New phenomena in the containment problem for simplicial arrangements

Abstract

We consider two simplicial arrangements of lines and ideals $I$ of intersection points of these lines. There are $127$ intersection points in both cases and the numbers $t_i$ of points lying on exactly $i$ configuration lines (points of multiplicity $i$) coincide. We show that in one of these examples the containment $I^{(3)}\subseteq I^2$ holds, whereas it fails in the other. We also show that the containment fails for a subarrangement of $21$ lines. The interest in the containment relation between $I^{(3)}$ and $I^2$ for ideals of points in ${\mathbb{P}}^2$ is motivated by a question posted by Hochster and Huneke in $2002$. Configurations of points with $I^{(3)}⊈I^2$ are quite rare. Our example reveals two particular features: All points are defined over $\mathbb{Q}$ and all intersection points of lines are involved. In examples studied by now only points with multiplicity $i\ge 3$ were considered. The novelty of our arrangements lies in the geometry of the element in $I^{(3)}$ which witness the noncontainment in $I^2$. In all previous examples such an element was a product of linear forms. Now, in both cases there is an irreducible curve of higher degree involved.