphism 
$q:E^{\vee}\to E\otimes L^{-1}$. By considering the determinant map 
$\det q:\det E^{\vee}\to \det(E\otimes L^{-1})$, we can consider that 
$\det q$ is an element of 
$H^0(B,(\det E)^{\otimes2}\otimes L^{-4})$. Since f is EH-general, we have 
$\det q\neq0$, and hence, 
$\det q$ defines an effective divisor 
${\rm Discr}(\frak{Q})$ over B. A general fibre of 
$\pi_{\frak{Q}}:\frak{Q}\to B$ is of rank 4, while for any point 
$p\in{\rm supp}{\rm Discr}(\frak{Q})$, the rank of the fibre 
$\pi_{\frak{Q}}^{-1}(p)$ is less than 4. 
${\rm Discr}(\frak{Q})$ is called the discriminant locus of 
$\frak{Q}$. The following is known:
Theorem 3. ([Reference Konno, Ashikaga and Yoshikawa10])
For 
$p\in B$, denote by 
${\rm mult}_p{\rm Discr}(\frak{Q})$ the coefficient of p in 
${\rm Discr}(\frak{Q})$ and by 
$\mathcal{T}_p$ the restriction of 
$\mathcal{T}$ to p. If we put
\begin{eqnarray}{\rm Ind}(f^{-1}(p)):={\displaystyle\frac{1}{4}}{\rm mult}_p{\rm Discr}(\frak{Q})+{\rm length}\mathcal{T}_p,
\end{eqnarray}then the following equality holds:
\begin{eqnarray}K_{S/B}^2={\displaystyle\frac{7}{2}}\chi_f+\sum_{p\in B}{\rm Ind}(f^{-1}(p))
\end{eqnarray}
2.2. H-index for EH-special case
In this subsection, we review the H-index for EH-special case. Although H-index is defined in [Reference Enokizono5] without any special condition, we use another definition in [Reference Takahashi14]. H-index in the latter case is defined under the assumption that the multiplication map 
${\rm Sym}^2E\to f_*\omega_{S/B}^{\otimes2}$ is surjective, and as we will see in § 4, the fibrations we consider in this paper satisfy this assumption.
Since a general fibre of 
$\pi_{\frak{Q}}:\frak{Q}\to B$ is a quadric cone, we obtain the relative vertex 
$B_0\subset\frak{Q}$. Since B 0 is a section of π, we have the short exact sequence
\begin{equation*}0\to E_0\to E\to M\to0\end{equation*} defining the embedding 
$B_0\subset\frak{W}$. E 0 is a locally free sheaf of rank 3 over B, and M is an invertible sheaf over B.
If 
$\rho:\widetilde{\frak{W}}\to\frak{W}$ is a blow-up along B 0, we obtain the following commutative diagram:
Put 
$\mathbb{E}:=\rho^{-1}(B_0)$ and let 
$\widetilde{Q}$ be the proper transform of 
$\frak{Q}$ by ρ and 
$T_{E_0}$ the tautological divisor of 
$\mathbb{P}(E_0)$. Since 
$\rho^*T\sim\widetilde{\pi}^*T_{E_0}+\mathbb{E}$, we have
\begin{equation*}\widetilde{Q}\sim\rho^*\frak{Q}-2\mathbb{E}\sim\widetilde{\pi}^*(2T_{E_0}-\zeta^*L),\end{equation*} namely, there exists a conic bundle 
$Q_0\in|2T_{E_0}-\zeta^*L|$ such that 
$\widetilde{Q}=\widetilde{\pi}^{-1}(Q_0)$. If 
$q_0\in H^0(\mathcal{O}_{\mathbb{P}(E_0)}(2T_{E_0}-\zeta^*L))$ defines Q 0, we can define the discriminant locus 
${\rm Discr}(Q_0)$ as in the previous subsection. For any 
$p\in{\rm supp}{\rm Discr}(Q_0)$, the fibre 
$\pi_{\frak{Q}}^{-1}(p)$ is a hyperquadric of 
$\mathbb{P}^3$ with rank less than 3.
Note that the restriction of the relative canonical image Sʹ to B 0 defines an effective divisor δ over B.
The following theorem is proved:
Theorem 4. ([Reference Takahashi14])
Let the notation and the conditions be as above. For any 
$p\in B$, put
\begin{eqnarray}{\rm Ind}(f^{-1}(p))={\displaystyle\frac{2}{7}}{\rm mult}_p\delta
+{\displaystyle\frac{3}{7}}{\rm mult}_p{\rm Discr}(Q_0).
\end{eqnarray}Then we have the following equality:
\begin{eqnarray}
K_{S/B}^2={\displaystyle\frac{24}{7}}\chi_f+\sum_{p\in B}{\rm Ind}(f^{-1}(p)).
\end{eqnarray}
3. Fibres with positive H-index
Let S be our K3 surface and 
$f:\widetilde{S}\to\mathbb{P}^1$ as in § 1. We have 
$K_{\widetilde{S}/\mathbb{P}^1}^2=18$ and 
$\chi_f=5$. Let us investigate the fibres of f with positive H-index. We use the same notation as in the previous section for the fibration 
$f:\widetilde{S}\to\mathbb{P}^1$.
3.1. EH-general case
If f is EH-general, then the sum of H-indices is 
$1/2$ by Equation (2.2). Hence, if 
${\rm Ind}(f^{-1}(p)) \gt 0$ for 
$p\in\mathbb{P}^1$, we have 
$\mathcal{T}_p=0$ and 
${\rm mult}_p{\rm Discr}(\frak{Q})=1$ or 2 by Equation (2.1). In either case, 
$\pi_{\frak{Q}}^{-1}(p)$ is a hyperquadric with rank less than 4.
If 
${\rm rk}(\pi_{\frak{Q}}^{-1}(p))=3$ and if S is sufficiently general, then 
$f^{-1}(p)$ is an EH-special nonhyperelliptic curve of genus 4. If 
${\rm rk}(\pi_{\frak{Q}}^{-1}(p))=2$, then 
$\pi_{\frak{Q}}^{-1}(p)$ is a sum of two distinct hyperplanes, and if S is sufficiently general, 
$f^{-1}(p)$ is the sum of two elliptic curves intersecting at three points transversally.
We have the following three cases:
(I-i)
${\rm Discr}(Q)=p_1+p_2$ for some $p_1,p_2\in\mathbb{P}^1$.(I-ii-a)
${\rm Discr}(Q)=2p_1$ for some $p_1\in\mathbb{P}^1$ and ${\rm rk}(\pi_Q^{-1}(p_1))=3$.(I-ii-b)
${\rm Discr}(Q)=2p_1$ for some $p_1\in\mathbb{P}^1$ and ${\rm rk}(\pi_Q^{-1}(p_1))=2$.
3.2. EH-special case
Assume f is EH-special. We obtain 
$K_{\widetilde{S}/B}^2-(24/7)\chi_f=6/7$ from Equation (2.4). Hence, by [Reference Takahashi15, Theorem 1.5], the multiplication map is surjective. The sum of the H-indices is 
$6/7$. Hence, by considering Equation (2.3), we obtain the following six possibilities:
(II-i-a)
$\delta=p_1+p_2+p_3\,(p_1,p_2,p_3\in\mathbb{P}^1,\,p_i\neq p_j\Leftrightarrow i\neq j\,(i,j=1,2,3))$, and ${\rm Discr}(Q_0)=0$.(II-i-b)
$\delta=p_1+2p_2\,(p_1,p_2\in\mathbb{P}^1,\,p_1\neq p_2)$ and ${\rm Discr}(Q_0)=0$.(II-i-c)
$\delta=3p_1\,(p_1\in\mathbb{P}^1)$ and ${\rm Discr}(Q_0)=0$.(II-ii-a) δ = 0 and
${\rm Discr}(Q_0)=p_1+p_2\,(p_1,p_2\in\mathbb{P}^1,\,p_1\neq p_2)$.(II-ii-b) δ = 0 and
${\rm Discr}(Q_0)=2p_1\,(p_1\in\mathbb{P}^1)$ with ${\rm rk}(\pi_{\frak{Q}}^{-1}(p))=2$.(II-ii-c) δ = 0 and
${\rm Discr}(Q_0)=2p_1\,(p_1\in\mathbb{P}^1)$ with ${\rm rk}(\pi_{\frak{Q}}^{-1}(p))=1$.
We investigate the details for each case. Note that the direct image 
$E:=f_*\omega_{\widetilde{S}/\mathbb{P}^1}$ satisfies
\begin{equation*}E\cong\mathcal{O}_{\mathbb{P}^1}(2)\oplus\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus3},\end{equation*}from the results of [Reference Fujita7]. See also [Reference Barja and Zucconi2].
3.2.1. The case where the discriminant locus is 0
First, consider the cases (II-i-a), (II-i-b) and (II-i-c). If E 0 and M are as in the previous section, we obtain 
$E_0\cong\mathcal{O}_{\mathbb{P}^1}^{\oplus3}(1)$, 
$M\cong\mathcal{O}_{\mathbb{P}^1}(2)$ and 
$L\cong\mathcal{O}_{\mathbb{P}^1}(2)$ by considering the argument of [Reference Takahashi14, § 1]. It is easily proved that 
$Q_0\cong\Sigma_0$ and 
$T_{E_0}|_{Q_0}\sim 2\Delta_0+\Gamma$, where Γ is the restriction of a fibre of 
$\mathbb{P}(E_0)\to\mathbb{P}^1$ to Q 0, and Δ0 is a fibre of the natural projection, which is different from the one whose fibre is Γ. (See Notation 1 also for Δ0.) Since 
$\widetilde{W}\cong\mathbb{P}(\mathcal{O}_{\mathbb{P}(E_0)}(T_{E_0})\oplus\mathcal{O}_{\mathbb{P}(E_0)}(2F^{\prime}))$, where Fʹ is a fibre of ζ, we have
\begin{equation*}\widetilde{Q}\cong\mathbb{P}(\mathcal{O}_{\Sigma_0}(2\Delta_0+\Gamma)\oplus
\mathcal{O}_{\Sigma_0}(2\Gamma)),\end{equation*}and hence,
\begin{eqnarray}
\widetilde{Q}\cong\mathbb{P}(\mathcal{O}_{\Sigma_0}(2\Delta_0-\Gamma)\oplus\mathcal{O}_{\Sigma_0}).
