Results on Containments and Resurgences, with a Focus on Ideals of Points in the Plane

Authors

Annika Denkert, University of Nebraska-LincolnFollow

Date of this Version

8-2013

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: August, 2013

Copyright (c) 2013 Annika Denkert

Abstract

Let K be an algebraically closed field and I ⊆ R=K[P^N] a nontrivial homogeneous ideal. We can describe ordinary powers I^r and symbolic powers I^(m) of I. One question that has been of interest over the past couple of years is that of when we have containment of I^(m) in I^r. Bocci and Harbourne defined the resurgence of I as rho(I)=sup_m,r{m/r | I^(m) is not contained in I^r}. Hence in particular I^(m) ⊆ I^r whenever m/r is at least rho(I). Results by Macaulay, Ein-Lazarsfeld-Smith, and Hochster-Huneke yield that rho(I) is always between 1 and N for all nontrivial homogeneous ideals I⊆K[P^N]. Like many other mathematicians working on this question, we start by examining ideals of points in P². We first consider a specific point configuration and analyze it using an approach mimicking monomial vector space bases. In particular, we find a vector space basis for K[P²] which is compatible with both symbolic powers and ordinary powers of I, and use it to look at several conjectures made by Harbourne and Huneke in the context of the chosen configuration of points. Finally, we compute some well-known values for our ideal, such as the Waldschmidt constant and the Castlenuovo-Mumford regularity. We then use this vector space approach to give a lower bound for the resurgence of another, closely related, point configuration. In the final chapter, we exhibit another method for computing an upper bound for the resurgence of any nontrivial homogeneous ideal I in K[P^N].

Adviser: Brian Harbourne