The resurgence of ideals of points and the containment problem

Abstract:

We relate properties of linear systems on $X$ to the question of when $I^r$ contains $I^{(m)}$ in the case that $I$ is the homogeneous ideal of a finite set of distinct points $p_1,\ldots ,p_n\in \mathbf {P}^2$, where $X$ is the surface obtained by blowing up the points. We obtain complete answers for when $I^r$ contains $I^{(m)}$ when the points $p_i$ lie on a smooth conic or when the points are general and $n\le 9$.