\end{eqnarray} If 
$T_{\widetilde{Q}}$ is the tautological divisor of 
$\widetilde{Q}$ under the consideration of Equation (3.1), and if 
$\widetilde{F}$ is a fibre of 
$\widetilde{Q}\to\mathbb{P}^1$, we have
\begin{equation*}S_1\sim3T_{\widetilde{Q}}+3\widetilde{F},\end{equation*} where S 1 is the preimage of S in 
$\widetilde{Q}$. Put 
$\mathbb{E}_0:=\mathbb{E}|_{\widetilde{Q}}$. Then 
$\mathbb{E}_0\cong Q_0$, and furthermore, the restriction 
$S_1|_{\mathbb{E}_0}$ consists of fibres of 
$(\zeta\circ\widetilde{\pi})|_{\mathbb{E}_0}$, and the image of the sum of these fibres by 
$\widetilde{Q}\to\mathbb{P}^1$ is δ. Namely, for a point 
$p\in{\rm supp}\,\delta$, the fibre of 
$f:\widetilde{S}\to\mathbb{P}^1$ over p is the one with positive H-index, and its value is one of 
$2/7$, 
$4/7$ and 
$6/7$. If we consider the fibre of 
$\widetilde{Q}\to\mathbb{P}^1$ over p as Σ2, then the restriction of 
$\mathbb{E}_0$ to Σ2 is 
$\Delta_{\infty}$, and hence, the restriction 
$S_1|_{\Sigma_2}$ is of the form of 
$\Delta_{\infty}+C$ with 
$C\sim 2\Delta_0+2\Gamma$. Hence, if 
$\mathcal{P}$ is generic in 
${\rm Gr}(3,1)$, this fibre consists of a rational curve and a hyperelliptic curve of genus three intersecting at two points.
3.2.2. The case where the discriminant locus is not 0
Next, consider the case (II-ii-a), (II-ii-b) and (II-ii-c). We have 
$E_0=\mathcal{O}_{\mathbb{P}^1}(2)\oplus\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus2}$, 
$M=\mathcal{O}_{\mathbb{P}^1}(1)$ and 
$L=\mathcal{O}_{\mathbb{P}^1}(2)$. We use the notation of the previous section under these consideration. Then we have
\begin{eqnarray}\widetilde{W}\cong\mathbb{P}(\mathcal{O}_{\mathbb{P}(E_0)}(T_{E_0})\oplus\mathcal{O}_{\mathbb{P}(E_0)}(F')),
\end{eqnarray} where Fʹ is a fibre. Let us consider the structure of 
$Q_0\,(\subset\mathbb{P}(E_0))$. It is easily proved that Q 0 is disjoint to the section 
$\widetilde{B}:=\mathbb{P}(E_0/\mathcal{O}_{\mathbb{P}^1}(2)\oplus\mathcal{O}_{\mathbb{P}^1}(1))$, and if 
$\widetilde{\mathbb{P}(E_0)}\to\mathbb{P}(E_0)$ is the blow-up along 
$\widetilde{B}$, we obtain the following commutative diagram:
We may consider 
$Q_0\subset\widetilde{\mathbb{P}(E_0)}$. We have 
$\widetilde{\mathbb{P}(E_0)}\cong\mathbb{P}(\mathcal{O}_{\Sigma_1}(\Delta_0+\Gamma)\oplus\mathcal{O}_{\Sigma_1}(\Gamma))$, and it is easily proved that Q 0 is a double cover of Σ1 branched along a divisor 
$\mathcal{B}$ that is linearly equivalent to 
$2\Delta_0$. Note that the inverse image of 
$\Delta_{\infty}$ by the double cover is the sum of two 
$(-1)$ curves unless 
$\Delta_{\infty}$ is contained in 
$\widetilde{B}$. Furthermore, we have 
$K_{Q_0}^2=6$.
3.2.2.1 If 
$\mathcal{B}$ is smooth, then Q 0 is isomorphic to the blown-up surface at two points q 1 and q 2 of Σ1 contained in some section 
$\Delta^{\prime}_0\in|\Delta_0|$. (Namely, Q 0 has the structure of the branched double cover over Σ1 and the structure of the blown-up surface of Σ1.) By the adjunction formula, we have 
$K_{Q_0}\sim-T_{E_0}|_{Q_0}$. Hence, by considering Equation (3.2), if we denote by 
$\nu:Q_0\to\Sigma_1$ the above blow-up, and if we put 
$\mathbb{E}_i=\nu^{-1}(q_i)$ 
$(i=1,2)$, we have
\begin{eqnarray}
\widetilde{Q}\cong\mathbb{P}(\mathcal{O}_{Q_0}(\nu^*(2\Delta_0)-\mathbb{E}_1-\mathbb{E}_2)\oplus\mathcal{O}_{Q_0}).
\end{eqnarray} If 
$p_i\in\mathbb{P}^1$ is the image of qi 
$(i=1,2)$, then we have 
${\rm Discr}(Q_0)=p_1+p_2$, and 
$\widetilde{S}$ has singular fibres of type (II-ii-a). The fibre of 
$\frak{Q}\to\mathbb{P}^1$ over pi is of rank 2, namely the fibre is a sum of two distinct hyperplanes, and hence, 
$f^{-1}(p_i)$ is a sum of two elliptic curves intersecting at three points transversally if 
$\mathcal{P}$ is sufficiently general in the Grasmannian 
${\rm Gr}(3,1)$.
3.2.2.2 We consider the case where 
$\mathcal{B}\,(\subset\Sigma_1)$ is written as 
$\mathcal{B}=\Delta_0+\Delta_0^{\prime}$ for 
$\Delta_0,\Delta_0^{\prime}\in|\Delta_0|$ with 
$\Delta_0\neq\Delta_0^{\prime}$. Let q be the intersection point of Δ0 and 
$\Delta_0^{\prime}$. Then Q 0 has a rational double point of type A 1 over q. Hence, 
$\widetilde{Q}$ has a singular locus along the fibre over the singularity of Q 0. Here, we consider the normalization of 
$\widetilde{Q}$.
Let 
$\widetilde{Q}_0$ be the minimal resolution of Q 0. If 
$p_1\in\mathbb{P}^1$ is the image of q, then the fibre of 
$\widetilde{Q}_0\to\mathbb{P}^1$ over p 1 consists of three rational curves 
$\ell_1, \ell_2$ and 
$\ell_3$. We may assume 
$\ell_1^2=\ell_3^2=-1$, 
$\ell_2^2=-2$, 
$\ell_1\ell_3=0$ and 
$\ell_1\ell_2=\ell_3\ell_2=1$. Moreover, if we denote by 
$\widetilde{\Delta}_{\infty}+
\widetilde{\Delta}^{\prime}_{\infty}$ the inverse image of 
$\Delta_{\infty}$ by the double cover 
$\widetilde{Q}_0\to\Sigma_1$ with 
$\widetilde{\Delta}_{\infty}\widetilde{\Delta}^{\prime}_{\infty}=0$, we may assume 
$\ell_1\widetilde{\Delta}_{\infty}=1$ and 
$\ell_3\widetilde{\Delta}^{\prime}_{\infty}=1$. Namely, 
$\widetilde{Q}_0$ is obtained as follows:
For some point 
$q_1\in\Sigma_1\setminus\Delta_{\infty}$, let Γ1 be a fibre containing 
$q_1, \nu_1:Q_1\to\Sigma_1$ a blow-up at q 1 and 
$\widetilde{\Gamma}_1$ the proper transform of Γ1. Put 
$\mathbb{E}_1:=\nu_1^{-1}(q_1)$ and let q 2 be a point of 
$\mathbb{E}_1\setminus\widetilde{\Gamma}_1$. Then by blowing-up at q 2, we obtain 
$\widetilde{Q}_0$. Let 
$\nu_2:\widetilde{Q}_0\to Q_1$ be the blow-up and put 
$\mathbb{E}_2:=\nu_2^{-1}(q_2)$. Put 
$\widehat{Q}:=\widetilde{Q}\times_{Q_0}\widetilde{Q}_0$. Then by the same argument as 3.2.2.1, we obtain
\begin{eqnarray}
\widehat{Q}\cong\mathbb{P}(\mathcal{O}_{\widetilde{Q}_0}(\nu_2^*(\nu_1^*(2\Delta_0)-\mathbb{E}_1)-\mathbb{E}_2)\oplus\mathcal{O}_{\widetilde{Q}_0}).
\end{eqnarray} Note that if 
$T_{\widehat{Q}}$ is the tautological divisor of 
$\widehat{Q}$, then we have the preimage of 
$\widetilde{S}$ in 
$\widehat{Q}$ is linearly equivalent to 
$3T_{\widehat{Q}}$ since δ = 0.
We have 
${\rm Discr}(Q_0)=2p_1$. Moreover, the rank of the fibre of 
$Q_0\to\mathbb{P}^1$ over p 1 is 2, and hence, the fibre of 
$\frak{Q}\to\mathbb{P}^1$ over p 1 is a sum of two hyperplanes. Namely, the fibre of 
$f:\widetilde{S}\to\mathbb{P}^1$ over p 1 is of type (II-ii-b).
