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

Grzegorz Malara; Justyna Szpond

doi:10.1016/j.jpaa.2017.09.015

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arXiv · 1702.02160

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

Grzegorz Malara·Justyna Szpond

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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.

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Grzegorz Malara, Justyna Szpond. 2017-02-07. Fermat-type configurations of lines in $\mathbb P^3$ and the containment problem. https://doi.org/10.1016/j.jpaa.2017.09.015

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