Mathematics, Department of

 

Date of this Version

5-2013

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Brian Harbourne. Lincoln, Nebraska: May, 2013

Copyright (c) 2013 Michael K. Janssen

Abstract

Let Ik[PN] 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 Ik[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/rN 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 to define the resurgence of I

ρ(I) = sup{m/r | I(m)Ir}.

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 P2 based on the difference t = α(I(2)) − α(I), where I = I(Z) and ZP2 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 P3.

Adviser: Brian Harbourne