Let 
$\Pi:\widehat{Q}\to\widetilde{Q}_0$ be the natural morphism (
$\mathbb{P}^1$-bundle) and put 
$\mathbb{F}_i=\Pi^{-1}(\ell_i)$ 
$(i=1,2,3)$. Then we have 
$\mathbb{F}_1\cong\mathbb{F}_3\cong\Sigma_1$ and 
$\mathbb{F}_2\cong\Sigma_0$. Furthermore, if S 1 is the preimage of 
$\widetilde{S}$ in 
$\widehat{Q}$, and if 
$\mathcal{P}$ is sufficiently general, then the restrictions 
$S_1|_{\mathbb{F}_1}$ and 
$S_1|_{\mathbb{F}_3}$ are both elliptic curves and 
$S_1|_{\mathbb{F}_2}$ is a sum of disjoint three rational curves that are 
$(-2)$-curves as the curves of S 1. Namely, the singular fibre of f over p 1 is of the form 
$C_1+C_2+\sum_{i=1}^3E_i$, where Cj 
$(j=1,2)$ is an elliptic curve with 
$C_1\cap C_2=\emptyset$, and Ei 
$(i=1,2,3)$ is a 
$(-2)$-curve with 
$E_i\cap E_{i^{\prime}}=\emptyset\,\Leftrightarrow i\neq i^{\prime}$ and 
$C_jE_i=1$ as curves of 
$\widetilde{S}$.
3.2.2.3 Assume 
$\widetilde{B}$ is of the form 
$\widetilde{B}=\Delta_0+\Delta_{\infty}+\Gamma$ for some 
$\Delta_0\in|\Delta_0|$ and some fibre Γ. Let 
$q^{\prime}_1$ and 
$q^{\prime}_2$ be the points such that 
$\Delta_0\cap\Gamma=\{q^{\prime}_1\}$ and 
$\Delta_{\infty}\cap\Gamma=\{q^{\prime}_2\}$. Furthermore, let 
$\gamma:\widetilde{\Sigma}_1\to\Sigma_1$ be the blow-up at 
$q^{\prime}_1$ and 
$q^{\prime}_2$, and 
$\widetilde{\Delta}_0, \widetilde{\Delta}_{\infty}$ and 
$\widetilde{\Gamma}$ the proper transforms of 
$\Delta_0, \Delta_{\infty}$ and Γ, respectively. Put 
$\mathcal{E}_i=\gamma^{-1}(q^{\prime}_i)$ 
$(i=1,2)$. If 
$\widehat{Q}_0$ is a normalization of 
$\widetilde{\Sigma}_1\times_{\Sigma_1}Q_0$ and if 
$p_1\in\mathbb{P}^1$ is the image of 
$q^{\prime}_i$ by 
$Q_0\to\mathbb{P}^1$, then the fibre of 
$\widetilde{Q}_0\to\mathbb{P}^1$ over p 1 consists of three rational components 
$\ell_1$, 
$\ell_2$ and 
$\ell_3$, where we may assume that 
$\ell_1$ dominates 
$\mathcal{E}_1$, 
$\ell_2$ dominates 
$\mathcal{E}_2$ and 
$\ell_3$ dominates 
$\widetilde{\Gamma}$. 
$\ell_1$ and 
$\ell_2$ are 
$(-2)$-curves and 
$\ell_3$ is a 
$(-1)$-curve. The curve that dominates 
$\widetilde{\Delta}_{\infty}$ is a 
$(-1)$-curve, and the self-intersection number of the curve dominating 
$\widetilde{\Delta}_0$ is 0. Hence, 
$\widehat{Q}_0$ is obtained as follows: For a point 
$q_1\in\Sigma_1\setminus\Delta_{\infty}$, let 
$\nu_1:Q_1\to\Sigma_1$ be the blow-up at q 1 and 
$\widehat{\Gamma}$ the proper transform of the fibre containing q 1. Put 
$\mathbb{E}_1:=\nu_1^{-1}(q_1)$, and let q 2 be the intersection point of 
$\widehat{\Gamma}$ and 
$\mathbb{E}_1$. By blowing-up at q 2, we obtain 
$\widehat{Q}_0$. Let 
$\nu_2:\widehat{Q}_0\to Q_1$ be the blow-up, and put 
$\mathbb{E}_2:=\nu_2^{-1}(q_2)$. Then by the same argument as in 3.2.2.1, if we put 
$\widehat{Q}:=\widetilde{Q}\times_{Q_0}\widehat{Q}_0$, we obtain
\begin{eqnarray}
\widehat{Q}\cong\mathbb{P}(\mathcal{O}_{\widetilde{Q}_0}(\nu_2^*(\nu_1^*(2\Delta_0)-\mathbb{E}_1)-\mathbb{E}_2)\oplus\mathcal{O}_{\widetilde{Q}_0}).
\end{eqnarray} Note that 
$\widehat{Q}_0$ is the minimal resolution of 
$Q_0\,(\subset\mathbb{P}(E_0))$ that has two rational double points of type A 1. If S 1 is the preimage of 
$\widetilde{S}$ in 
$\widehat{Q}$, and if 
$\mathcal{P}$ is sufficiently general, then the fibre of 
$S_1\to\mathbb{P}^1$ over p 1 can be written as 
$2C+\sum_{i=1}^6E_i$, where C is an elliptic curve and Ei is 
$(-2)$-curve such that 
$CE_i=1$ and 
$E_iE_j=0$ 
$(i\neq j)$.
We have 
${\rm Discr}(Q_0)=2p_1$. Moreover, the rank of the fibre of 
$Q_0\to\mathbb{P}^1$ is 1, and hence, the fibre of 
$\frak{Q}\to\mathbb{P}^1$ over p 1 is a double of a hyperplane. Namely, the fibre of 
$f:\widetilde{S}\to\mathbb{P}^1$ over p 1 is of type (II-ii-c).
Remark 4. In either case of 3.2.2.1, 3.2.2.2 and 3.2.2.3, the singular fibre with positive H-index has the components of elliptic curves. Let C be one of the elliptic curves. By Zariski’s lemma (cf. e.g., [Reference Barth, Hulek, Peters and Van de Ven3, (8.2) Lemma]), 
$C\mathcal{F}=0$ for a fibre of f, which leads us to 
$C^2=-3$. By the adjunction formula, we have 
$K_{\widetilde{S}}C=3$. On the other hand, since S is a K3 surface, 
$K_{\widetilde{S}}$ consists of six exceptional curves of the blow-up 
$\widetilde{S}\to S$. Namely, C intersects with 3 of them.
4. Classification of subpencils
Let Sʹ be the surface obtained as the complete intersection of a hyperquadric Q and a hypercubic Y. Assume Sʹ satisfies the Assumption 1. Let S be a desingularization of Sʹ.
For Q, the following is known (see [Reference Fujita6], [Reference Harris8]):
(4-1) The case where
${\rm rk}(Q)=3$.Put
$E_3:=\mathcal{O}_{\mathbb{P}^1}(2)\oplus\mathcal{O}_{\mathbb{P}^1}^{\oplus2}$, and let $\pi_3:W_3:=\mathbb{P}(E_3)\to\mathbb{P}^1$ be the $\mathbb{P}^2$-bundle, T 3 the tautological divisor of W 3 and F a fibre of π. For the rational map $\Phi_{|T_3|}:W_3\to\mathbb{P}^4$, we have $Q\cong\Phi_{|T_3|}(W_3)$. Let $T_{3,0}\subset W_3$ be the relative hyperplane with $T_{3,0}\sim T_3-2F$. Then we have $T_{3,0}\cong\mathbb{P}^1\times\mathbb{P}^1$. Let $\beta_i:T_{3,0}\to\mathbb{P}^1$ $(i=1,2)$ be the natural projection. We may assume $\beta_1=\pi_3|_{T_{3,0}}$. If we put $Z:=\Phi_{|T_3|}(T_{3,0})$, then Z is a line in $\mathbb{P}^4$, and we have $\Phi_{|T_3|}|_{T_{3,0}}=\beta_2$. Z is a compound rational double point of type A 1 of Q.(4-2) The case where
${\rm rk}(Q)=4$.Put
$E_4:=\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus2}\oplus\mathcal{O}_{\mathbb{P}^1}$, and let $\pi_4:W_4:=\mathbb{P}(E_4)\to\mathbb{P}^1$ be the $\mathbb{P}^2$-bundle, T 4 the tautological divisor of W 4 and F a fibre of π 4. We have $Q\cong\Phi_{|T_4|}(W_4)$. For a section $B_0=\mathbb{P}(E_4/\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus2})$ of π 4, we have that $\Phi_{|T_4|}(B_0)$ is a point. If we put $q_0:=\Phi_{|T_4|}(B_0)$, then Q has a three-dimensional rational double point of type A 1 at q 0.(4-3) The case where
${\rm rk}(Q)=5$.For a point
$q\in\mathbb{P}^4\setminus Q$, let $\nu:\widetilde{P}\to\mathbb{P}^4$ be the blow-up at q. Then we have $\widetilde{P}\cong\mathbb{P}(\mathcal{O}_{\mathbb{P}^3}(1)\oplus\mathcal{O}_{\mathbb{P}^3})$. Let $\Pi:\widetilde{P}\to\mathbb{P}^3$ be the $\mathbb{P}^1$-bundle, $\widetilde{H}$ the tautological divisor of $\widetilde{P}, H_0\subset\mathbb{P}^3$ a hyperplane and put $\mathbb{E}:=\nu^{-1}(q)$. We have $\widetilde{H}\sim\mathbb{E}+\Pi^*H_0$. If we consider $Q\subset\widetilde{P}$, we have $Q\sim 2\widetilde{H}$. If we put $\Pi_Q:=\Pi|_Q$, then the morphism $\Pi_Q:Q\to\mathbb{P}^3$ is a double cover branched along some smooth hyperquadric.$\Pi_Q$ is also considered as follows: For a hyperplane section $\widetilde{H}_Q$ of Q, we have $\dim|\widetilde{H}_Q|=4$. There exists a base point free three-dimensional subspace $V\subset|\widetilde{H}_Q|$ such that $\Pi_Q=\Phi_V$. For a hyperplane $H_0\subset\mathbb{P}^4$, $\Pi_Q^*H_0$ is a hyperplane section of Q. However, all the hyperplane sections cannot be written as above. On the other hand, we have the following:Lemma 2. For any hyperplane section
$\widetilde{H}_Q\in|\widetilde{H}_Q|$, there exist a base point free three-dimensional subspace $V\subset|\widetilde{H}_Q|$ and a hyperplane $H_0\subset\mathbb{P}^3$ such that $\widetilde{H}_Q=\Phi_V^*H_0$.Proof. If we take V as
$\widetilde{H}_Q\in V$ and if we put H 0 the image of $\widetilde{H}_Q$, then we obtain the desired equality.
