2. Proof of theorem

In this section, we prove Theorem 1.1. Let $f\colon S\to B$ be a relatively minimal E-H special fibration of genus 4. Since the general fiber F of f is non-hyperelliptic, the multiplicative map $\textrm{Sym}^{2}f_{*}\omega_f\to f_{*}\omega_f^{\otimes 2}$ is generically surjective from Noether’s theorem. Thus, we have the following exact sequences of sheaves of $\mathcal{O}_B$ -modules:

(2.1) \begin{equation} 0\to \mathcal{L}\to \textrm{Sym}^{2}f_{*}\omega_f\to f_{*}\omega_f^{\otimes 2}\to \mathcal{T}\to 0,\end{equation}

where the kernel $\mathcal{L}$ is a line bundle on B and the cokernel $\mathcal{T}$ is a torsion sheaf on B. Then, the first injection defines a section $q\in H^{0}(B,\textrm{Sym}^{2}f_{*}\omega_f\otimes \mathcal{L}^{-1})=H^{0}(\mathbb{P}_{B}(f_{*}\omega_f),2T-\pi^{*}\mathcal{L})$ , where $\pi\colon \mathbb{P}_{B}(f_{*}\omega_f)\to B$ is the projection and $T=\mathcal{O}_{\mathbb{P}_{B}(f_{*}\omega_f)}(1)$ is the tautological line bundle on $\mathbb{P}_{B}(f_{*}\omega_f)$ . The section q can be regarded as a relative quadratic form $q\colon (f_{*}\omega_f)^{*}\to f_{*}\omega_f\otimes \mathcal{L}^{-1}$ , which defines the determinant $\textrm{det}(q)\colon \textrm{det}(f_{*}\omega_f)^{-1}\to \textrm{det}(f_{*}\omega_f)\otimes \mathcal{L}^{-4}$ . Note that for a non-hyperelliptic fibration f of genus 4, $\textrm{det}(q)=0$ if and only if f is E-H special. On the other hand, $Q=(q)\in |2T-\pi^{*}\mathcal{L}|$ is regarded as the unique relative quadric on $\mathbb{P}_{B}(f_{*}\omega_f)$ containing the image of the relative canonical map $\Phi_f\colon S\dashrightarrow \mathbb{P}_{B}(f_{*}\omega_f)$ . Since f is E-H special, the general fiber of $\pi|_{Q}\colon Q\to B$ is a quadric of rank 3 on $\mathbb{P}(H^{0}(F,K_F))=\mathbb{P}^3$ . The closure of the set of vertexes of general fibers of $\pi|_{Q}$ defines a section $v\colon B\to Q$ , which corresponds to some quotient line bundle $\mathcal{F}$ of $f_{*}\omega_f$ . Let $\mathcal{E}$ be the kernel of the surjection $f_{*}\omega_f\to \mathcal{F}$ and put $P=\mathbb{P}_{B}(f_{*}\omega_f)$ and $P^{\prime}=\mathbb{P}_{B}(\mathcal{E})$ . Let $\tau\colon \widetilde{P}\to P$ be the blow-up of P along the section v(B). Then, the relative projection $P\dashrightarrow P^{\prime}$ from the section v(B) extends to the morphism $\tau^{\prime}\colon \widetilde{P}\to P^{\prime}$ with

\begin{equation*}\tau^{\prime*}T^{\prime}=\tau^{*}T-E,\end{equation*}

where $T^{\prime}=\mathcal{O}_{\mathbb{P}_{B}(\mathcal{E})}(1)$ is the tautological line bundle of $\mathbb{P}_{B}(\mathcal{E})$ and E is the exceptional divisor of $\tau$ . Let $\widetilde{Q}$ denote the proper transform of Q on $\widetilde{P}$ . It follows that in $\textrm{Pic}(\widetilde{P})$ ,

\begin{equation*}\widetilde{Q}=\tau^{*}Q-2E=\tau^{\prime*}(2T^{\prime}-\pi^{\prime*}\mathcal{L}),\end{equation*}

