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.

8 pages, typos corrected, accepted to appear

References in corpus (2)

• On the containment problem
• Arrangements of curves and algebraic surfaces

Cited by in corpus (5)

• Expected resurgences and symbolic powers of ideals
• Singular loci of reflection arrangements and the containment problem
• The Initial Degree of Symbolic Powers of Fermat-like Ideals of Planes and Lines Arrangements
• Containment problem and combinatorics
• On the fattening of ACM arrangements of codimension 2 subspaces in P^N