4.1. The case ${\rm rk}(Q)=3$ (The proof for (1) of Theorem 1)
Let 
$Q\subset\mathbb{P}^4$ be a hyperquadric of rank 3. Let the notation be as in (4-1). If 
$Y\subset\mathbb{P}^4$ is a general hypercubic, then we may assume that Y is smooth and that the intersection 
$S^{\prime}:=Q\cap Y$ has rational double points over Z and no other singularity. Let 
$S_1\subset W_3$ be the preimage of S 0 by 
$\Phi_{|T_3|}$. We have 
$S_1\sim 3T_3$, and there exists a divisor δ 0 of degree 3 over 
$\mathbb{P}^1$ such that 
$S_1|_{T_{3,0}}\sim\beta_2^*\delta_0$. If Y is sufficiently general, then δ 0 is reduced and S 1 is nonsingular, namely, if S is the minimal resolution of Sʹ, then we have 
$S_1=S$. In order to classify subpencils of the complete linear system of the hyperplane section, it is sufficient to classify those of S 1. The hyperplane section of S 1 is written as 
$T_3|_{S_1}$. Therefore, in order to achieve our goal, it is sufficient to classify the subpencil of 
$|T_3|$.
Let 
$X_0,\,X_1\in H^0(\mathcal{O}_{W_3}(T_3))$ and 
$X_2\in H^0(\mathcal{O}_{W_3}(T_3-2F))$ be global sections defining the homogeneous coordinates of each fibre of π 3. Any member 
$\Psi\in H^0(\mathcal{O}_{W_3}(T_3))$ can be written as
\begin{eqnarray}
\Psi=c_0X_0+c_1X_1+\psi_2X_2,\quad
(c_0,\;c_1\in\mathbb{C},\;\;\psi_2\in H^0(\mathcal{O}_{\mathbb{P}^1}(2))).
\end{eqnarray} The divisor 
$(\Psi)$ is irreducible if and only if 
$(c_0,c_1)\neq(0,0)$, and we obtain the following:
Lemma 3. There exist three types of subpencil 
$\widehat{\mathcal{P}}$ of 
$|T_3|$ as follows:
(3-i) Any member of
$\widehat{\mathcal{P}}$ is irreducible.(3-ii)
$\widehat{\mathcal{P}}$ has only one reducible member.(3-iii) Any member of
$\widehat{\mathcal{P}}$ is reducible.
There exists no other type of subpencils.
Proof. Let 
$T_3,\,T^{\prime}_3\in|T_3|$ be two distinct members. Assume that the global sections defining T 3 and 
$T^{\prime}_3$ are written as 
$c_0X_0+c_1X_1+\psi_2X_2$ and 
$c^{\prime}_0X_0+c^{\prime}_2X_1+\psi^{\prime}_2X_2$, respectively. Let 
$\widehat{\mathcal{P}}\subset|T_3|$ be the subpencil generated by T 3 and 
$T^{\prime}_3$. If 
$c_0c^{\prime}_2\neq c_1c^{\prime}_0$, then any member of 
$\widehat{\mathcal{P}}$ is irreducible. If 
$c_0c^{\prime}_1=c_1c^{\prime}_0$ and that at least one of 
$(c_0,c_1)$ and 
$(c^{\prime}_0,c^{\prime}_1)$ is not equal to 
$(0,0)$, then reducible members of 
$\widehat{\mathcal{P}}$ is only the one defined by the global section of the form 
$\psi^{\prime\prime}_2X_2$. If 
$(c_0,c_1)=(c^{\prime}_0,c^{\prime}_1)=(0,0)$, then any member of 
$\widehat{\mathcal{P}}$ is reducible.
If T 3 is irreducible, then we have 
$T_3\cong\Sigma_2$ as a variety. If we put 
$\Delta_0:=T_3|_{T_3}$, then we have 
$\Delta_0^2=2$. If Γ is a fibre of the ruling 
$T_3\to\mathbb{P}^1$, and if 
$\Delta_{\infty}$ is the section with 
$\Delta_{\infty}\sim\Delta_0-2\Gamma$, then we have 
$T_{3,0}|_{T_3}\sim\Delta_{\infty}$.
Assume that 
$T_3,\,T^{\prime}_3\in|T_3|$ are irreducible and that the restriction 
$T^{\prime}_3|_{T_3}$ is written as 
$\Delta_{\infty}+\Gamma_1+\Gamma_2$. In this case, we have 
$T_3|_{T_{3,0}}=T^{\prime}_3|_{T_{3,0}}$, and this is the case (3-ii) of Lemma 3. Hence, in the case of (3-i), the restriction 
$T^{\prime}_3|_{T_3}$ is irreducible.
Since 
$S_1|_{T_3}=Y|_{T_3}\sim3\Delta_0$, a general member of 
$\mathcal{P}$ is EH-special nonhyperelliptic curve of genus 4 in the cases where 
$\widehat{\mathcal{P}}$ is of type(3-i) or (3-ii).
4.1.1. Type (3-i)
Let 
$\widehat{\mathcal{P}}$ be a subpencil of type (3-i). Then the base locus 
${\rm Bs}\widehat{\mathcal{P}}$ is an irreducible rational curve. For the pencil 
$\mathcal{P}$ over S corresponding to 
$\widehat{\mathcal{P}}$, the base locus 
${\rm Bs}\mathcal{P}$ consists of six points scheme theoretically. If 
$\mathcal{P}$ is generic in 
${\rm Gr}(3,1)$, then 
${\rm Bs}\mathcal{P}$ consists of six points set-theoretically also, and distinct two members of 
$\mathcal{P}$ intersect at these points transversally.
If 
$\eta_3:\widetilde{W}_3\to W_3$ is a blow-up along 
${\rm Bs}\widehat{\mathcal{P}}$, we obtain the following commutative diagram:
If we denote by Δ a fibre of β 1 and by Γ a fibre of β 2, then we have
\begin{equation*}\widetilde{W}_3\cong\mathbb{P}(\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(\Gamma)\oplus\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(2\Delta)).\end{equation*} If S 2 is the proper transform of S 1 by η 3, then 
$\beta_1\circ\xi_3:S_2\to\mathbb{P}^1$ is an elliptic fibration, and 
$\beta_2\circ\xi_3:S_2\to\mathbb{P}^1$ is an EH-special nonhyperelliptic fibration of genus 4. If 
$\mathbb{E}$ is an exceptional divisor of η 3, then we have 
$\mathbb{E}\cong\mathbb{P}^1\times\mathbb{P}^1$. We may consider that 
$T_{3,0}$ is contained in 
$\widetilde{W}_3$. The morphism 
$\Phi_{|2\eta_3^*T_3-\mathbb{E}|}$ maps 
$T_{3,0}$ onto a rational curve by contracting each fibre of β 2 onto a point. We obtain the following commutative diagram:
where 
$Q^{\prime}\to Q$ is the blow-up along the image of 
${\rm Bs}\widehat{\mathcal{P}}$ by 
$\Phi_{|T_3|}$. Qʹ has a structure of a quadric cone bundle 
$\widetilde{\xi}_3:Q^{\prime}\to\mathbb{P}^1$. Furthermore, if we consider 
$\mathbb{E}\cong T_{3,0}\,(\cong\mathbb{P}^1\times\mathbb{P}^1)$, the restriction of a fibre of 
$\widetilde{\xi}_3$ to 
$\mathbb{E}$ is a fibre of β 2. We have 
$\widetilde{W}_3\cong\mathbb{P}(\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}
(2\Delta-\Gamma))$, which coincides with Equation (3.1). Namely, the morphism 
$\widetilde{W}_3\to Q^{\prime}$ coincides with the morphism 
$\widetilde{Q}\to\frak{Q}$ in § 2.2. If 
$\widetilde{T}_3$ is the tautological divisor of 
$\widetilde{W}_3$ under this consideration, we obtain 
$S_2\sim 3\widetilde{T}_3+3\xi_3^*\Gamma$. Therefore, the image S 3 of S 2 in Qʹ intersects with the relative vertex of Qʹ at three points scheme theoretically. If 
$\widetilde{S}\to S_3$ is the minimal resolution, and if 
$f:\widetilde{S}\to\mathbb{P}^1$ is the naturally obtained EH-special nonhyperelliptic fibration of genus 4, then f has singular fibres of type (II-i-a), (II-i-b) or (II-i-c), and no other fibre with positive H-index. 
$S^{\prime}\subset Q$ is the image of S 3 by 
$Q^{\prime}\to Q$, and it is proved that the complete linear system of the hyperplane section of the minimal resolution S of Sʹ has the subpencil of type R3-1 of Theorem 1.
4.1.2. Type (3-ii)
Let us consider the case (3-ii). We use the same notation as in the proof of Lemma 3.
