The resurgence of ideals of points and the containment problem

Abstract

We relate properties of linear systems on X X to the question of when I r I^r contains I ( m ) I^{(m)} in the case that I I is the homogeneous ideal of a finite set of distinct points p 1 , … , p n ∈ P 2 p_1,\ldots ,p_n\in \mathbf {P}^2 , where X X is the surface obtained by blowing up the points. We obtain complete answers for when I r I^r contains I ( m ) I^{(m)} when the points p i p_i lie on a smooth conic or when the points are general and n ≤ 9 n\le 9 .