A Bound for the Castelnuovo-Mumford Regularity of Log Canonical Varieties

In this note, we give a bound for the Castelnuovo-Mumford regularity of a homogeneous ideal $I$ in terms of the degrees of its generators. We assume that $I$ defines a local complete intersection with log canonical singularities.Comment: 14 pages; Title changed; Typos corrected; To appear in Journal of Pure and Applied Algebr

A Vanishing Theorem and Asymptotic Regularity of Powers of Ideal Sheaves

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.Comment: 13 pages; Corrected typos, added references, improve one of the main theorem