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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,...,p_n in 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's lie on a smooth conic or when the points are general and n ≤ 9

The resurgence of ideals of points and the containment problem