Since 
$c_0c^{\prime}_1=c_1c^{\prime}_0$, we can change one of the basis of the two-dimensional subspace of 
$H^0(\mathcal{O}_{W_3}(T_3))$ defining 
$\mathcal{P}$ to one of the form 
$\widetilde{\psi}_2X_2$ 
$(\widetilde{\psi}_2\in H^0(\mathcal{O}_{\mathbb{P}^1}(2)))$. Assume that another basis s is written as 
$s=c_2X_0+c_1X_1+\psi_2X_2$ 
$(c_0,c_1\in\mathbb{C},\;\psi_2\in H^0(\mathcal{O}_{\mathbb{P}^1}(2)))$. Let 
$q_1,q_2\in\mathbb{P}^1$ be points with 
$(\widetilde{\psi}_2)=q_1+q_2$. Put 
$\widehat{T}_3:=(s)$ and 
$B_{3,0}=\widehat{T}_3\cap T_{3,0}$. If we further put 
$F_i:=\pi_3^{-1}(q_i)$ and 
$\ell_i:=F_i\cap\widehat{T}_3$ 
$(i=1,2)$, we have
\begin{equation*}{\rm Bs}\widehat{\mathcal{P}}=B_{3,0}\cup\ell_1\cup\ell_2.\end{equation*} We have 
$\widehat{T}_3\cong\Sigma_2$, and if we consider that 
$B_{3,0}$ is a divisor of 
$\widehat{T}_3$, then 
$B_{3,0}=\Delta_{\infty}$ holds. On the other hand, if we denote by 
$\widehat{Y}$ the pull back of the hypercubic 
$Y\,(\subset\mathbb{P}^3)$ defining our surface Sʹ, then we have 
$\widehat{Y}|_{\widehat{T}_3}\sim3\Delta_0$. Hence, if we denote by S 1 the pull back of 
$S^{\prime}\,(=Q\cap Y)$ to W 3, one of 
$B_{3,0}\subset S_1$ and 
$B_{3,0}\cap S_1=\emptyset$ holds. The former case is excluded because 
$\mathcal{P}$ does not have a fixed component.
From now on, we consider the case (3-ii) by dividing it into two cases 
$q_1\neq q_2$ and 
$q_1=q_2$.
(3-ii-a) The case 
$q_1\neq q_2$
Assume 
$q_1\neq q_2$. We transform W 3 birationally as follows:
Step 1 Note that 
$B_{3,0}$ can be written as 
$B_{3,0}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(2)\oplus\mathcal{O}_{\mathbb{P}^1}^{\oplus2}/\mathcal{O}_{\mathbb{P}^1}(2)\oplus\mathcal{O}_{\mathbb{P}^1})$. Let 
$\rho_1:\widetilde{W}_3\to W_3$ be the blow-up along 
$B_{3,0}$. We have 
$\widetilde{W}_3\cong\mathbb{P}(\mathcal{O}_{\Sigma_2}(\Delta_0)\oplus\mathcal{O}_{\Sigma_2})$ and the following commutative diagram:
Put 
$\zeta:\widetilde{W}_3\to\Sigma_2$. Then ζ is a 
$\mathbb{P}^1$-bundle. If we let 
$\widetilde{T}_3$ be the proper transform of 
$\widehat{T}_3$ by ρ 1, then 
$\widetilde{T}_3$ can be written as 
$\widetilde{T}_3=\zeta^*\Delta_0^{\prime}$ for some 
$\Delta_0^{\prime}\in|\Delta_0|$. Namely, there exists a subpencile 
$\mathcal{P}_0\subset|\Delta_0|$ such that the proper transform of any member of 
$\mathcal{P}$ is the pull back of some member of 
$\mathcal{P}_0$. Note that there exist two base points of 
$\mathcal{P}_0$, say 
$\widetilde{q}_1$ and 
$\widetilde{q}_2$. For the ruling 
$\mu:\Sigma_2\to\mathbb{P}^1$, we may assume 
$\mu(\widetilde{q}_i)=q_i$ 
$(i=1,2)$. If we consider 
$\ell_i\subset\widetilde{W}_3$, then 
$\ell_i=\zeta^{-1}(\widetilde{q}_i)$ holds. Furthermore, if we put 
$\Gamma_i=\mu^{-1}(q_i)$, then we have 
$\Delta_{\infty}+\Gamma_1+\Gamma_2\in\mathcal{P}_0$ and its pull back by ζ is the divisor defined by 
$\widetilde{\psi}_2X_2$.
Step 2 Let 
$\rho_2:\widehat{W}_3\to\widetilde{W}_3$ be the blow-up along 
$\ell_1\cup\ell_2$. If we let 
$\widetilde{\rho}_2:\widetilde{\Sigma}\to\Sigma_2$ be the blow-up at 
$\widetilde{q}_1$ and 
$\widetilde{q}_2$, then we obtain the following commutative diagram:
Furthermore, we obtain the morphism 
$\widetilde{\mu}:\widetilde{\Sigma}\to\mathbb{P}^1$ whose fibre is a proper transform of a member of 
$\mathcal{P}_0$ by 
$\widetilde{\rho}_2$. Put 
$\mathcal{E}_i:=\widetilde{\rho}_2^{-1}(\widetilde{q}_i)$ 
$(i=1,2)$, and denote by 
$\widetilde{\Gamma}_i$ the proper transform of 
$\Gamma_i$ by 
$\widetilde{\rho}_2$ 
$(i=1,2)$. Let 
$\gamma_1:\widetilde{\Sigma}\to\Sigma^{\prime}$ be the blow-down of Γ1, and 
$\gamma_2:\Sigma^{\prime}\to\Sigma_1$ the blow-down of the image 
$\Delta^{\prime}_{\infty}$ of 
$\Delta_{\infty}$ by γ 1. If 
$\Delta^{\prime}_0$ is the section of 
$\Sigma_1\to\mathbb{P}^1$ with 
${\Delta^{\prime}}_0^2=1$, we obtain
\begin{equation*}\widetilde{\rho}_2^*\Delta_0\sim\gamma_1^*(\gamma_2^*(2\Delta^{\prime}_0)-\Delta^{\prime}_{\infty})-\Gamma_1,\end{equation*} and hence, 
$\widehat{W}_3$ coincides with Equation (3.4). The only singular fibre of the morphism 
$\widetilde{\mu}\circ\widetilde{\zeta}$ is the inverse image of 
$\Delta_{\infty}+\Gamma_1+\Gamma_2$ by ζ, where 
$\widetilde{\zeta}$ is as in the above commutative diagram. The preimage 
$S_3\subset\widetilde{W}_3$ of 
$S^{\prime}\subset\frak{Q}$ has a singular fibre with H-index 
$6/7$, which is the case (II-ii-b). Then it has been proved that Λ has a subpencil of type R3-2 of Theorem 1.
(3-ii-b) The case 
$q_1=q_2$
Assume 
$q_1=q_2$. Let 
$\rho_1:\widetilde{W}_3\to W_3$ be as in (3-ii-a). Furthermore, let 
$\rho_2:\widehat{W}_3\to\widetilde{W}_3$ be the blow-up along 
$\ell_1\,(=\ell_2)$. If 
$\widetilde{T}_3$ is as in Step 1 of (3-ii-a), then the base locus of 
$|\widetilde{T}_3|$ is a rational curve 
$\widetilde{\ell}$, and any two general member of 
$|\widetilde{T}_3|$ intersect along 
$\widetilde{\ell}$ transversally. Let 
$\rho_3:\overline{W}_3\to\widehat{W}_3$ be a blow-up along 
$\widetilde{\ell}$.
If 
$\widetilde{q}_1\in\Sigma_2$ is the point with 
$\zeta^{-1}(\widetilde{q}_1)=\ell_1$, then the subpencil 
$\mathcal{P}_0$ of 
$|\Delta_0|$ corresponding to 
$\widehat{\mathcal{P}}$ satisfies 
${\rm Bs}\mathcal{P}_0=\{\widetilde{q}_1\}$. Any two general member of 
$\mathcal{P}_0$ contact at 
$\widetilde{q}_1$ with intersection multiplicity 2. Let 
$\widetilde{\rho}_2:\widehat{\Sigma}\to\Sigma_2$ be a blow-up at 
$\widetilde{q}_1$ and 
$\widetilde{\mathcal{P}}_0$ the pencil consisting with the proper transforms of the members of 
$\mathcal{P}_0$. Then we have 
${\rm Bs}\widetilde{\mathcal{P}}_0=\{q^{\prime}_1\}$ for some point 
$q^{\prime}_1$ on the exceptional curve of 
$\widetilde{\rho}_2$. Any two general members of 
$\widetilde{\mathcal{P}}_0$ intersect at 
$q^{\prime}_1$ transversally. If 
$\widetilde{\rho}_3:\overline{\Sigma}\to\widehat{\Sigma}$ is a blow-up at 
$q^{\prime}_1$, then we obtain the following commutative diagram:
Note that 
$\overline{W}_3\cong\mathbb{P}(\mathcal{O}_{\overline{\Sigma}}(\widetilde{\rho}_3^*\,\widetilde{\rho}_2^*\Delta_0)
\oplus\mathcal{O}_{\overline{\Sigma}})$ holds. Put 
$E_1:=\widetilde{\rho}_2^{-1}(\widetilde{q}_1)$ and 
$\widetilde{E}_2:=\widetilde{\rho}_3^{-1}(q^{\prime}_1)$, and let Γ1 be the fibre of µ containing 
$\widetilde{q}_1$. Furthermore, let 
$\Delta_0\in|\Delta_0|$ be a member containing 
$\widetilde{q}_1$, and 
$\widetilde{\Delta}_0, \widetilde{\Gamma}_1$ the proper transform of Δ0 and Γ1, respectively. Moreover, put 
$E_2:=\widetilde{\rho}_3^{-1}(q^{\prime}_1)$, and let 
$\Delta^{\prime}_0$ and 
$\widetilde{E}_1$ be the proper transform of 
$\widetilde{\Delta}_0$ and E 1, respectively.
If we consider 
$\widetilde{\Gamma}_0\subset\overline{\Sigma}$, then 
$\widetilde{\Gamma}_0$ is a 
$(-1)$-curve, and we obtain the blow-down 
$\rho^{\prime}_3:\overline{\Sigma}\to\Sigma^{\prime}$. If we consider that 
$\Delta_{\infty}$ and 
$\widetilde{E}_1$ are the curves of 
$\Sigma^{\prime}$, then they are 
$(-1)$-curves. By blowing down 
$\widetilde{E}_1$, we obtain the birational morphism 
$\rho^{\prime}_2:\Sigma^{\prime}\to\mathbb{P}^1\times\mathbb{P}^1$.
