The resurgence of ideals of points and the containment problem

Abstract

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 , . . . , 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.

The effect of points fattening on postulation

“…In other words, I is the intersection m 1 ∩ · · · ∩ m n where the m i 's are distinct maximal homogeneous ideals. Notice that I (2) is then equal to m 2 1 ∩ · · · ∩ m 2 n . Definition 2.1.…”

“…In more detail, let Z be a configuration of points in P 2 and let I = I(Z ) be the homogeneous ideal. Put I (2) = I(2Z ). We say that Z has type (d − t, d) if the generators of I and I (2) have minimal degrees d − t and d respectively.…”

On the containment hierarchy for simplicial ideals

“…[2], [3], [10]. The first counterexample to the I (3) ⊂ I 2 containment for an ideal of points in P 2 announced in [6] has prompted another series of papers [1], [11], [13].…”

“…Bocci and Harbourne introduced in [3] an interesting invariant, the resurgence ρ(I) measuring in effect the asymptotic discrepancy between symbolic and ordinary powers of a given ideal (see Definition 2.3). This is a delicate invariant and the family of ideals for which it is known is growing slowly, see e.g.…”

“…It is a pleasure to thank Institute of Mathematics in Freiburg for providing nice working conditions and the graduate school Graduiertenkolleg 1821 "Cohomological Methods in Geometry" for some additional financial support. We would like to thank Marcin Dumnicki and Tomasz Szemberg for inspiring discussions, helpful remarks and bringing reference [3] to our attention. We are greatly obliged to the anonymous referee for an extensive list of very helpful remarks which greatly improved the present note.…”