A remark on the Castelnuovo-Mumford regularity of powers of ideal sheaves

TL;DR

This paper establishes that the Castelnuovo-Mumford regularity bound for powers of ideal sheaves is sharp only for complete intersections, extending previous results to a broader class of varieties.

Contribution

It generalizes the known sharpness condition of regularity bounds to varieties cut out scheme-theoretically by hypersurfaces, beyond complete intersections.

Findings

Regularity bound is sharp only for complete intersections.

Extension of Bertram-Ein-Lazarsfeld's result to more general varieties.

Provides a criterion for the sharpness of regularity bounds.

Abstract

We show that a bound of the Castelnuovo-Mumford regularity of any power of the ideal sheaf of a smooth projective complex variety $X\subseteq\mathbb{P}^r$ is sharp exactly for complete intersections, provided the variety $X$ is cut out scheme-theoretically by several hypersurfaces in $\mathbb{P}^r$. This generalizes a result of Bertram-Ein-Lazarsfeld.