Zariski's multiplicity question and aligned singularities
type: Item
instance of: scholarly article
title: Zariski's multiplicity question and aligned singularities
zbMATH Open document ID: 1089.32019
author: C. Eyral
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2006-03-20T00:00:00Z
review text: Zariski's famous multiplicity conjecture [cf. \textit{O. Zariski}, Bull. Am. Math. Soc. 77, 481--491 (1971; Zbl 0236.14002)] can be briefly formulated as follows. Let \(f_t:({\mathbb C}^n, 0)\to (\mathbb C, 0)\) define a family of hypersurface singularities. Assume the family is topologically constant (i.e., in case of isolated singularities the Milnor number \(\mu(f_t)=\dim_\mathbb C \mathbb Cx_1, \ldots, x_n/\langle \frac{\partial f_t}{\partial x_1}, \ldots \frac{\partial f_t}{\partial x_n}\rangle\) is constant). Does this imply that the multiplicity \(m(f_t)=\text{ord}_x f_t\) is constant? This conjecture is proved only for very special cases. The author considers Zariski's question for a special class of non-isolated singularities and gives a proof.
Mathematics Subject Classification ID: 32S15
Mathematics Subject Classification ID: 32S25
Mathematics Subject Classification ID: 32S50
zbMATH DE Number: 5013362
zbMATH Keywords: Zariski conjecture
zbMATH Keywords: multiplicity
zbMATH Keywords: Milnor number
zbMATH Keywords: constant topological type
reviewed by: Gerhard Pfister
MaRDI profile type: Publication
full work available at URL: j.crma.2005.12.008
OpenAlex ID: W2024354360
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DOI: 10.1016/J.CRMA.2005.12.008
description: scientific article; zbMATH DE number 5013362