The Semiring of Values of an Algebroid Curve

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 results of Zariski and Waldi that characterize the topological type of the curve.

The Apéry Set of a Good Semigroup

“…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, ((…”

“…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.…”

“…, 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 ).…”

Partition of complement of good ideals and Apéry sets

“…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.…”

“…, 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).…”