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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The multiplier ideals of a sum of ideals


Author: Mircea Mustata
Journal: Trans. Amer. Math. Soc. 354 (2002), 205-217
MSC (2000): Primary 14B05; Secondary 14F17
Published electronically: August 29, 2001
MathSciNet review: 1859032
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Abstract: We prove that if $\underline{\mathbf{a}}$, $\underline{\mathbf{b}}\subseteq\mathcal{O}_X$ are nonzero sheaves of ideals on a complex smooth variety $X$, then for every $\gamma\in{\mathbb Q}_+$we have the following relation between the multiplier ideals of $\underline{\mathbf{a}}$, $\underline{\mathbf{b}}$ and $\underline{\mathbf{a}}+\underline{\mathbf{b}}$:

\begin{displaymath}\mathcal{I}\left(X,\gamma\cdot(\underline{\mathbf{a}}+ \under... ...thbf{a}})\cdot\mathcal{I}(X,\beta\cdot \underline{\mathbf{b}}).\end{displaymath}

A similar formula holds for the asymptotic multiplier ideals of the sum of two graded systems of ideals.

We use this result to approximate at a given point arbitrary multiplier ideals by multiplier ideals associated to zero dimensional ideals. This is applied to compare the multiplier ideals associated to a scheme in different embeddings.


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Additional Information

Mircea Mustata
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720 and Institute of Mathematics of The Romanian Academy, Bucharest, Romania
Email: mustata@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02867-7
PII: S 0002-9947(01)02867-7
Keywords: Multiplier ideals, log resolutions, monomial ideals
Received by editor(s): March 1, 2001
Published electronically: August 29, 2001
Article copyright: © Copyright 2001 American Mathematical Society