Sous-monoïdes d'intersection complète de $N$

Delorme, Charles

doi:10.24033/asens.1307

Cited by 90 publications

(119 citation statements)

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“…We need an extension to the equivariant reducible case of the theory of numerical semigroups and monomial curves developed by Delorme, Herzog, Kunz, Watanabe and the authors [4,9,10,21,28]. The necessary parts of this theory are developed in Sections 3-7.…”

Section: Our Main Results Ismentioning

confidence: 99%

The end curve theorem for normal complex surface singularities

Neumann

Wahl

2010

J. Eur. Math. Soc.

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Abstract. We prove the "End Curve Theorem," which states that a normal surface singularity (X, o) with rational homology sphere link is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree.An "end curve function" is an analytic function (X, o) → (C, 0) whose zero set intersects in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf.A "splice quotient singularity" (X, o) is described by giving an explicit set of equations describing its universal abelian cover as a complete intersection in C t , where t is the number of leaves in the resolution graph for (X, o), together with an explicit description of the covering transformation group.Among the immediate consequences of the End Curve Theorem are the previously known results: (X, o) is a splice quotient if it is weighted homogeneous (Neumann 1981), or rational or minimally elliptic (Okuma 2005).Keywords. Surface singularity, splice quotient singularity, rational homology sphere, complete intersection singularity, abelian cover, numerical semigroup, monomial curve, linking pairing We consider normal surface singularities whose links are rational homology spheres (QHS for short). The QHS condition is equivalent to the condition that the resolution graph of a minimal good resolution be a rational tree, i.e., is a tree and all exceptional curves are genus zero.Among singularities with QHS links, splice quotient singularities are a broad generalization of weighted homogeneous singularities. We recall their definition briefly here and in more detail in Section 1. Full details can be found in [20].Recall first that the topology of a normal complex surface singularity is determined by and determines the minimal resolution graph . Let t be the number of leaves of . For i = 1, . . . , t, we associate the coordinate function x i of C t to the i-th leaf. This leads to a natural action of the "discriminant group" D = H 1 ( ) by diagonal matrices on C t (see Section 1).

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Section: Our Main Results Ismentioning

confidence: 99%

The end curve theorem for normal complex surface singularities

Neumann

Wahl

2010

J. Eur. Math. Soc.

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“…The translations of the above coordinates in terms of numerical semigroups are { 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 33 , 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 34, 23 , 12, 13, 16, 17, 18, 19, 20, 21, 22, 23 , 12, 15, 17, 18, 20, 21, 22, 23, 25, 26, 28, 43 , 12, 13, 17, 18, 21, 22, 23, 26, 27, 28, 31, 44 , 12, 15, 17, 18, 21, 23, 26, 28, 31, 32, 34, 49 }. Now, by solving problem (SC m (D)), 12,13,14,15,16,17,18,19,20,21,34, 23 is discarded. Then, the decomposition using our methodology is given by five 12-irreducible numerical semigroups: S = 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 33 ∩ 12, 13, 16, 17, 18, 19, 20, 21, 22, 23 ∩ 12, 15, 17, 18, 20, 21, 22, 23, 25, 26, 28, 43 ∩ 12, 13, 17, 18, 21, 22, 23, 26, 27, 28, 31, 44 ∩ 12, 15, 17, 18, 21, 23, 26, 28, 32, 34 . However, this decomposition is not minimal since S = 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 26, 39 ∩ 12, 15, 17, 18, 20, 21, 22, 23, 25, 26, 28, 43 ∩ 12, 13, 17, 18, 21, 22, 23, 26, 27, 28, 31, 44 ∩ 12, 15, 16, 17, 18, 23, 26, 31, 32, 33, 34, 49 is a decomposition into mirreducible numerical semigroups using a smaller number of terms.…”

Section: Lemma 26 Let Y Be An Optimal Solution Of (Ip M (X H)) Thenmentioning

confidence: 99%

“…Note that the simplest numerical semigroup is Z + . Numerical semigroups were first considered while studying the set of nonnegative solutions of Diophantine equations and their investigation is closely related to the analysis of monomial curves (see [20]). Because of these connections with algebraic geometry, some terminology has been exported to the theory of numerical semigroups, for instance, the multiplicity, the genus, or the embedding dimension of a numerical semigroup.…”

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confidence: 99%

An Application of Integer Programming to the Decomposition of Numerical Semigroups

Blanco¹,

Puerto²

2012

SIAM J. Discrete Math.

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Abstract. This paper addresses the problem of decomposing a numerical semigroup into mirreducible numerical semigroups. The problem originally stated in algebraic terms is translated, introducing the so-called Kunz-coordinates, to resolve a series of several discrete optimization problems. First, we prove that finding a minimal m-irreducible decomposition is equivalent to solve a multiobjective linear integer problem. Then, we restate that problem as the problem of finding all the optimal solutions of a finite number of single objective integer linear problems plus a set covering problem. Finally, we prove that there is a suitable transformation that reduces the original problem to find an optimal solution of a compact integer linear problem. This result ensures a polynomial time algorithm for each given multiplicity m. We have implemented the different algorithms and have performed some computational experiments to show the efficiency of our methodology.

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“…In this direction it is worth highlighting [Ramírez Alfonsín 2000;≥ 2005], where a review of this problem is given, with many references. In the literature one can also find a large number of publications devoted to the study of one-dimensional analytically irreducible local domains via their value semigroups, which are numerical semigroups; see, for instance, [Apéry 1946;Barucci et al 1997;Bertin and Carbonne 1977;Delorme 1976;Fröberg et al 1987;Kunz 1970;Teissier 1973;Watanabe 1973]. As a consequence of this study, some interesting kinds of numerical semigroups arise, such as symmetric and pseudo-symmetric numerical semigroups.…”

Section: Introductionmentioning

confidence: 99%