where $\pi^{\prime}\colon P^{\prime}\to B$ is the projection. Let $Q^{\prime}=\tau^{\prime}(\widetilde{Q})$ be the image of $\widetilde{Q}$ via $\tau^{\prime}$ . It follows that $Q^{\prime}\in |2T^{\prime}-\pi^{\prime*}\mathcal{L}|$ and $\widetilde{Q}=\tau^{\prime*}Q^{\prime}$ . The general fiber of $\pi^{\prime}|_{Q^{\prime}}\colon Q^{\prime}\to B$ is a conic on $\mathbb{P}(H^{0}(F,\mathcal{E}|_{F}))=\mathbb{P}^2$ of rank 3, which is isomorphic to $\mathbb{P}^1$ . Note that the composite $\tau^{\prime}\circ \Phi_f\colon S\dashrightarrow Q^{\prime}\subset P^{\prime}$ of the relative canonical map $\Phi_f\colon S\dashrightarrow P$ and the projection $\tau^{\prime}\colon P\dashrightarrow P^{\prime}$ determines the unique trigonal structure of the general fiber F of f. Let $q^{\prime}\in H^{0}(P^{\prime},2T^{\prime}-\pi^{\prime*}\mathcal{L})=H^{0}\!\left(B,\textrm{Sym}^{2}\mathcal{E}\otimes \mathcal{L}^{-1}\right)$ be a section which defines $Q^{\prime}=(q^{\prime})$ . Then q ^′ can be regarded as a relative quadratic form $q^{\prime}\colon \mathcal{E}^{*}\to \mathcal{E}\otimes \mathcal{L}^{-1}$ , which has non-zero determinant $\textrm{det}(q^{\prime})\colon \textrm{det}(\mathcal{E})^{-1}\to \textrm{det}(\mathcal{E})\otimes \mathcal{L}^{-3}$ since Q ^′ is of rank 3. Thus, $\textrm{det}(q^{\prime})\in H^{0}(B,\textrm{det}(\mathcal{E})^{\otimes 2}\otimes \mathcal{L}^{-3})$ defines an effective divisor $\Delta_{Q^{\prime}}=(\textrm{det}(q^{\prime}))$ on B. The degree of $\Delta_{Q^{\prime}}$ is

(2.2) \begin{equation} \textrm{deg}\Delta_{Q^{\prime}}=2\textrm{deg}\mathcal{E}-3\textrm{deg}\mathcal{L}.\end{equation}

Let $\rho\colon \widetilde{S}\to S$ be the minimal desingularization of the rational map $\tau^{-1}\circ \Phi_f\colon S\dashrightarrow \widetilde{P}$ and $\widetilde{\Phi}\colon \widetilde{S}\to \widetilde{P}$ the induced morphism. Put $\Phi=\tau\circ\widetilde{\Phi}\colon \widetilde{S}\to P$ , $\Phi^{\prime}=\tau^{\prime}\circ\widetilde{\Phi}\colon \widetilde{S}\to P^{\prime}$ , $M=\Phi^{*}T$ and $M^{\prime}=\Phi^{\prime*}T^{\prime}$ . Then we can write $\rho^{*}K_f=M+Z$ for some effective vertical divisor Z on $\widetilde{S}$ . Since $M^{\prime}=M-\widetilde{\Phi}^{*}E$ , we can also write $\rho^{*}K_f=M^{\prime}+Z^{\prime}$ , where $Z^{\prime}=Z+\widetilde{\Phi}^{*}E$ is also an effective vertical divisor on $\widetilde{S}$ . Since $\Phi^{\prime}$ is of degree 3 onto the image Q ^′, we have $\Phi^{\prime}_{*}\widetilde{S}=3Q^{\prime}$ as cycles. It follows that

\begin{align*}M^{\prime 2}&=(\Phi^{\prime*}T^{\prime})^{2}\widetilde{S}=T^{\prime 2}\Phi^{\prime}_{*}\widetilde{S} \\&=3T^{\prime 2}Q^{\prime}=3T^{\prime 2}(2T^{\prime}-\pi^{\prime*}\mathcal{L}) \\&=6\textrm{deg}\mathcal{E}-3\textrm{deg}\mathcal{L},\end{align*}

while we have

\begin{equation*}M^{\prime 2}=\left(\rho^{*}K_f-Z^{\prime}\right)^2=K_f^2-\left(\rho^{*}K_f+M^{\prime}\right)\!Z^{\prime}.\end{equation*}

