on $C$ with $\deg \mathcal H\geq 2$ and let $k$ be any positive integer. Then the sheaf $$f_*\mathcal \omega^{\otimes k}_{X/C}\otimes \omega_C\otimes \mathcal H$$ is generated by global sections.

Here, we give a detailed proof of Theorem \[z-thm1.4\] in order to explain our idea.

If $f_*\omega^{\otimes k}_{X/C}=0$, then there are nothing to prove. So we assume that $f_*\omega^{\otimes k}_{X/C}
\ne 0$. We take any closed point $P$.

\[z-claim\] $H^1(C, f_*\omega^{\otimes k}_{X/C}\otimes \omega_C
\otimes \mathcal H\otimes \mathcal O_C(-P))=0$.

By Viehweg’s weak positivity theorem, $f_*\omega^{\otimes k}_{X/C}$ is a nef locally free sheaf since $C$ is a smooth projective curve. Therefore, $\mathcal E:=f_*\omega^{\otimes k}_{X/C}\otimes 
\mathcal H\otimes \mathcal O_C(-P)$ is ample. If $H^1(C, \mathcal E\otimes \omega_C)\ne 0$, then we get $H^0(C, \mathcal E^*)\ne 0$ by Serre duality. This implies that there is a nontrivial inclusion $0\to \mathcal O_C\to\mathcal E^*$. By taking the dual of this inclusion, we have the following surjection $\mathcal E\to \mathcal O_C\to 0$. This is a contradiction since $\mathcal E$ is ample. Hence we have $H^1(C, \mathcal E\otimes \omega_C)=0$.

By Claim, the natural restriction map $$H^0(C, f_*\omega^{\otimes k}_{X/C}\otimes 
\omega_C\otimes \mathcal H)\to 
f_*\omega^{\otimes k}_{X/C}\otimes 
\omega_C\otimes \mathcal H\otimes \mathbb C(P)$$ is surjective. This means that $f_*\omega^{\otimes k}_{X/C}\otimes 
\omega_C\otimes \mathcal H$ is generated by global 