O_Y(K_Y+lH^\dag)
\quad \text{and}\quad 
\left(\bigotimes ^s f_*\mathcal O_X(L)\right)^{**} 
\otimes \mathcal O_Y(K_Y+lH^\dag)$$ are generically generated by global sections for all integers $s\geq 1$ and $l \geq n^2+\min\{2, k\}$.

Let us quickly explain the idea of the proof of Theorem \[z-thm1.5\], which is mainly due to Nakayama (see [@nakayama]). Let $f:X\to Y$ be a surjective morphism between smooth projective varieties and let $H$ be an ample Cartier divisor on $Y$ such that $|H|$ is free. We fix a positive integer $k\geq 2$. Then we can construct a surjective morphism $g:Z\to Y$ from a smooth projective variety $Z$ and a direct summand $\mathcal F$ of $g_*\mathcal O_Z(K_Z)$ such that there exists a generically isomorphic injection $$\mathcal F\hookrightarrow f_*\omega^{\otimes k}_{X/Y}\otimes 
\omega_Y\otimes \mathcal O_Y(H).$$ By Kollár’s vanishing theorem, we have $$H^i(Y, \mathcal F\otimes \mathcal O_Y((n+1-i)H))=0$$ for every $i>0$, where $n=\dim Y$. Therefore, by Castelnuovo–Mumford regularity, $\mathcal F\otimes \mathcal O_Y((n+1)H)$ is generated by global sections. This implies that $$f_*\omega^{\otimes k}_{X/Y}\otimes 
\omega_Y\otimes \mathcal O_Y((n+2)H)$$ is generically generated by global sections. Note that we do not try to establish any vanishing theorem for $f_*\omega^{\otimes k}_{X/Y}\otimes \omega_Y\otimes 
\mathcal O_Y(H)$ directly. Anyway, it is natural to consider:

\[y-def1.9\] A torsion-free coherent sheaf $\mathcal F$ on a normal quasi-projective variety $W$ is 