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

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.