Abstract:We introduce the semiring of values Γ with respect to the tropical operations associated to an algebroid curve. As a set, Γ determines and is determined by the well known semigroup of values S and we prove that Γ is always finitely generated in contrast to S. In particular, for a plane curve, we present a straightforward way to obtain Γ in terms of the semiring of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytical case, this allows us to connect directly the re… Show more
“…Now, applying the same argument as above, one of them has to be in A i+1 . (7) Any element α = (α 1 , α 2 ) with α 2 > γ 2 (respectively, α 1 > γ 1 ) belongs to S if and only if (α 1 , γ 2 + 1) ∈ S (respectively, ((…”
Section: Figurementioning
confidence: 99%
“…However good semigroups present some problems that make difficult their study; first of all they are not finitely generated as monoid (even if they can be completely determined by a finite set of elements (see [15], [7] and [10]) and they are not closed under finite intersections. Secondly, the behavior of the good ideals of good semigroups (e.g.…”
Section: Introductionmentioning
confidence: 99%
“…, t e 2 ∈ S 2 . Now, notice that, by Lemma 3.2 (7), if an infinite line is contained in Ap(S), then its elements must be contained eventually in a level A i . Moreover, by Lemma 3.2(8), A i cannot contain more than two infinite lines and, if it contains two of them, they must be one of the form ∆ 1 (s 1 , r) and the other of the form ∆ 2 (q, s 2 ).…”
We study the Apéry Set of good subsemigoups of N 2 , a class of semigroups containing the value semigroups of curve singularities with two branches. Even if this set in infinite, we show that, for the Apéry Set of such semigroups, we can define a partition in "levels" that allows to generalize many properties of the Apéry Set of numerical semigroups, i.e. value semigroups of one-branch singularities.MSC: 13A18, 14H99, 13H99, 20M25.
“…Now, applying the same argument as above, one of them has to be in A i+1 . (7) Any element α = (α 1 , α 2 ) with α 2 > γ 2 (respectively, α 1 > γ 1 ) belongs to S if and only if (α 1 , γ 2 + 1) ∈ S (respectively, ((…”
Section: Figurementioning
confidence: 99%
“…However good semigroups present some problems that make difficult their study; first of all they are not finitely generated as monoid (even if they can be completely determined by a finite set of elements (see [15], [7] and [10]) and they are not closed under finite intersections. Secondly, the behavior of the good ideals of good semigroups (e.g.…”
Section: Introductionmentioning
confidence: 99%
“…, t e 2 ∈ S 2 . Now, notice that, by Lemma 3.2 (7), if an infinite line is contained in Ap(S), then its elements must be contained eventually in a level A i . Moreover, by Lemma 3.2(8), A i cannot contain more than two infinite lines and, if it contains two of them, they must be one of the form ∆ 1 (s 1 , r) and the other of the form ∆ 2 (q, s 2 ).…”
We study the Apéry Set of good subsemigoups of N 2 , a class of semigroups containing the value semigroups of curve singularities with two branches. Even if this set in infinite, we show that, for the Apéry Set of such semigroups, we can define a partition in "levels" that allows to generalize many properties of the Apéry Set of numerical semigroups, i.e. value semigroups of one-branch singularities.MSC: 13A18, 14H99, 13H99, 20M25.
“…For example, the well-known result by Kunz (see [13]) that a one-dimensional analytically irreducible local domain is Gorenstein if and only if its value semigroup is symmetric can be generalized to analytically unramified rings (see [3] and also [10]). However, good semigroups present some problems that make difficult their study; first of all they are not finitely generated as monoid (even if they can be completely determined by a finite set of elements (see [11], [4] and [6])) and they are not closed under finite intersections.…”
Section: Introductionmentioning
confidence: 99%
“…, 2, 6),(1,2,7),(1,2,8),(2,3,3),(2,3,6),(2,3,7),(2,4,3),(2,4,6), (2, 4, 9)(3, 3, 3),(3,3,6),(3,3,7),(3,5,3),(3,5,6), (3, 5, 9)} It has conductor c = (3, 5, 9) and γ = (2, 4, 8).…”
Good semigroups form a class of submonoids of N d containing the value semigroups of curve singularities. In this article, we describe a partition of the complements of good semigroup ideals, having as main application the description of the Apéry sets of good semigroups. This generalizes to any d ≥ 2 the results of [7], which are proved in the case d = 2 and only for the standard Apéry set with respect to the smallest nonzero element. Several new results describing good semigroups in N d are also provided.
In this paper we present an algorithm to compute a Standard Basis for a fractional ideal I of the local ring O of an n-space algebroid curve with several branches. This allows us to determine the semimodule of values of I. When I = O, we may obtain a (finite) set of generators of the semiring of values of the curve, which determines its classical semigroup. In the complex context, identifying the Kähler differential module Ω O/C of a plane curve with a fractional ideal of O and applying our algorithm, we can compute the set of values of Ω O/C , which is an important analytic invariant associated to the curve.
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