# Fermat-type configurations of lines in $\mathbb P^3$ and the containment problem

## TL;DR

This paper presents new examples of line arrangements in projective 3-space that defy the usual containment between symbolic and ordinary powers of ideals, expanding understanding of the containment problem.

## Contribution

It introduces novel Fermat-type line configurations in P^3 that serve as counterexamples to the containment I^{(3)} ⊆ I^2, previously known only for point-supported ideals.

## Key findings

- New counterexamples to the containment problem in P^3
- Line arrangements resembling Fermat configurations
- Potential implications for the containment problem theory

## Abstract

The purpose of this note is to show a new series of examples of homogeneous ideals $I$ in ${\mathbb K}[x,y,z,w]$ for which the containment $I^{(3)}\subset I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb P}^3$, which resemble Fermat configurations of points in ${\mathbb P}^2$, see \cite{NagSec16}. All examples exhibiting the failure of the containment $I^{(3)}\subseteq I^2$ constructed so far have been supported on points or cones over configurations of points. Apart of providing new counterexamples, these ideals seem quite interesting on their own.