Jumping coefficients of multiplier ideals

Ein, Lawrence; Lazarsfeld, Robert; Smith, Karen E.; Varolin, Dror

doi:10.1215/s0012-7094-04-12333-4

Cited by 98 publications

(124 citation statements)

References 33 publications

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“…They extend in a natural way the information given by the log-canonical threshold, the smallest jumping number (see [3] for example). They are periodic, completely determined by the jumping numbers less than 1, but otherwise difficult to compute in general, even if a set of candidates is easy to provide, cf.…”

Section: Introductionmentioning

confidence: 84%

Jumping numbers of a unibranch curve on a smooth surface

Naie

2008

manuscripta math.

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A formula for the jumping numbers of a curve unibranch at a singular point is established. The jumping numbers are expressed in terms of the Enriques diagram of the log resolution of the singularity, or equivalently in terms of the canonical set of generators of the semigroup of the curve at the singular point.The jumping numbers of a curve on a smooth complex surface are a sequence of positive rational numbers revealing information about the singularities of the curve. They extend in a natural way the information given by the log-canonical threshold, the smallest jumping number (see [3] for example). They are periodic, completely determined by the jumping numbers less than 1, but otherwise difficult to compute in general, even if a set of candidates is easy to provide, cf. [9, Lemma 9.3.16].The aim of this paper is to give a formula for the jumping numbers of a curve unibranch at a singular point. A curve C will be said to be unibranch at a point P , if the analytic germ of C at P is irreducible. The formula is expressed in terms of the Enriques diagram associated to the singularity, or equivalently (see Theorem 3.1) in terms of a minimal set of generators (β 0 , β 1 , . . . , β g ) of the semigroup S(C, P ) of C at P :where the m j are defined below. HereThe semigroup is defined by S(C) = {ord P s | s ∈ O C,P }, the order of the local section s being computed using a normalization of C. It is finitely generated and a minimal set of generators (β 0 , β 1 , . . . , β g ) is constructed as follows (see [13, Theorem 4.3.5]): β 0 is the least element of S(C); set m 1 = β 0 ; β j is the least element of S(C) not divisible by m j and m j+1 = gcd(m j , β j ).To prove (1) we use the notion of relevant divisors of the minimal log resolution of C at P , notion introduced in [12], and previously in [6] from the point of view of valuations corresponding to Puiseux exponents: a relevant divisor is an irreducible exceptional divisor that intersects at least three other components of the total transform of C through the resolution. When C is unibranch at P , we show that the relevant divisors account for all the jumping numbers. This is the content of Proposition 2.5 and represents the key step of the proof. In 1

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Section: Introductionmentioning

confidence: 84%

Jumping numbers of a unibranch curve on a smooth surface

Naie

2008

manuscripta math.

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“…This argument is essentially repeated in Example 3.6 of [ELSV04], and discussed at greater length in Section 9.3.C of [Laz04]. Note that since this curve is analytically irreducible, the result also follows from [Jär06].…”

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confidence: 86%

“…The values of λ where the multiplier ideals change are known as jumping numbers. These discrete local invariants were studied systematically in [ELSV04], after appearing indirectly in [Lib83], [LV90], [Vaq92], and [Vaq94]. The jumping numbers encode algebraic information about the ideal, and geometric properties of the associated closed subscheme.…”

Section: Introductionmentioning

confidence: 99%

Jumping numbers on algebraic surfaces with rational singularities

Tucker

2009

Trans. Amer. Math. Soc.

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Abstract. In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored.

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“…These ideals and the invariants arising from them have been widely studied. See [BL04] for an introduction and [ELSV04] for some applications. One of the main invariants defined in terms of multiplier ideals is the log canonical threshold.…”

Section: Introductionmentioning

confidence: 99%

A procedure for computing the log canonical threshold of a binomial ideal

Blanco¹,

Encinas²

2017

manuscripta math.

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Abstract. We present a procedure for computing the log-canonical threshold of an arbitrary ideal generated by binomials and monomials.The computation of the log canonical threshold is reduced to the problem of computing the minimum of a function, which is defined in terms of the generators of the ideal. The minimum of this function is attained at some ray of a fan which only depends on the exponents of the monomials appearing in the generators of the ideal.

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Jumping coefficients of multiplier ideals

Cited by 98 publications

References 33 publications

Jumping numbers of a unibranch curve on a smooth surface

Jumping numbers of a unibranch curve on a smooth surface

Jumping numbers on algebraic surfaces with rational singularities

A procedure for computing the log canonical threshold of a binomial ideal

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