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Jumping numbers of a simple complete ideal in a two-dimensional regular local ring


About this Title

Tarmo Järvilehto, Pääskykuja 5, 04620 Mäntsälä, Finland

Publication: Memoirs of the American Mathematical Society
Publication Year 2011: Volume 214, Number 1009
ISBNs: 978-0-8218-4811-1 (print); 978-1-4704-0626-4 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2011-00597-6
Published electronically: April 4, 2011
MathSciNet review: 2856648
Keywords: Multiplier ideals, log-canonical threshold, plane curve singularity
MSC (2010): Primary 13H05; Secondary 14B05

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Preliminaries on complete ideals
  • Chapter 3. Arithmetic of the point basis
  • Chapter 4. The dual graph
  • Chapter 5. Multiplier ideals and jumping numbers
  • Chapter 6. Main theorem
  • Chapter 7. Proof of main theorem
  • Chapter 8. Jumping numbers of a simple ideal
  • Chapter 9. Jumping numbers of an analytically irreducible plane curve

Abstract


The multiplier ideals of an ideal in a regular local ring form a family of ideals parameterized by non-negative rational numbers. As the rational number increases the corresponding multiplier ideal remains unchanged until at some point it gets strictly smaller. A rational number where this kind of diminishing occurs is called a jumping number of the ideal.

In this manuscript we shall give an explicit formula for the jumping numbers of a simple complete ideal in a two-dimensional regular local ring. In particular, we obtain a formula for the jumping numbers of an analytically irreducible plane curve. We then show that the jumping numbers determine the equisingularity class of the curve.

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