Under the above consideration, we obtain
\begin{equation*}\widetilde{\rho}_3^*\,\widetilde{\rho}_2^*\Delta_0\sim{\rho^{\prime}}_3^*({\rho^{\prime}}_2^*(2E_2+\Delta^{\prime}_0)-\widetilde{E}_1)-\widetilde{\Gamma}_0,\end{equation*} which leads us to the fact that 
$\overline{W}_3$ coincides with Equation (3.5). Hence, if we put 
$p_1=\mu(\widetilde{q}_1)$, then 
$\widetilde{S}$ has a singular fibre of type (II-ii-c) over p 1 and no other degenerate fibre with positive H-index. It has been proved that Λ has a subpencil of type R3-3 of Theorem 1.
Denote by 
$\mathcal{F}=2C+\sum_{i=1}^6E_i$ the degenerate fibre with H-index 
$6/7$ as in 3.2.2.3. By checking the birational transformation of W 3 in detail, we obtain the following. Namely, three members of 
$\{E_i\}_{i=1}^6$ (say 
$E_1, E_2$ and E 3) are contracted to the rational double points on the singular locus of 
$\frak{Q}$. In 
$\widetilde{S}, E_i$ 
$(i=4,5,6)$ intersects with a 
$(-1)$-curve 
$\mathcal{E}_i$ which intersects every fibre of f. By contracting 
$\mathcal{E}_i$ 
$(i=4,5,6)$, Ei changes to a 
$(-1)$-curve, and other fibres intersect at three points transversally. By contracting Ei, we obtain the pencil whose base locus consists of three points set theoretically, and any two members contact at these points with intersection multiplicity 2. The image of C intersects with other member transversally at these three points.
4.2. The case ${\rm rk}(Q)=4$ (The proof for (2) of Theorem 1)
Let 
$Q\subset\mathbb{P}^4$ be a hyperquadric of rank 4. Let the notation be as in (4-2).
Consider the following two short exact sequence:
\begin{equation*}\begin{array}{l}
0\to\mathcal{O}_{\mathbb{P}^1}\to E_4\to\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus2}\to0,\\[2mm]
0\to\mathcal{O}_{\mathbb{P}^1}\to E_4\to\mathcal{O}_{\mathbb{P}^1}(2)\oplus\mathcal{O}_{\mathbb{P}^1}\to0.
\end{array}\end{equation*} We have two types of the tautological divisor. The first one is isomorphic to 
$\Sigma_0=\mathbb{P}^1\times\mathbb{P}^1$ and the second one is isomorphic to Σ2. These are obtained as follows:
Let 
$X_0,\,X_1\in H^0(\mathcal{O}_{W_4}(T_4-F))$ and 
$X_2\in H^0(\mathcal{O}_{W_4}(T_4))$ be global sections defining the homogeneous coordinates of each fibre of π 4. Any 
$\Psi\in H^0(\mathcal{O}_{W_4}(T_4))$ can be written as
\begin{equation*}\Psi=\psi_0X_0+\psi_1X_1+c_2X_2\quad(\psi_0,\psi_1\in H^0(\mathcal{O}_{\mathbb{P}^1}(1)),\;c_2\in\mathbb{C}.)\end{equation*}The following lemma is trivial:
Lemma 4. Let the notation be as above. Then 
$(\Psi)$ is isomorphic to 
$\mathbb{P}^1\times\mathbb{P}^1$ if and only if 
$c_2\neq0$. 
$(\Psi)$ is isomorphic to Σ2 if and only if 
$c_2=0$ and ψ 0 and ψ 1 have no common zero. 
$(\Psi)$ is reducible if and only if 
$c_2=0$ and ψ 0 and ψ 1 have a common zero.
Definition 1. In the above notation, let us call 
$T_4\in|T_4|$ the tautological divisor of type 
$({\rm t}_0)$ if 
$T_4\cong\mathbb{P}^1\times\mathbb{P}^1$ and the tautological divisor of type 
$({\rm t}_2)$ if 
$T_4\cong\Sigma_2$. Note that any member of 
$|T_4-F|$ is irreducible and isomorphic to Σ1. For any member 
$T^{\prime}\in|T_4-F|$, let us call the divisor 
$T^{\prime}+F$ the tautological divisor of type 
$({\rm t}_1)$.
Lemma 5. Let the notation be as above. The classification of subpencils 
$\widehat{\mathcal{P}}$ of 
$|T_4|$ is as follows:
(4-i)
$\widehat{\mathcal{P}}$ is generated by the tautological divisors of type $({\rm t}_0)$ and $({\rm t}_2)$.(4-ii)
$\widehat{\mathcal{P}}$ is generated by the tautological divisors of type $({\rm t}_0)$ and $({\rm t}_1)$.(4-iii)
$\widehat{\mathcal{P}}$ is generated by two tautological divisors of type $({\rm t}_1)$.(4-iv)
$\widehat{\mathcal{P}}$ is generated by the tautological divisors of type $({\rm t}_2)$ and $({\rm t}_1)$.
Proof. Let Ψ and 
$\Psi^{\prime}$ be the global sections of 
$\mathcal{O}_{W_4}(T_4)$ generating the two-dimensional subspace 
$V\subset H^0(\mathcal{O}_{W_4}(T_4))$ corresponding to 
$\widehat{\mathcal{P}}$. Assume Ψ and 
$\Psi^{\prime}$ are written as 
$\Psi=\psi_0X_0+\psi_1X_1+c_2X_2$ and 
$\Psi^{\prime}=\psi^{\prime}_0X_0+\psi^{\prime}_1X_1+c^{\prime}_2X_2$.
First, assume 
$c_2\neq0$. Put 
$\Psi^{\prime\prime}:=\Psi^{\prime}-(c^{\prime}_2/c_2)\Psi$. Assume it is written as 
$\Psi^{\prime\prime}=\psi^{\prime\prime}_0X_0+\psi^{\prime\prime}_1X_1$. If 
$\psi^{\prime\prime}_0$ and 
$\psi^{\prime\prime}_1$ are linearly independent, then 
$\widehat{\mathcal{P}}$ is of type (4-i). If 
$\psi^{\prime\prime}_0$ and 
$\psi^{\prime\prime}_1$ are linearly dependent, then 
$\widehat{\mathcal{P}}$ is of type (4-ii).
Next, assume 
$c_2=c^{\prime}_2=0$. If at least one of Ψ and 
$\Psi^{\prime}$ is of type 
$({\rm t}_1)$, then there is nothing to prove. So we may assume ψ 0 and ψ 1 do not have a common zero and neither do 
$\psi^{\prime}_0$ and 
$\psi^{\prime}_1$. If ψ 0 and 
$\psi^{\prime}_0$ are linearly dependent, then 
$\widehat{\mathcal{P}}$ is of type (4-iv). Assume that the pairs 
$(\psi_0,\psi^{\prime}_0)$ and 
$(\psi_1,\psi^{\prime}_1)$ are both linearly independent. Furthermore, assume that it is written as 
$\psi_0=A\psi_1+B\psi^{\prime}_1$ and 
$\psi^{\prime}_0=C\psi_1+D\psi^{\prime}_1$ for 
$A,B,C,D\in\mathbb{C}$. Let m be the solution of the quadric equation
\begin{eqnarray}
\left|\begin{array}{cc}m-A&-C\\ B&-m+D\end{array}\right|=0,
\end{eqnarray} and 
$(x,y)=(k,l)$ the nonzero solution of
\begin{equation*}\left(\begin{array}{cc}m-A&-C\\ B&-m+D\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)
=\left(\begin{array}{c}0\\ 0\end{array}\right).\end{equation*} Then we have 
$k\psi_0+l\psi^{\prime}_0=m(k\psi_1+l\psi^{\prime}_1)$ and the tautological divisor defined by 
$k\Psi+l\Psi^{\prime}$ is of type 
$({\rm t}_1)$. If Equation (4.2) has two distinct solutions, then 
$\widehat{\mathcal{P}}$ is of type (4-iii). If Equation (4.2) has a multiple solution, then 
$\widehat{\mathcal{P}}$ is of type (4-iv).
4.2.1. Type (4-i)
We may assume S is contained in W 4 since we assume that S does not go through the vertex of Q. Let 
$\widehat{\mathcal{P}}$ be the pencil of type (4-i) defining 
$\mathcal{P}$. Then there exists a tautological divisor 
$T_{4,0}\in\widehat{\mathcal{P}}$ of type 
$({\rm t}_2)$, and all the other members are the tautological divisor of type 
$({\rm t}_0)$. Since we assume that a general member of 
$\mathcal{P}$ is smooth, it is an EH-general curve of genus 4. Moreover, there exists a member that is an EH-special curve of genus 4. If 
$\mathcal{P}$ is generic in 
${\rm Gr}(3,1)$, then the base locus of 
$\mathcal{P}$ consists of six points and any two distinct member intersects at these points transversally. Hence, 
$\mathcal{P}$ is of type R4-1 of Theorem 1.
Remark 5. Let 
$T_4,T^{\prime}_4\in\widehat{\mathcal{P}}$ be of type 
$({\rm t}_0)$. For the image 
$Q\subset\mathbb{P}^4$ of W 4, we may consider as 
$T_4,T^{\prime}_4\subset Q$. There exist hyperplanes 
$H,H^{\prime}\subset\mathbb{P}^4$ with 
$H|_Q=T_4$ and 
$H^{\prime}|_Q=T^{\prime}_4$. Put 
$P:=H\cap H^{\prime}$. Then we have 
$P\cong\mathbb{P}^2$. Let 
$\alpha:\frak{X}\to\mathbb{P}^4$ be the blow-up along P. We have 
$\frak{X}\cong\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(2)\oplus\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus3})$. Let 
$\frak{T}$ be the tautological divisor of 
$\frak{X}$, 
$\frak{F}$ a fibre of the 
$\mathbb{P}^3$-bundle 
$\beta:\frak{X}\to\mathbb{P}^1$ and 
$\frak{Q}$ the proper transform of Q by α. Then 
$\frak{Q}\sim2\frak{T}-2\frak{F}$ holds. Namely, 
$\frak{X}$ and 
$\frak{Q}$ are the latter of the case of [Reference Takahashi16, Remark 3].
