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

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.