Containment problem for points on a reducible conic in $\mathbb{P}^2$

Abstract

Given an ideal $I$ in a Noetherian ring, one can ask the containment question: for which $m$ and $r$ is the symbolic power $I^{(m)}$ contained in the ordinary power $I^r$? C. Bocci and B. Harbourne study the containment question in a geometric setting, where the ideal $I$ is in a polynomial ring over a field. Like them, we will consider special geometric constructs. In particular, we obtain a complete solution in two extreme cases of ideals of points on a pair of lines in $\mathbb{P}^2$; in one case, the number of points on each line is the same, while in the other all the points but one are on one of the lines.