  Let $\mathscr{I}$ be an ideal sheaf on $P^n$ defining a subscheme $X$.
Associated to $\mathscr{I}$ there are two elementary invariants: the invariant
$s$ which measures the positivity of $\mathscr{I}$, and the minimal number $d$
such that $\mathscr{I}(d)$ is generated by its global sections. It is now clear
that the asymptotic behavior of $\reg \mathscr{I}^t$ is governed by $s$ but
usually not linear. In this paper, we first describe the linear behavior of the
asymptotic regularity by showing that if $s=d$, i.e., $s$ reaches its maximal
value, then for $t$ large enough $\reg \mathscr{I}^t=dt+e$ for some positive
constant $e$. We then turn to concrete geometric settings to study the
asymptotic regularity of $\mathscr{I}$ in the case that $X$ is a nonsingular
variety embedded by a very ample adjoint line bundle. Our approach also gives
regularity bounds for $\mathscr{I}^t$ once we know $\reg \mathscr{I}$ and
assume that $X$ is a local complete intersection.
