The resurgence of ideals of points and the containment problem

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 ∈ 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 ≤ 9.

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