Simple singularities in positive characteristic
type: Item
instance of: scholarly article
title: Simple singularities in positive characteristic
zbMATH Open document ID: 0715.14001
DOI: 10.1007/BF02570742
author: H. Kröning
author: Gert-Martin Greuel
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174133
review text: Simple singularities (also called ADE because the different types \(A_ k\), \(D_ k\) and \(E_ 6\), \(E_ 7\), \(E_ 8\) in which they are classified) appear in a natural way in problems of classification of singularities from different points of view. In dimension 1, simple singularities have been characterized in terms of their resolution procedure by \textit{W. P. Barth}, \textit{C. A. M. Peters} and \textit{A. J. H. M. Van de Ven} [``Compact complex surfaces'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 4 (1984; Zbl 0718.14023)] in characteristic 0 and in any characteristic by \textit{K. Kiyek} and \textit{G. Steinke} [Arch. Math. 45, 565-573 (1985; Zbl 0553.14012)] who obtained the normal forms of them. In dimension 2 simple singularities are just the rational double points [see \textit{M. Artin}'s paper in Complex Anal. Algebr. Geom., Collect. Papers dedic. K. Kodaira, 11-22 (1977; Zbl 0358.14008)]. On the other hand in characteristic 0 and arbitrary dimension \textit{V. I. Arnol'd} showed [cf. Funct. Anal. 6(1972), 254-272 (1973); translation from Funkts. Anal. Prilozh. 6, No.4, 3-25 (1972; Zbl 0278.57011)] that they are exactly the hypersurface singularities of finite deformation type (i.e. singularities from which one can obtain, by deformation, only a finite number of nonequivalent singularities). According to previous works of \textit{H. Knörrer} [Invent. Math. 88, 153- 164 (1987; Zbl 0617.14033)] and \textit{R.-O. Buchweitz}, \textit{G.-M. Greuel} and \textit{F.-O. Schreyer} [ibid. 165-182 (1987; Zbl 0617.14034)] it is known that the simple singularities are the hypersurface singularities with finite Cohen-Macaulay type (i.e. singularities for which there exists only a finite number of non isomorphic indecomposable maximal Cohen-Macaulay modules over their local ring). However, the relationship with the point of view of Arnol'd (using deformation theory) remains essentially unknown in arbitrary characteristic. This point constitutes th…
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14B12
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 32S25
zbMATH DE Number: 4177241
zbMATH Keywords: ADE singularities
zbMATH Keywords: hypersurface singularities of finite deformation type
zbMATH Keywords: hypersurface singularities with finite Cohen-Macaulay type
zbMATH Keywords: normal forms for simple singularities
reviewed by: F. Delgado
describes a project that uses: classifyCeq.lib
MaRDI profile type: Publication
cites work: Rational Singularities and Almost Split Sequences
cites work: Cohen-Macaulay modules on hypersurface singularities. II
cites work: Reflexive modules on quotient surface singularities.
cites work: Einfache Kurvensingularitäten und torsionsfreie Moduln
cites work: Einfache Kurvensingularitäten in beliebiger Charakteristik
cites work: Cohen-Macaulay modules on hypersurface singularities. I
cites work: Ein neuer Zusammenhang zwischen einfachen Gruppen und einfachen Singularitäten. (A new connection between simple groups and simple singularities)
cites work: Rational singularities, with applications to algebraic surfaces and unique factorization
cites work: Hypersurface Singularities of Finite Cohen-Macaulay Type
description: scientific article; zbMATH DE number 4177241