A vanishing theorem and symbolic powers of planar point ideals

TL;DR

This paper introduces a vanishing theorem for linear series with fat points ideals and uses it to partially prove a conjecture on the containment relations between ordinary and symbolic powers of planar point ideals.

Contribution

It presents a new vanishing theorem and applies it to advance understanding of the relationship between symbolic and ordinary powers of planar point ideals.

Findings

Established a vanishing theorem for linear series with fat points ideals.

Provided a partial proof of a conjecture on symbolic and ordinary power containments.

Enhanced understanding of algebraic properties of planar point ideals.

Abstract

The purpose of this note is twofold. We present first a vanishing theorem for families of linear series with base ideal being a fat points ideal. We apply then this result in order to give a partial proof of a conjecture raised by Bocci, Harbourne and Huneke concerning containment relations between ordinary and symbolic powers of planar point ideals.