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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Length, multiplicity, and multiplier ideals

Author: Tommaso de Fernex
Journal: Trans. Amer. Math. Soc. 358 (2006), 3717-3731
MSC (2000): Primary 14B05; Secondary 13H05, 14B07, 13H15
Published electronically: March 24, 2006
MathSciNet review: 2218996
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Abstract: Let $ (R,\mathfrak{m})$ be an $ n$-dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an $ \mathfrak{m}$-primary ideal $ \mathfrak{a}$ of $ R$, the relationship between the singularities of the scheme defined by $ \mathfrak{a}$ and those defined by the multiplier ideals $ \mathcal{J}(\mathfrak{a}^c)$, with $ c$ varying in $ \mathbb{Q}_+$, are quantified in this paper by showing that the Samuel multiplicity of $ \mathfrak{a}$ satisfies $ e(\mathfrak{a}) \ge (n+k)^n/c^n$ whenever $ \mathcal{J}(\mathfrak{a}^c) \subseteq \mathfrak{m}^{k+1}$. This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustata and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

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Additional Information

Tommaso de Fernex
Affiliation: Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, Michigan 48109-1109
Address at time of publication: School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540

Keywords: Multiplier ideal, Samuel multiplicity, monomial ideal
Received by editor(s): June 21, 2004
Received by editor(s) in revised form: September 20, 2004
Published electronically: March 24, 2006
Additional Notes: The author’s research was partially supported by the University of Michigan Rackham Research Grant and Summer Fellowship, and by the MIUR of the Italian Government in the framework of the National Research Project “Geometry on Algebraic Varieties" (Cofin 2002)
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.