On the containment problem for fat points ideals

TL;DR

This paper investigates the containment problem for fat points ideals, confirming Harbourne's conjecture in specific cases and exploring its validity for ideals related to line arrangements in projective spaces.

Contribution

It proves Harbourne's conjecture for symbolic powers of point ideals and extends the validation to stable versions for very general and generic points, also linking these results to line arrangements.

Findings

Harbourne's conjecture holds for symbolic powers of point ideals.

Stable version of the conjecture is valid for very general and generic points.

Conjectures are confirmed for ideals from line arrangements in projective planes.

Abstract

In this note we show that Harbourne's conjecture is true for symbolic powers of ideals of points, we check that the stable version of this conjecture is valid for ideals of very general points (resp. generic points) in $\mathbb P_{\mathbb K}^N$ (resp. $\mathbb P_{\mathbb K(\underline{z})}^N$). We also show that this conjecture and the Harbourne-Huneke conjecture are true for a class of ideals $I$ defining fat points obtained from line arrangements in $\mathbb P_{\mathbb K}^2$.