  In this paper, we investigate the problem of simultaneously generations of
$r$-jets of $\omega_X\otimes L^{\otimes m}$ when $L$ is ample and base point
free. It turns out that in this case, the bound of $m$ is optimistic, i.e.
$m\geq f(r)=n+r+1$, which is linear in $r$. Our results work over arbitrary
characteristics. We also treat the same problem when $X$ is singular where
$\omega_X$ is replaced by the pushforward of canonical sheaves or the
cohomology sheaves of dualizing complexes.
