Deformationen isolierter Kurvensingularitäten mit eingebetteten Komponenten. (Deformations of isolated curve singularities with embedded components)
type: Item
instance of: scholarly article
title: Deformationen isolierter Kurvensingularitäten mit eingebetteten Komponenten. (Deformations of isolated curve singularities with embedded components)
zbMATH Open document ID: 0728.32016
DOI: 10.1007/BF02568363
author: Christian Brücker
author: Gert-Martin Greuel
published in: Manuscripta Mathematica
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155574
review text: \textit{R.-O. Buchweitz} and the second author studied deformations of reduced isolated curve singularities in their paper in Invent. Math. 58, 241-281 (1980; Zbl 0458.32014). The main invariants in that paper are the Milnor number \(\mu\) and the number of virtual double points \(\delta\). In families of complex spaces it is quite common, that special fibres contain embedded components. This has the effect that the generic fibre can consist of (isolated) fat points and disjoint 1-dimensional components. This is the main theme in the paper under review. The authors define a new invariant \(\epsilon\), which measures the non-reduced structure and they generalize the definitions of \(\mu\) and \(\delta\). The new definitions behave very well with respect to the topological properties of the singularities. As an example: the jump in \(\mu\) is equal to the change in the Euler characteristic of the fibre. This is similar to the reduced case, but in the non-reduced case also may new phenomena occur. The paper is well-written and contains all the technical details, which are necessary to understand the situation.
Mathematics Subject Classification ID: 32S30
Mathematics Subject Classification ID: 32S50
zbMATH DE Number: 4202700
zbMATH Keywords: deformations
zbMATH Keywords: reduced isolated curve singularities
zbMATH Keywords: invariants
zbMATH Keywords: Milnor number
zbMATH Keywords: number of virtual double points
zbMATH Keywords: non-reduced structure
reviewed by: Dirk Siersma
MaRDI profile type: Publication
cites work: The Milnor number and deformations of complex curve singularities
cites work: Q5560333
cites work: Complex analytic geometry
cites work: Q3824508
OpenAlex ID: W1991327966
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description: scientific article; zbMATH DE number 4202700