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The multiplier ideals of a sum of ideals

Authors:	Mircea Mustata

Institution:	Department of Mathematics, University of California, Berkeley, California 94720 and Institute of Mathematics of The Romanian Academy, Bucharest, Romania

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 , 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.

Keywords:	Multiplier ideals log resolutions monomial ideals

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