Saturation bounds for smooth varieties

TL;DR

This paper establishes bounds on the saturation degrees of ideals defining smooth complex projective varieties, extending classical results and providing specific bounds for curves based on regularity.

Contribution

It generalizes Macaulay's classical bounds from zero-dimensional complete intersections to higher-dimensional smooth varieties and relates saturation degrees to regularity for curves.

Findings

Bounds on saturation degrees for smooth varieties

Extension of Macaulay's classical results

Bounds for powers of ideals in terms of regularity

Abstract

We prove bounds on the saturation degrees of homogeneous ideals (and their powers) defining smooth complex projective varieties. For example, we show that a classical statement due to Macualay for zero-dimensional complete intersection ideals holds for any smooth variety. For curves, we bound the saturation degree of powers in terms of the regularity.