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Symbolic powers of ideals in k[PN]
Let I ⊆ k[P N] be a homogeneous ideal and k an algebraically closed field. Of particular interest over the last several years are ideal containments of symbolic powers of I in ordinary powers of I of the form I(m ) ⊆ Ir, and which ratios m/r guarantee such containment. A result of Ein-Lazarsfeld-Smith and Hochster-Huneke states that, if I ⊆ k[PN], where k is an algebraically closed field, then the symbolic power I (Ne) is contained in the ordinary power Ie, and thus, whenever m/r ≥ N, we have the containment I (m) ⊆ Ir. Therefore, for each ideal J, there is a number a ≤ N such that m/r > a implies J( m) ⊆ Jr. This led Bocci and Harbourne [BH10a] to define the resurgence of I [special characters omitted] In particular, if m/r > ρ(I), then I(m ) ⊆ Ir. An interesting problem, then, is to compute ρ(I) for various classes of ideals. Much of the work that has been done on this question involves examining ideals of points in PN. In Chapter 2 we investigate such questions for an ideal defining a certain configuration of points in P2 using a certain k-vector space basis of k[P2 compatible with I(m ) and Ir. We are also able to use this approach to verify several conjectures of Harbourne-Huneke and Bocci-Cooper-Harbourne for our particular class of ideals, and compute some well-known invariants of these ideals, such as α(I( m)), γ(I), the Castelnuovo-Mumford regularity and the saturation degree. In Chapter 3, we consider a question raised in Bocci and Chiantini's paper which is related to the computation of γ(I). Bocci and Chiantini classify configurations of points in P 2 based on the difference t = α(I (2)) − α(I), where I = I(Z) and Z ⊆ P2 is a finite set of points. When t = 1, Z is either a set of collinear points or a star configuration of points. We extend that result to configurations of lines in P 3.
Janssen, Michael K, "Symbolic powers of ideals in k[PN]" (2013). ETD collection for University of Nebraska - Lincoln. AAI3558614.