Let $\mathscr{ I}$ be an ideal sheaf on $P^n$. In the first part of this paper, we bound the asymptotic regularity of powers of $\mathscr{ I}$ as $ps-3\leq \reg \mathscr{ I}^p\leq ps+e$, where $e$ is a constant and $s$ is the $s$-invariant of $\mathscr{ I}$. We also give the same upper bound for the asymptotic regularity of symbolic powers of $\mathscr{ I}$ under some conditions. In the second part, by using multiplier ideal sheaves, we give a vanishing theorem of powers of $\mathscr{ I}$ when it defines a local complete intersection subvariety with log canonical singularities.
Categories: math.AG math.AC