Algorithms for computing multiplier ideals
type: Item
instance of: scholarly article
title: Algorithms for computing multiplier ideals
zbMATH Open document ID: 1233.14015
author: Takafumi Shibuta
published in: Journal of Pure and Applied Algebra
publication date: 2011-12-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/0807.4302
review text: Let \(X\) be a smooth complex variety and \(\mathfrak{a}\subseteq \mathcal{O}_X\) an ideal sheaf. Let \(\pi:\widetilde{X}\to X\) be a log resolution of \(\mathfrak{a}\). Then the multiplier ideal of \(\mathfrak{a}\) with exponent \(c\in \mathbb{R}_{\geq 0}\) is defined as \(J(\mathfrak{a}^c)=J(c\cdot \mathfrak{a})=\pi_\ast \mathcal{O}_{\widetilde{X}}(K_{\widetilde{X}|X}-\lfloor cF\rfloor)\subseteq \mathcal{O}_X\). Here \(K_{\widetilde{X}|X}\) is the relative canonical divisor of \(\pi\) and \(\mathfrak{a}\mathcal{O}_{\widetilde{X}}=\mathcal{O}_{\widetilde{X}}(-F)\) for an effective divisor \(F\) on \(\widetilde{X}\) such that Supp \(F\cup\text{Exc}(\pi)\) is a divisor with simple normal crossing. \(\text{Exc} (\pi)\) is the exceptional locus of \(\pi\). Algorithms are given for computing multiplier ideals using the theory of \(D\)-modules, i.e. Gröbner bases in Weyl algebras are used. Several examples are given.
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 14F18
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 14Q15
zbMATH DE Number: 5986577
zbMATH Keywords: multiplier ideal
zbMATH Keywords: log resolution
zbMATH Keywords: \(\mathcal D\)-modules
zbMATH Keywords: Weyl algebra
zbMATH Keywords: Gröbner basis
describes a project that uses: Risa/Asir
describes a project that uses: Kan
MaRDI profile type: Publication
OpenAlex ID: W2170292350
arXiv ID: 0807.4302
cites work: Q3784384
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cites work: Q3760849
cites work: MULTIPLICITY ONE PROPERTY AND THE DECOMPOSITION OF b-FUNCTIONS
DOI: 10.1016/J.JPAA.2011.04.002
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description: scientific article