On the containment problem for fat points

TL;DR

This paper investigates the containment problem for specific classes of fat point ideals, focusing on points on a line and three nonlinear points in projective space, to understand their symbolic and ordinary power relations.

Contribution

It provides new insights into the containment problem for fat point ideals, specifically for points on a line and three nonlinear points in projective space.

Findings

Characterization of containment relations for points on a line.

Results on the containment problem for three nonlinear points.

Advancement in understanding symbolic and ordinary power containments.

Abstract

Given an ideal $I$, the containment problem is concerned about finding the values $m$ and $n$ such that the $m$-th symbolic power of $I$ is contained in its $n$-th ordinary power. In this paper we consider this problem focusing on two classes of ideals of fat points. In particular we study the ideal of $n$ points on a line in $\mathbb{P}^N$ and the ideal of three nonlinear points in $\mathbb{P}^N$ for $N\geq2$.