×

The value semiring of an algebroid curve. (English) Zbl 1451.14093

Let \(\mathbb{K}\) be an algebraically closed field. And let \(F=\mathbb{K}[\![x_1,\dots,x_n]\!]\) be the ring of formal series on the variables \(x_1,\dots,x_n\) and with coefficients in \(\mathbb{K}\). An algebroid curve in \(\mathbb{K}^n\) is a ring \(\mathcal{O}=F/Q\), where \(Q=\bigcap_{i=1}^rP_i\) is a proper radical ideal of \(F\) such that \(\mathcal{O}_i=F/P_i\) has Krull dimension one for all \(i\in\{1,\dots,r\}\) (called the branches of the curve \(\mathcal{O}\)). The authors consider non-degenerate curves.
The integral closure of \(\mathcal{O}_i\) in its quotient field, \(\overline{\mathcal{O}}_i\), is a discrete valuation domain. Denote by \(v_i\colon \overline{\mathcal{O}}_i\to \mathbb{N}\cup\{\infty\}\) the normalized discrete valuation of \(\overline{\mathcal{O}}_i\) (\(v_i(0)=\infty\)). For \(g\in \mathcal{O}\), set \(v(g)=(v_1(g),\dots,v_r(g))\). Then \(S=\{v(g): g \text{ is a nonzero divisor of } \mathcal{O} \}\) is a subsemigroup under addition of \(\mathbb{N}^r\), and \(\Gamma = \{v(g): g\in \mathcal{O}\}\) is a subsemigroup of \((\mathbb{N}\cup \{\infty\})^r\) (where \(\infty+n=\infty\) for any \(n\in \mathbb{N}\cup\{\infty\}\)). These semigroups are also closed under the infimum operation: \((x_1,\dots,x_r)\wedge (y_1,\dots,y_r)= (\min(x_1,y_1),\dots,\min(x_r,y_r))\). The aim of this paper is to prove that the semiring \((\Gamma, \wedge, +)\) (called the value semiring of \(\mathcal{O}\)) is finitely generated, that is, there exists a finite subset \(A\) of \(\Gamma\) such that every element in \(\Gamma\) is expressed as an infimum of finitely many sums with finitely many summands in \(A\).
There has been several approaches in the literature to describe \(S\), and some have considered the semiring structure. The novelty here is to include nonzero divisors, and thus \(\infty\) as a coordinate, and also to prove that systems of generators come from standard bases of \(\mathcal{O}\).
A finite subset \(G\) of the maximal ideal of \(\mathcal{O}\) is said to be a standard basis of \(\mathcal{O}\) is for any \(g\in \mathcal{O}\setminus\{0\}\) there exists a finite product \(p\) of elements in \(G\) and \(k\in \mathbb{K}\) such that \(v(g-kp)>v(g)\) (here \(\le\) denotes the usual cartesian partial ordering on \((\mathbb{N}\cup\{\infty\})^r\)). Basically, what the authors prove is that \(\mathcal{O}\) admits a standard basis, and that if \(G\) is a standard basis for \(\mathcal{O}\), then \(v(G)\) is a system of generators of the semiring \(\Gamma\). As the authors have a way to compute standard bases of \(\mathcal{O}\), they have a procedure to calculate systems of generators of \(\Gamma\).

MSC:

14H20 Singularities of curves, local rings
14H50 Plane and space curves
14Q05 Computational aspects of algebraic curves
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bayer, V., Semigroup of two irreducible algebroid plane curves, Manuscr. Math, 49, 3, 207-241 (1985) · Zbl 0581.14021 · doi:10.1007/BF01215247
[2] Carvalho, E.; Hernandes, M. E., Standard bases for fractional ideals of the local ring of an algebroid curve, J. Algebra, 551, 1, 342-361 (2020) · Zbl 1441.14104 · doi:10.1016/j.jalgebra.2020.01.018
[3] Castellanos, A.; Castellanos, J., Algorithm for the semigroup of a space curve singularity, Semigroup Forum., 70, 1, 44-60 (2005) · Zbl 1105.14037 · doi:10.1007/s00233-004-0143-z
[4] Cotterill, E., Feital, L., Martins, R. V. Dimension counts for singular rational curves via semigroups. arXiv:1511.08515v4 [math.AG]. · Zbl 1454.14079
[5] D’Anna, M., The canonical module of a one-dimensional reduced local ring, Commun. Algebra, 25, 9, 2939-2965 (1997) · Zbl 0889.13006
[6] D’Anna, M.; García-Sánchez, P. A.; Micale, V.; Tozzo, L., Good subsemigroup of \(####\), Int. J. Algebra Comput., 28, 2, 179-206 (2018) · Zbl 1454.14089 · doi:10.1142/S0218196718500091
[7] Delgado, F., The semigroup of values of a curve singularity with several branches, Manuscr. Math, 59, 347-374 (1987) · Zbl 0611.14025
[8] Delgado, F., Gorenstein curves and symmetry of the semigroup of values, Manuscr. Math, 61, 285-296 (1988) · Zbl 0692.13017
[9] Garcia, A., Semigroups associated to singular points of plane curves, J. Reine Angew. Math, 336, 165-184 (1982) · Zbl 0484.14008
[10] Hefez, A.; Hernandes, M. E., Standard bases for local rings of branches and their modules of differentials, J. Symb. Comput, 42, 1-2, 178-191 (2007) · Zbl 1121.14048 · doi:10.1016/j.jsc.2006.02.008
[11] Waldi, R., Wertehalbgruppe und Singularität einer ebenen algebraischen Kurve (1972), Regensburg
[12] Zariski, O., Studies in equisingularity I, Amer. J. Math, 87, 2, 507-536 (1965) · Zbl 0132.41601 · doi:10.2307/2373019
[13] Zariski, O., Studies in equisingularity II, Amer. J. Math, 87, 4, 972-1006 (1965) · Zbl 0146.42502 · doi:10.2307/2373257
[14] Zariski, O., Studies in equisingularity III, Amer. J. Math, 90, 3, 961-1023 (1968) · Zbl 0189.21405 · doi:10.2307/2373492
[15] Zariski, O., The Moduli Problem for Plane Branches. University Lecture Series (2006), Providence, RI: AMS, Providence, RI · Zbl 1107.14021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.