4.2.2. Type (4-ii)
Similarly to 4.2.1, if 
$\widehat{\mathcal{P}}$ is of type (4-ii) and generic in 
${\rm Gr}(3,1)$, then the corresponding pencil 
$\mathcal{P}$ on S is of type R4-2 of Theorem 1.
4.2.3. Type (4-iii)
Let Ψ0 and Ψ1 be the basis of two-dimensional subspace of 
$H^0(\mathcal{O}_{W_4}(T_4))$ corresponding to 
$\widehat{\mathcal{P}}$. Assume that 
$\Psi_i$ is written as 
$\Psi_i=\psi_iX_i$ 
$(\psi_i\in H^0(\mathcal{O}_{\mathbb{P}^1}(1))$ for 
$i=0,1$. Let 
$p_i\in\mathbb{P}^1$ be the point such that 
$\psi_i(p_i)=0$, and put 
$F_i=\pi_4^{-1}(p_i)$. Assume 
$p_0\neq p_1$. Denote by B 0 the section of π 4 defined by 
$X_0=X_1=0$. For 
$i=0,1$, denote by 
$\ell_i$ the intersection of Fi and 
$X_{\sigma(i)}$, where σ is the permutation of 0 and 1. Then we have 
${\rm Bs}\widehat{\mathcal{P}}=B_0\cup\ell_0\cup\ell_1$.
Let 
$\rho_1:\widetilde{W}_4\to W_4$ be the blow-up along B 0. Consider that 
$\ell_i$ 
$(i=0,1)$ is the curve of 
$\widetilde{W}_4$, and let 
$\rho_2:\widehat{W}_4\to\widetilde{W}_4$ be the blow-up along 
$\ell_0\cup\ell_1$. Then by the same argument as in case (3-ii-a) of § 4.1.2, we obtain that 
$\widehat{W}_4$ coincides with Equation (3.3) and that the subpencil of type R4-3 of Theorem 1 exists.
4.2.4. Type (4-iv)
We may assume that the basis Ψ and 
$\Psi^{\prime}$ of the subspace corresponding 
$\widehat{\mathcal{P}}$ are written as 
$\Psi=\psi_0X_0+\psi_1X_1$ and 
$\Psi^{\prime}=\psi^{\prime}_1X_1$. Hence, in the proof of Lemma 5, we have 
$C=D=0$. If the quadric equation (4.2) has the multiple solution, we have A = 0. Hence, we may assume 
$\psi^{\prime}_1=\psi_0$.
Let B 0 be as in the previous subsection, and 
$\ell_0$ be the curve defined by 
$\psi_0=X_1=0$. Then we have 
${\rm Bs}\widehat{\mathcal{P}}=B_0\cup\ell_0$. Let 
$\rho_1:\widetilde{W}_4\to W_4$ be as in the previous subsection and 
$\rho_2:\widehat{W}_4\to\widetilde{W}_4$ the blow-up along 
$\ell_0$. Then 
$\widetilde{W}_4$ has the structure of the 
$\mathbb{P}^1$-bundle 
$\zeta_4:\widetilde{W}_4\to\mathbb{P}^1\times\mathbb{P}^1$. Furthermore, if we put 
$q_0:=\zeta_4(\ell_0)$, and if we denote by 
$\nu_1:\widetilde{\Sigma}\to\mathbb{P}^1\times\mathbb{P}^1$ the blow-up at q 0, then we have 
$\widehat{W}_4\cong\widetilde{W}_4\times_{\mathbb{P}^1\times\mathbb{P}^1}\widetilde{\Sigma}$. Let 
$\widetilde{\zeta}_4:\widehat{W}_4\to\widetilde{\Sigma}$ be the obtained 
$\mathbb{P}^1$-bundle. For the proper transform 
$\widehat{T}_4$ of T 4 by 
$\rho_1\circ\rho_2$, there exists an infinitely near point 
$\widetilde{q}_0\in\widetilde{\Sigma}$ with 
${\rm Bs}\left|\widehat{T}_4\right|=\widetilde{\zeta}_4^{-1}(\widetilde{q}_0)$. Put 
$\ell_1:=\widetilde{\zeta}_4^{-1}(\widetilde{q}_0)$.
Let 
$\overline{\rho}_4:\overline{W}_4\to\widehat{W}_4$ be the blow-up along 
$\ell_1$. By the same argument as before, we obtain that 
$\overline{W}_4$ coincides with Equation (3.4).
In this case, the only degenerate fibre of f with positive H-index is of the form 
$C_1+C_2+\sum_{i=1}^3E_i$, where C 1 and C 2 are elliptic curves, and Ei is a 
$(-2)$-curve. f has three sections 
$\mathbb{E}_i$ 
$(i=1,2,3)$ as 
$(-1)$-curves. We may assume 
$\mathbb{E}_iE_i=1$ and 
$\mathbb{E}_iE_j=0$ if and only if i ≠ j. Furthermore, we have 
$\mathbb{E}_iC_j=0$. Let 
$\nu_1:\widetilde{S}\to S_1$ be the blow-down of 
$\mathbb{E}_1, \mathbb{E}_2$ and 
$\mathbb{E}_3$. Then the images of general fibres of f by ν 1 intersect at three points. Let 
$\nu_2:S_1\to S$ be the blow-down of the image of 
$E_1+E_2+E_3$ by ν 1. Then the images of general fibres of f by 
$\nu_2\circ\nu_1$ contact at three points with intersection multiplicity 2, while the image of Cj intersects the images at these points transversally. Hence, the obtained subpencil is of type R4-4 of Theorem 1.
4.3. The case ${\rm rk}(Q)=5$ (The proof for (3) of Theorem 1)
Let Q be a hyperquadric of rank 5. By the similar argument to Lemma 2, we obtain the following:
Lemma 6. Let the notation be as in Lemma 2. For any subpencil 
$\mathcal{P}_Q\subset|\widetilde{H}_Q|$, there exists a double cover 
$\gamma:Q\to\mathbb{P}^3$ such that any member of 
$\mathcal{P}_Q$ is mapped onto a hyperplane by γ as a branched double cover.
Let 
$S\subset Q$ be our surface and 
$\mathcal{P}$ the subpencil of the complete linear system of the hyperplane sections. Let 
$\widehat{\mathcal{P}}$ be the subpencil of the complete linear system of the hyperplane sections of Q whose restriction to S is 
$\mathcal{P}$. Let 
$\gamma:Q\to\mathbb{P}^3$ be the double cover of Lemma 6. If 
$\widetilde{\mathcal{P}}$ is the subpencil of the hyperplanes of 
$\mathbb{P}^3$ corresponding to 
$\mathcal{P}$, then 
$\ell:={\rm Bs}\widetilde{\mathcal{P}}$ is a line of 
$\mathbb{P}^3$. Let 
$Q_0\subset\mathbb{P}^3$ be the branch locus of γ. We have 
$Q_0\cong\mathbb{P}^1\times\mathbb{P}^1$.
There are following three cases:
(5-i)
$\ell$ and Q 0 intersect at two distinct points.(5-ii)
$\ell$ and Q 0 contact at a point.(5-iii)
$\ell\subset Q_0$ holds. In this case, $\ell$ is a fibre of one of the natural projection for the direct product.
4.3.1. The case (5-i)
Let us consider the case (5-i).
Let 
$\mathcal{P}_0$ be the restriction of 
$\widetilde{\mathcal{P}}$ to Q 0. If 
$q_1,q_2\in Q_0$ are the points with 
$\ell\cap Q_0=\{q_1,q_2\}$, then we have 
${\rm Bs}\mathcal{P}_0=\{q_1,q_2\}$. Let 
$\eta_i:Q_0\to\mathbb{P}^1$ be the natural projection for the ith element 
$(i=1,2)$, 
$\Delta_{1j}$ the fibre of η 1 with 
$q_j\in\Delta_{1j}$ and 
$\Gamma_{2j}$ the fibre of η 2 with 
$q_j\in\Gamma_{2j}$. We have 
$\Delta_{11}+\Gamma_{21},\Delta_{12}+\Gamma_{22}\in\mathcal{P}_0$. Furthermore, any member of 
$\mathcal{P}_0\setminus\{\Delta_{11}+\Gamma_{21},\Delta_{12}+\Gamma_{22}\}$ is irreducible and non-singular. Hence, 
$\widehat{\mathcal{P}}$ has two quadric cones and any other member is a smooth hyperquadric. If we consider 
${\rm Bs}\widehat{\mathcal{P}}$ as a divisor of a general member, then it is a smooth diagonal divisor, while if we consider it as a divisor of the quadric cone, then it is smooth section not going through the vertex. If we put 
$\widetilde{\ell}:={\rm Bs}\widehat{\mathcal{P}}$, then 
$\widetilde{\ell}$ is an irreducible and non-singular rational curve. Let 
$\overline{\mathcal{P}}$ be the pencil of the complete linear system of hyperplanes of 
$\mathbb{P}^4$ corresponding to 
$\widehat{\mathcal{P}}$. We have 
$\mathbb{H}:={\rm Bs}\overline{\mathcal{P}}$ is a two-dimensional subspace with 
$\mathbb{H}|_Q=\widetilde{\ell}$.
