Some Results on Asymptotic Regularity of Ideal Sheaves

Let $\mathcal{I}$ be an ideal sheaf on $P^n$ defining a subscheme $X$. Associated to $\mathcal{I}$ there are two elementary invariants: the invariant $s$ which measures the positivity of $\mathcal{I}$, and the minimal number $d$ such that $\mathcal{I}(d)$ is generated by its global sections. It is now clear that the asymptotic behavior of $\text{reg}{\mathcal{I}}^t$ 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 $\text{reg}{\mathcal{I}}^t=dt+e$ for some positive constant $e$. We then turn to concrete geometric settings to study the asymptotic regularity of $\mathcal{I}$ in the case that $X$ is a nonsingular variety embedded by a very ample adjoint line bundle. Our approach also gives regularity bounds for ${\mathcal{I}}^t$ once we know $\text{reg}\mathcal{I}$ and assume that $X$ is a local complete intersection.