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# Symbolic powers of ideals in k[P N]

#### 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*, and which ratios

^{r}*m/r*guarantee such containment. A result of Ein-Lazarsfeld-Smith and Hochster-Huneke states that, if

*I*⊆

*k*[

**P**

*], where*

^{N}*k*is an algebraically closed field, then the symbolic power

*I*

^{(}

^{Ne}^{)}is contained in the ordinary power

*I*, and thus, whenever

^{e}*m/r*≥

*N*, we have the containment

*I*

^{(}

^{m}^{)}⊆

*I*Therefore, for each ideal

^{r}.*J*, there is a number

*a*≤

*N*such that

*m/r*>

*a*implies

*J*

^{(}

^{ m}^{)}⊆

*J*This led Bocci and Harbourne [BH10a] to define the resurgence of

^{r}.*I*rI=sup m/r:I

^{ m}⊈I

^{r}. In particular, if

*m/r*> ρ(

*I*), then

*I*

^{(}

^{m}^{ )}⊆

*I*An interesting problem, then, is to compute ρ(

^{r}.*I*) for various classes of ideals. Much of the work that has been done on this question involves examining ideals of points in

**P**

*In Chapter 2 we investigate such questions for an ideal defining a certain configuration of points in*

^{N}.**P**

^{2}using a certain

*k*-vector space basis of

*k*[

**P**

^{2}compatible with

*I*

^{(}

^{m}^{ )}and

*I*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(

^{r}.*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}.^

#### Subject Area

Mathematics

#### Recommended Citation

Janssen, Michael K, "Symbolic powers of ideals in k[P N]" (2013). *ETD collection for University of Nebraska - Lincoln*. AAI3558614.

https://digitalcommons.unl.edu/dissertations/AAI3558614