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Results on containments and resurgences, with a focus on ideals of points in the plane
Let K be a field of arbitrary characteristic and let I be a nontrivial homogeneous ideal in R = K[ PN ]. Then we can take two different kinds of powers of I - ordinary powers of the form Ir and symbolic powers of the form I(m ) = R ∩ ⋂ P∈AssIIm RP . A question that has been of particular interest to the mathematical community over the last two decades is that of the relationship between Ir and I( m). How do these two notions compare?^ It can be shown that Ir ⊆ I(m) if and only if r ≥ m, so the question that remains is what we can say about the containment of I (m) in I r.^ Results by Ein/Lazarsfeld/Smith and Hochster/Huneke showed that I(m) ⊆ Ir whenever m ≥ Nr. Moreover, it can be shown that I( m) ⊈ Ir whenever r > m, and, in addition, improvements on the bound m ≥ Nr have been made in many specific situations. However, the question of exactly when I(m ) ⊆ Ir holds for r ≤ m < Nr remains open in general.^ An asymptotic variant of this Containment Question is the Resurgence Problem posed by Bocci and Harbourne: Define the resurgence of a homogeneous ideal I by ρ(I) = sup m,r &cubl0;mr&vbm0;I m⊈Ir&cubr0; . What can we say about its value in general or for certain classes of ideals?^ Most work in this direction has been done in the geometric setting of ideals of points in PN .^ In this thesis we will address both the Containment Question and the Resurgence Problem for a family of ideals of points in P2 . We start in chapter 2 with a point configuration for which we can give a complete answer to both questions. The key to comparing the symbolic and ordinary powers of I in these cases is to find a vector space basis for the homogeneous coordinate ring of the plane such that subsets of this basis give bases for all powers and symbolic powers of I. In chapter 3, we describe a way to obtain a lower bound for ρ( I) for point configurations for which the vector space basis approach does not apply. We connect the configuration of points under consideration in this chapter to the one we examine in the preceding chapter to demonstrate how to obtain partial results through the vector space approach. In chapter 4, we develop computational methods for estimating resurgences arbitrarily accurately for nontrivial homogeneous ideals in K[ PN ] whenever there is an m ∈ N such that powers of I( m) are symbolic, i.e. I (mt) = (I (m)) t for all t ∈ N . Our main result here is Theorem 4.1.5, which gives for such ideals I a computational method for determining ρ(I) to any desired accuracy. We also demonstrate the application in several examples. ^
Denkert, Annika, "Results on containments and resurgences, with a focus on ideals of points in the plane" (2013). ETD collection for University of Nebraska - Lincoln. AAI3590312.