Irrelevant exceptional divisors for curves on a smooth surface

Smith, Karen E.; Thompson, Howard M

doi:10.1090/conm/448/08669

Cited by 26 publications

(57 citation statements)

References 9 publications

(8 reference statements)

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“…Note that this is an extension of Definition 5 from [ST06], where Smith and Thompson introduced jumping number contribution for prime divisors. Further, if a jumping number is contributed by a prime divisor E, this contribution is automatically critical.…”

Section: Jumping Numbers Contributed By Divisorsmentioning

confidence: 99%

“…Our techniques build upon the work of Smith and Thompson in [ST06]. Roughly speaking, multiplier ideals are defined by a finite number of divisorial conditions on a given resolution.…”

Section: Introductionmentioning

confidence: 99%

“…Distinct prime divisors often have common candidate jumping numbers. In some cases, as the next example from [ST06] shows, these prime divisors may separately contribute the same jumping number. In others, collections of these divisors may be needed to capture a jump in the multiplier ideals.…”

mentioning

confidence: 98%

“…According to Definition 3.1, E 0 +E 2 +E 2 critically contributes 1 2 . Further, it is shown in [ST06] that 9 10 is a jumping number contributed by either E 2 or E 2 . One may even argue there is a sense in which the collection E 2 +E 2 is responsible for this jump.…”

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confidence: 99%

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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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Section: Jumping Numbers Contributed By Divisorsmentioning

confidence: 99%

“…Our techniques build upon the work of Smith and Thompson in [ST06]. Roughly speaking, multiplier ideals are defined by a finite number of divisorial conditions on a given resolution.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 98%

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confidence: 99%

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Jumping numbers on algebraic surfaces with rational singularities

Tucker

2009

Trans. Amer. Math. Soc.

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“…. , 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.…”

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confidence: 99%

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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Constancy regions of mixed multiplier ideals in two‐dimensional local rings with rational singularities

Alberich-Carramiñana

Montaner

Dachs-Cadefau

2017

Mathematische Nachrichten

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The aim of this paper is to study mixed multiplier ideals associated with a tuple of ideals in a two-dimensional local ring with a rational singularity. We are interested in the partition of the real positive orthant given by the regions where the mixed multiplier ideals are constant. In particular we reveal which information encoded in a mixed multiplier ideal determines its corresponding jumping wall and we provide an algorithm to compute all the constancy regions, and their corresponding mixed multiplier ideals, in any desired range. K E Y W O R D SMixed multiplier ideals, jumping walls, rational singularities M S C ( 2 0 1 0 ) 14F18, 14H20

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Irrelevant exceptional divisors for curves on a smooth surface

Abstract: Given a singular curve on a smooth surface, we determine which exceptional divisors on the minimal resolution of that curve contribute toward its jumping numbers.

Cited by 26 publications

References 9 publications

Jumping numbers on algebraic surfaces with rational singularities

Jumping numbers on algebraic surfaces with rational singularities

Jumping numbers of a unibranch curve on a smooth surface

Constancy regions of mixed multiplier ideals in two‐dimensional local rings with rational singularities

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