A Vanishing Theorem and Asymptotic Regularity of Powers of Ideal Sheaves

TL;DR

This paper establishes bounds on the asymptotic regularity of powers of ideal sheaves on projective space and proves a vanishing theorem for such powers under certain singularity conditions, advancing understanding of their algebraic and geometric properties.

Contribution

It provides new bounds on the regularity of powers and symbolic powers of ideal sheaves and introduces a vanishing theorem using multiplier ideals for sheaves defining log canonical singularities.

Findings

Bounds on asymptotic regularity of ideal sheaf powers

Upper bounds for symbolic powers under specific conditions

Vanishing theorem for powers of ideal sheaves with log canonical singularities

Abstract

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.