On the trees associated to plane curve singularities and ideals
type: Item
instance of: scholarly article
title: On the trees associated to plane curve singularities and ideals
zbMATH Open document ID: 0909.14016
DOI: 10.1007/BF02844514
author: Augusto Nobile
publication date: 1998-01-11T00:00:00Z
review text: A possible approach to study the complexity of singularity of a reduced plane curve, or that of an ideal in a regular two dimensional local ring (primary to the maximal ideal) is to consider its associated tree of ``infinitely near points''. Each vertex of the tree corresponds to a point infinitely near to the singularity (or the support of the ideal), two vertices being joined by an edge if one is obtained from the other by means of a single quadratic transformation. The tree itself does not say much about the singularity (or ideal), but one may enrich it by assigning integral ``weights'' to the vertices. One obtains different notions of equivalence, or equisingularity, for curves (or ideals) by requiring that the various weighted trees that appear be isomorphic (by means of an isomorphism respecting the weights). However, the situation is different when we are dealing with curves which are members of the same family. In this case, several different natural notions of equivalence agree. More precisely, in this note we discuss, along the lines just mentioned, what we call the notions of \(R\)- and \(C\)-equivalence of ideals. The concept of \(R\)-equivalence was studied by \textit{J.-J. Risler} in Bull. Soc. Math. Fr. 101, 3-16 (1973; Zbl 0256.14006) and by \textit{A. Nobile} in Pac. J. Math. 170, No. 2, 543-566 (1995; Zbl 0903.14002). Also, there are discussed the notions of \(Z\)- and \(M\)-equivalence of reduced curves [\(Z\)-equivalence being the same as the well-known concept of Zariski-equivalence; see \textit{O. Zariski}, Am. J. Math. 87, 507-536 (1965; Zbl 0132.41601)]. In section 3 (theorem 2), it is checked that given a family of curves \(\{C_t\}\), \(t\) in a connected parameter space \(T\), then the different fibers are \(Z\)-equivalent to each other if and only if they are \(M\)-equivalent (although in general only one implication is valid). Concerning families of ideals, to require that all members be \(C\)-equivalent is strictly stronger that requirin…
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 1089786
zbMATH Keywords: complexity of singularity
zbMATH Keywords: reduced plane curve
zbMATH Keywords: equisingularity
MaRDI profile type: Publication
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cites work: Q4486505
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cites work: Sur les déformations équisingulières d'idéaux
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cites work: Studies in Equisingularity II. Equisingularity in Codimension 1 (and Characteristic Zero)
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1089786