Some results on the local moduli of non-singular normalizations of surfaces with ordinary singularities
type: Item
instance of: scholarly article
title: Some results on the local moduli of non-singular normalizations of surfaces with ordinary singularities
zbMATH Open document ID: 0653.32026
author: Syôji Tsuboi
author: Masaki Ôkawa
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1988-01-01T00:00:00Z
review text: Let S be a compact irreducible analytic surface embedded in a compact threefold W. Assume that S has (with respect to that embedding) at most ordinary singularities, i.e. double points, triple points or cusps. Denote by X the normalization of S, and by \(\Phi_ S\) the sheaf of infinitesimal displacements in W. The authors give a sufficient condition for the connecting homomorphism \[ \delta:\quad H^ 0(S,\Phi_ S)\quad \to \quad H^ 1(X,{\mathcal T}_ X) \] to be surjective. This condition, expressed in terms of the sheaf cohomology of the surfaces S, X and of the ambient threefold W, represents a sharpening of some previous results of the authors [cf. the first author, Sci. Rep. Kagoshima Univ. 32, 23-46 (1983; Zbl 0568.14019)]. Applying this surjectivity criterion in a deformation-theoretical context, the authors compute the Kuranishi families of deformations of complex structures for certain algebraic surfaces of general type, namely for normalizations of surfaces with ordinary singularities in \({\mathbb{P}}^ 3({\mathbb{C}})\). The non- singularity of these Kuranishi families allows to compute the number of moduli for those particular surfaces.
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 32G05
Mathematics Subject Classification ID: 32J25
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14J15
zbMATH DE Number: 4065418
zbMATH Keywords: local moduli
zbMATH Keywords: Kuranishi space
zbMATH Keywords: analytic surface
zbMATH Keywords: singularities
zbMATH Keywords: sheaf cohomology
zbMATH Keywords: deformations of complex structures
zbMATH Keywords: algebraic surfaces
reviewed by: Werner Kleinert
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228030
OpenAlex ID: W2018077995
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DOI: 10.2748/TMJ/1178228030
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description: scientific article; zbMATH DE number 4065418