Hence, we get

(2.3) \begin{equation} K_f^2=6\textrm{deg}\mathcal{E}-3\textrm{deg}\mathcal{L}+\left(\rho^{*}K_f+M^{\prime}\right)\!Z^{\prime}.\end{equation}

From (2.2) and (2.3), we can delete the term $\textrm{deg}\mathcal{E}$ and then we have

(2.4) \begin{equation} \textrm{deg}\mathcal{L}=\frac{1}{6}K_f^2-\frac{1}{6}\!\left(\rho^{*}K_f+M^{\prime}\right)\!Z^{\prime}-\frac{1}{2}\textrm{deg}\Delta_{Q^{\prime}}.\end{equation}

On the other hand, taking the degree of (2.1), we get

(2.5) \begin{equation} K_f^2=4\chi_f-\textrm{deg}\mathcal{L}+\textrm{length}\mathcal{T}.\end{equation}

Substituting (2.4) in the equation (2.5), we get

\begin{equation*}K_f^2=\frac{24}{7}\chi_f+\frac{1}{7}\left(\rho^{*}K_f+M^{\prime}\right)Z^{\prime}+\frac{3}{7}\textrm{deg}\Delta_{Q^{\prime}}+\frac{6}{7}\textrm{length}\mathcal{T}.\end{equation*}

For a fiber germ $f^{-1}(p)$ , we define $\textrm{Ind}\!\left(f^{-1}(p)\right)$ by

\begin{equation*}\textrm{Ind}\!\left(f^{-1}(p)\right)=\frac{1}{7}\left(\rho^{*}K_f+M^{\prime}\right)Z^{\prime}_p+\frac{3}{7}\textrm{mult}_p\Delta_{Q^{\prime}}+\frac{6}{7}\textrm{length}_p\mathcal{T},\end{equation*}

where $Z=\sum_{p\in B}Z_p$ is the natural decomposition with $(f\circ \rho)(Z_p)=\{p\}$ for any $p\in B$ . For the definitions of M ^′, Z ^′, etc., we do not use the completeness of the base B. Thus, we can modify the definition of Ind for any fiber germ of relatively minimal E-H special fibrations of genus 4 which is invariant under holomorphically equivalence. Thus, we can define the Horikawa index $\textrm{Ind}\colon \mathcal{A}\to \mathbb{Q}_{\ge 0}$ such that

\begin{equation*}K_f^2=\frac{24}{7}\chi_f+\sum_{p\in B}\textrm{Ind}\!\left(f^{-1}(p)\right).\end{equation*}

The non-negativity of $\textrm{Ind}\!\left(f^{-1}(p)\right)$ is as follows. From the nefness of $K_f$ , we have $\rho^{*}K_fZ^{\prime}_p\ge 0$ . For a sufficiently ample divisor $\mathfrak{a}$ on B, the linear system $|M^{\prime}+(f\circ \rho)^{*}\mathfrak{a}|$ is free from base points. Thus, by Bertini’s theorem, there is a smooth horizontal member $C\in |M^{\prime}+(f\circ \rho)^{*}\mathfrak{a}|$ and then $M^{\prime}Z^{\prime}_p=(M^{\prime}+(f\circ \rho)^{*}\mathfrak{a})Z^{\prime}_p=CZ^{\prime}_p\ge 0$ .

Once the Horikawa index is introduced, we can define the local signature since $\textrm{Sign}(S)=K^2_{f}-8\chi_f$ and $e_f=12\chi_f-K_f^{2}$ is localized by using the topological Euler numbers of the singular fibers (cf. [Reference Ashikaga and Konno1, Section 2]). Indeed, we put

\begin{equation*}\sigma\!\left(f^{-1}(p)\right)=\frac{7}{15}\textrm{Ind}\!\left(f^{-1}(p)\right)-\frac{8}{15}e_f\!\left(f^{-1}(p)\right),\end{equation*}

where $e_f\!\left(f^{-1}(p)\right)=e_{\textrm{top}}\!\left(f^{-1}(p)\right)+6$ is the Euler contribution at $p\in B$ . Then we have $\textrm{Sign}(S)=\sum_{p\in B}\sigma\!\left(f^{-1}(p)\right)$ .