Equisingularity theory for plane curves with embedded points
type: Item
instance of: scholarly article
title: Equisingularity theory for plane curves with embedded points
zbMATH Open document ID: 0903.14002
author: Augusto Nobile
published in: Pacific Journal of Mathematics
publication date: 1996-08-28T00:00:00Z
full work available at URL: nyjm.albany.edu
review text: In this article the author studies different notions of equisingularity for plane curves which are not necessarily reduced, i.e. for generically reduced curves with embedded points. This theory is related with both the theory of equisingularity for plane reduced curves and that for ideals with finite support. In the first part there is a review of known results on the theory for reduced curves. We can associate to an algebroid plane reduced curve \(C\), with branches \(\gamma=(\gamma_1,\ldots,\gamma_s)\) a weighted tree \(T_1(C,\gamma)\) whose vertices are the infinitely near points of \(C\), and each vertex has the weight \((m_1,\ldots,m_r)\) of the multiplicities of the strict transforms of the irreducible components at this point. Then two curves are equisingular if they have the same tree. In the same way we can associate a finite tree \(\tau({\mathcal I})\) to each \({\mathcal M}\)-primary ideal \({\mathcal I}\) of \(kx,y\), each vertex corresponding to a proper transform of \({\mathcal I}\) and its weight being the multiplicity of the corresponding ideal. In the same way we say that two \({\mathcal M}\)-primary ideals are equisingular if they have the same tree. We note that \(\tau({\mathcal I})=\tau(\overline{\mathcal I})\), where \(\overline{\mathcal I}\) is the integral closure of \({\mathcal I}\). An algebroid plane curve \(C\) with embedded point is defined by an ideal \({\mathcal I}\) of \(R=kx,y\), with \({\mathcal I}=(f).{\mathcal N}\), where \(f\in R\) and either the radical \(r({\mathcal N})\) is the maximal ideal \({\mathcal M}\) or \({\mathcal N}=R\). Let \(\pi_1\colon X_1\to{\mathcal A}^2=\text{Spec}R\) be the blowing-up of center \({\mathcal M}\), then the intersection \(E\cap C_1\) of the exceptional divisor \(E\) with the strict transform \(C_1\) of the curve \(C\) is a finite set of points \(\{ P_1,\ldots,P_r\}\), which are by definition the points in the first neighborhood of \(C\). At each point \(P_i\), we have either a curve \(C_i\…
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 910242
zbMATH Keywords: equisingularity
zbMATH Keywords: non-reduced curve
zbMATH Keywords: complete ideal
zbMATH Keywords: algebroid plane curve
reviewed by: Michel Vaquié
MaRDI profile type: Publication
OpenAlex ID: W1597942619
DOI: 10.2140/PJM.1995.170.543
description: scientific article; zbMATH DE number 910242