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 $I^{(3)}\subset I<sup>2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb P}<sup>3$, which resemble Fermat configurations of points in ${\mathbb P}<sup>2$, see \cite{NagSec16}. All examples exhibiting the failure of the containment $I^{{(3)}\subseteq} I<sup>2$ 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.