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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, 2004and, 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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