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MULTIPLIER IDEALS OF HYPERPLANE ARRANGEMENTS

TRANSACTIONS OF THE

AMERICAN MATHEMATICAL SOCIETY

Volume 358,Number 11,November 2006,Pages 5015–5023

S 0002-9947(06)03895-5

Article electronically published on June 13,2006

MULTIPLIER IDEALS OF HYPERPLANE ARRANGEMENTS

MIRCEA MUSTAT ¸ˇA

Abstract.In this note we compute multiplier ideals of hyperplane arrange-

ments.This is done using the interpretation of multiplier ideals in terms of

spaces of arcs by Ein,Lazarsfeld,and Mustat ¸ˇa (2004).

Introduction

If X is a complex smooth variety,and if a ⊆O X is a coherent sheaf of ideals on X ,then for every λ∈Q ∗+one can define the multiplier ideal of exponent λdenoted I (X,a λ).The interest in these ideals comes from the fact that on one hand,they encode properties of the singularities of the subscheme defined by a ,and on the other hand,they appear naturally in vanishing theorems.This led to various applications in recent years,in higher-dimensional birational geometry as well as in local algebra (see [Laz]for a thorough presentation).

The definition of multiplier ideals is given in terms of a log resolution of singu-larities for the pair (X,a ).This makes the computation of specific examples quite delicate,so there are not many classes of ideals for which the corresponding mul-tiplier ideals are known.One notable exception is the case of multiplier ideals for monomial ideals in the affine space,which have been computed by Howald using toric methods in [How2].Most of the other examples where a general formula is known proceed by reduction to the monomial case under suitable genericity as-sumptions (see [How1]).

In this note we give a formula for the multiplier ideals of hyperplane arrange-ments.This is achieved using the description of multiplier ideals in terms of spaces of arcs from [ELM].

Let A be a hyperplane arrangement in a vector space V C n with n ≥1,i.e.A is the (reduced)union of affine hyperplanes H 1,...,H d ⊂V .If h i is an equation defining H i ,then A is defined by f = i h i ,and our goal is to compute I (V,f λ).

We will assume that A is central,i.e.that all hyperplanes H i pass through the origin.This is not an important restriction:if A is an arbitrary arrangement,and if we want to compute multiplier ideals in a neighbourhood of a point x ∈V ,then it is enough to do the computation for the subarrangement consisting of those hyperplanes passing through x .Therefore after a suitable translation,we are in the case of a central arrangement.

Received by the editors February 20,2004a
nd,in revised form,November 5,2004.

Received by the editors February 20,2004and,in revised form,November 5,2004.

2000Mathematics Subject Classification.Primary 14B05;Secondary 52C35.

Key words and phrases.Arcs,multiplier ideals,hyperplane arrangements.

The author served as a Clay Mathematics Institute Research Fellow while this research was conducted.

c

2006American Mathematical Society Reverts to public domain 28years from publication

5015

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