On the containment $I^{(3)} \subseteq I^2$ and configurations of triple points in B\"{o}r\"{o}czky line arrangements

TL;DR

This paper investigates the containment problem for symbolic and ordinary powers of ideals in the specific context of B"{o}r"{o}czky line arrangements, identifying the minimal counterexample configuration with 12 lines.

Contribution

It demonstrates that the smallest counterexample to the containment $I^{(3)} subseteq I^2$ in B"{o}r"{o}czky arrangements occurs at 12 lines, providing new insights into the structure of these arrangements.

Findings

Counterexample at 12 lines for the containment problem.

B"{o}r"{o}czky arrangements' triple points are key to understanding containment.

Minimal counterexample identified at 12 lines.

Abstract

We study sets of triple points of B\"{o}r\"{o}czky's arrangements of lines in the context of the containment problem proposed by Harbourne and Huneke. We show that in the class of those arrangements, the smallest counterexample to the containment $I^{(3)} \subseteq I^2$ is obtained when the number of lines is equal to 12.