The core of an ideal and its relationship to the adjoint and coefficient ideals

Doctoral Dissertation
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Abstract

Given an ideal $I$ in a Noetherian ring $R$, the core of $I$ is the intersection of all ideals contained in $I$ with the same integral closure as $I$. The core naturally arises in the context of the Brianc{c}on-Skoda theorem as an ideal which contains the adjoint of a certain power of $I$. As the arbitrary-characteristic analog of the multiplier ideal, the adjoint is an important tool in the study of resolutions of singularities. The question of when the core and the adjoint of a power of $I$ are equal has been tied to a celebrated conjecture of Kawamata about the non-vanishing of sections of line bundles. We show for certain classes of monomial ideals in the polynomial ring $k[x1,ldots,xd]$ over a field of characteristic zero , $core(I)=adj(I^d)$ if and only if $core(I)$ is integrally closed. In order to prove our main result, we further develop the theory of coefficient ideals in regular local rings of dimension two and study the combinatorial properties of the core of a monomial ideal via the symmetry of its exponent set.

Attributes

Attribute NameValues
URN
  • etd-04152010-111950

Author Angela Kohlhaas
Advisor J. Chris Howk
Contributor Claudia Polini, Committee Member
Contributor Nero Budur, Committee Member
Contributor J. Chris Howk, Committee Chair
Contributor Juan Migliore, Committee Member
Contributor Yu Xie, Committee Member
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name PhD
Defense Date
  • 2010-03-31

Submission Date 2010-04-15
Country
  • United States of America

Subject
  • exponent set

  • commutative algebra

  • birational geometry

Publisher
  • University of Notre Dame

Language
  • English

Record Visibility Public
Content License
  • All rights reserved

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