Let 
$\alpha:\frak{X}\to\mathbb{P}^4$ be the blow-up along 
$\mathbb{H}$. Then 
$\frak{X}\cong\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(2)\oplus\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus3})$. Let 
$\beta:\frak{X}\to\mathbb{P}^1$ be the natural projection, 
$\frak{T}$ the tautological divisor of 
$\frak{X}$ and 
$\frak{F}$ a fibre of β. If we put 
$\mathbb{E}:=\alpha^{-1}(\mathbb{H})$, then we have 
$\mathbb{E}\cong\mathbb{P}^1\times\mathbb{P}^2$ and 
$\mathbb{E}\sim\frak{T}-2\frak{F}$. If 
$\frak{Q}$ is the proper transform of Q by α, then we have 
$\frak{Q}\sim 2\frak{T}-2\frak{F}$. This situation is similar to that of 4.2.1. The differences from 4.2.1 are that 
$\frak{Q}$ is smooth and that the fibration 
$\frak{Q}\to\mathbb{P}^1$ has two degenerate fibres of rank 3. Namely, the discriminant locus of 
$\frak{Q}$ is a reduced divisor of degree 2, and hence, f is of type (I-i). In a general case, 
${\rm Bs}\mathcal{P}$ consists of six points over 
$\widetilde{\ell},$ and general members intersect at these points transversally. We obtain the pencil of type R5-1 of Theorem 1.
4.3.2. The case (5-ii)
The similar argument to the case (5-i) is applied. We use the same notation. The base locus 
${\rm Bs}\mathcal{P}_0$ consists of one point, say 
$q\in Q_0$. Let Δ3 and Γ3 be fibres of the natural projection η 1 and η 2, respectively, such that 
$\Delta_3\cap\Gamma_3=\{q\}$. 
$\mathcal{P}_0$ contains 
$\Delta_3+\Gamma_3$ and all the other members of 
$\mathcal{P}_0$ are smooth diagonal divisors contacting at q with intersection multiplicity 2. 
$\widehat{\mathcal{P}}$ contains a quadric cone, and any other member is a smooth hyperquadric. Since 
${\rm Bs}\widetilde{\mathcal{P}}$ is a line in 
$\mathbb{P}^3$ contacting with Q 0 at q, the base locus 
${\rm Bs}\widehat{\mathcal{P}}$ is a union of two rational curves intersecting at a point. Put 
$\widetilde{\ell}_1+\widetilde{\ell}_2={\rm Bs}\widehat{\mathcal{P}}$.
Let 
$\alpha:\frak{X}\to\mathbb{P}^4$ be as in the case (iii-a). Moreover, we use the same other notation related to 
$\frak{X}$. Then 
$\frak{Q}$ is a relative hyperquadric and a general member of 
$\frak{Q}\to\mathbb{P}^1$ is isomorphic to 
$\mathbb{P}^1\times\mathbb{P}^1$. There exists only one degenerate fibre that is isomorphic to the quadric cone. Namely, the discriminant locus of 
$\frak{Q}$ is a non-reduced divisor of degree 2 and we obtain the subpencil of type R5-2 of Theorem 1. If 
$\mathcal{P}$ is generic, then 
${\rm Bs}\mathcal{P}$ consists of six points. Three of them are on 
$\ell_1$, and the rest three points are on 
$\ell_2$. General members of 
$\mathcal{P}$ intersect at these points transversally one another.
4.3.3. The case (5-iii)
Let us consider the case (5-iii). All the members of 
$\widehat{\mathcal{P}}$ are quadric cones, and they contact along a generating line to one another. On the other hand, the vertices of any two distinct cones are the distinct points on the generating line. Let 
$\ell$ be the generating line, and 
$\alpha:\frak{M}\to\mathbb{P}^4$ the blow-up along 
$\ell$. We have 
$\frak{M}\cong\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}(1)\oplus\mathcal{O}_{\mathbb{P}^2}^{\oplus2})$, which is a 
$\mathbb{P}^2$-bundle over 
$\mathbb{P}^2$. Let 
$\eta:\frak{M}\to\mathbb{P}^2$ be the projection. Denote by 
$T_{\frak{M}}$ the tautological divisor of 
$\frak{M}$. Let 
$L\in\mathbb{P}^2$ be a line, and put 
$\mathbb{E}:=\alpha^{-1}(\ell)$. We have 
$T_{\frak{M}}\sim\mathbb{E}+\eta^*L$ and 
$\mathbb{E}\cong\mathbb{P}^1\times\mathbb{P}^2$. If 
$\widetilde{Q}$ is the proper transform of Q by α, then we have 
$\widetilde{Q}\sim T_{\frak{M}}+\eta^*L$. Denote by 
$\eta_{\mathbb{E}}$ the natural projection 
$\mathbb{E}\to\mathbb{P}^2$ and by β the natural projection 
$\mathbb{E}\to\mathbb{P}^1$. If we consider the restriction 
$\widetilde{Q}|_{\mathbb{E}}$ as a divisor of 
$\mathbb{E}$, then we have
\begin{equation*}\widetilde{Q}|_{\mathbb{E}}\sim T_{\mathbb{E}}+\eta_{\mathbb{E}}^*L,\end{equation*} where 
$T_{\mathbb{E}}$ is a fibre of β. The restricted morphism 
$\eta_{\widetilde{Q}}:\widetilde{Q}|_{\mathbb{E}}\to\mathbb{P}^2$ coincides with the blow-up of 
$\mathbb{P}^2$ at a point, namely, we obtain 
$\widetilde{Q}|_{\mathbb{E}}\cong\Sigma_1$. Put 
$\widetilde{\mathbb{E}}:=\mathbb{E}|_{\widetilde{Q}}$. Then 
$\widetilde{\mathbb{E}}$ is the exceptional divisor of the blow-up 
$\alpha_{\widetilde{Q}}:\widetilde{Q}\to Q$. If 
$\widehat{H}_Q$ is the proper transform of 
$\widetilde{H}_Q\in\widehat{\mathcal{P}}(\subset|\widetilde{H}_Q|)$, then we have 
$\widehat{H}_Q\cong\Sigma_2$ and all the members of 
$|\widehat{H}_Q|$ intersect along some fibre (say Γ0) transversally. Moreover, 
$\widehat{H}_Q$ and 
$\widetilde{\mathbb{E}}$ intersect along Γ0 transversally. The restriction of 
$\widehat{H}_Q$ to 
$\widetilde{\mathbb{E}}$ is the 
$(-1)$-curve, and hence, it is the section of the ruling 
$\widetilde{\mathbb{E}}(\cong\Sigma_1)\to\mathbb{P}^1$. If 
$\widetilde{\alpha}:\widehat{Q}\to\widetilde{Q}$ is the blow-up along Γ0, we obtain the following commutative diagram:
If we consider 
$\widehat{H}_Q$ as a divisor of 
$\widehat{Q}$, then 
$\widehat{H}_Q$ is the pull back of a fibre of µ 1, and the restricted morphism 
$\widehat{\eta}|_{\widetilde{\mathbb{E}}}:\widetilde{\mathbb{E}}\to\Sigma_1$ is an isomorphism. 
$\widehat{Q}$ has the structure of Σ2-bundle over 
$\mathbb{P}^1$, and the discriminant locus of 
$\widehat{Q}$ is zero.
For our K3 surface S, if we assume 
$\ell\subset S$, then 
$\mathcal{P}$ has a fixed component, and this case is excluded by the assumption that a general member of 
$\mathcal{P}$ is smooth. Hence, we may assume 
$\ell\not\subset S$. Then S and 
$\ell$ intersect at three points scheme theoretically. Put 
$\{p_1,p_2,p_3\}=S\cap\ell$. For 
$i=1,2,3$, there exists a member 
$\widetilde{H}_{Q,i}\in|\widetilde{H}_Q|$ such that pi is the vertex of 
$\widetilde{H}_{Q,i}$. The curve 
$\widetilde{\alpha}^{-1}(p_i)$ is mapped onto a fibre of µ 1 isomorphically. Namely, the set of degenerate members of 
$\mathcal{P}$ is one of (II-i-a), (II-i-b) and (II-i-c), and 
$\mathcal{P}$ is the subpencil of type R5-3 of Theorem 1.
From now on, we assume that 
$p_i\neq p_j$ if and only if i ≠ j. Let 
$\mathcal{F}_i$ be the member of 
$\mathcal{P}$ contained in 
$\widetilde{H}_{Q,i}$ 
$(i=1,2,3)$. General members of 
$\mathcal{P}$ are irreducible and non-singular and contact at 
$p_1, p_2$ and p 3 with intersection multiplicity 2. On the other hand, 
$\mathcal{F}_i$ intersects with a general member at 
$p_{\sigma(i)}$ and 
$p_{\sigma^2(i)}$ transversally and contacts at p 1 with intersection multiplicity 2, where σ is one of the nontrivial cyclic permutation of 
$1, 2$ and 3 of order 3.
Let 
$f:\widetilde{S}\to S$ be as before. Then f has three sections 
$C_1, C_2$ and C 3 that are 
$(-1)$-curves. Any degenerate fibre of f with positive H-index is of the form 
$\mathcal{E}_i+D_i$, where 
$\mathcal{E}_i$ is a 
$(-2)$-curve that is the preimage of 
$\widetilde{\alpha}^{-1}(p_i)$, and Di is a hyperelliptic curve of genus 3. We may assume 
$C_i\mathcal{E}_i=1$ and that 
$C_i\mathcal{E}_j=0$ if and only if i ≠ j. If C 1, C 2 and C 3 are contracted to smooth points, then the images of 
$\mathcal{E}_1$, 
$\mathcal{E}_2$ and 
$\mathcal{E}_3$ are 
$(-1)$-curves. If these three 
$(-1)$-curves are contracted to smooth points, we obtain S and 
$\mathcal{P}$. The images of any two distinct general members contact at three points with intersection multiplicity 2, while the image of Di has an ordinary node at pi and contact at 
$p_{\sigma(i)}$ and 
$p_{\sigma^2(i)}$ with intersection multiplicity 2.
