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Some Results on Asymptotic Regularity of Ideal Sheaves

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.