sections.

The following theorem supports Conjecture \[z-conj1.3\].

\[z-thm1.5\] Let $f:X\to Y$ be a surjective morphism between smooth projective varieties and let $H$ be an ample divisor on $Y$ such that $|H|$ is free. We put $\dim Y=n$. Then $$f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal O_Y(lH)$$ is generically generated by global sections for all integers $k\geq 1$ and $l\geq n+1$. Moreover, $$\left(\bigotimes ^s (f_*\omega^{\otimes k}_{X/Y})\right)^{**} 
\otimes \omega_Y\otimes \mathcal O_Y(lH)$$ is generically generated by global sections for all integers $k\geq 1$, $s\geq 1$, and $l\geq n+1$.

Let $H^\dag$ be an ample divisor on $Y$ such that $|H^\dag|$ is not necessarily free. Then $$f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes \mathcal O_Y(lH^\dag)$$ is generically generated by global sections for all integers $k\geq 1$ and $l\geq n^2+\min\{2, k\}$. Moreover, the sheaf $$\left(\bigotimes ^s (f_*\omega^{\otimes k}_{X/Y})\right)^{**} 
\otimes \omega_Y\otimes \mathcal O_Y(lH^\dag)$$ is generically generated by global sections for all integers $k\geq 1$, $s\geq 1$, and $l\geq n^2+\min\{2, k\}$.

The author learned the following remark from Masataka Iwai.

\[y-rem1.6\] Let $f:X\to Y$ and $H^\dag$ be as in Theorem \[z-thm1.5\]. Let $U$ be the largest Zariski open set of $Y$ such that $f$ is smooth over $U$. Then, by the argument in [@iwai], we can check that the natural map $$H^0(Y, f_*\omega^{\otimes k}_{X/Y}\otimes 
\omega_Y\otimes 