## Mathematics, Department of

#### Date of this Version

5-2013

#### Abstract

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)} ⊆ *I*^{r}, and which ratios *m/r* guarantee such containment. A result of Ein-Lazarsfeld-Smith and Hochster-Huneke states that, if *I* ⊆ *k*[**P**^{N}], where *k* is an algebraically closed field, then the symbolic power *I*^{(Ne)} is contained in the ordinary power *I*^{e}, and thus, whenever *m/r* ≥ *N* we have the containment *I*^{(m)} ⊆ *I*^{r}. Therefore, for each ideal *J*, there is a number *a ≤ N* such that *m/r > a* implies *J*^{(m)} ⊆ *J*^{r}. This led Bocci and Harbourne to define the resurgence of *I*

ρ(*I*) = sup{*m/r* | *I*^{(m)} ⊈ *I*^{r}}.

In particular, if *m/r* > ρ(*I*), then *I*^{(m)} ⊆ *I*^{r}. 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 **P**^{N}. In Chapter 2 we investigate such questions for an ideal defining a certain configuration of points in **P**^{2} using a certain *k*-vector space basis of *k*[**P**^{2}] compatible with *I*^{(m)} and *I*^{r}. 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* ⊆ **P**^{2} 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}.

Adviser: Brian Harbourne

## 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