Local moduli and singularities. Appendix (by B. Martin and G. Pfister): An algorithm to compute the kernel of the Kodaira-Spencer map for an irreducible plane curve singularity
type: Item
instance of: scholarly article
title: Local moduli and singularities. Appendix (by B. Martin and G. Pfister): An algorithm to compute the kernel of the Kodaira-Spencer map for an irreducible plane curve singularity
zbMATH Open document ID: 0657.14005
author: Gerhard Pfister
author: Olav Arnfinn Laudal
published in: Lecture Notes in Mathematics
publication date: 1993-06-05T00:00:00Z
review text: The central concept discussed in this research monograph is that of ``local moduli suite'' of an algebro-geometric object X. Roughly speaking this is a collection \(\{M_{\tau}\}\) of algebraic spaces each \(M_{\tau}\) prorepresenting the \(\tau\)-constant deformations of X occurring in the family \(\pi_{\tau}\) obtained by restricting an algebraisation \(\pi:\quad \tilde X\to H\) of the formal versal family of X to the ``\(\tau\)-constant stratum''. An abstract theorem is proved first which guarantees the existence of the local moduli suite in the presence of certain axioms on the formal versal family. Results along this line were independently obtained by Palamodov and Saito. A conjecture of \textit{J. Wahl} [cf. Topology 20, 219-246 (1981; Zbl 0484.14012)] on the dimension of a smoothing component is then proved [this was independently proved by \textit{G.-M. Greuel} and \textit{E. Looijenga} in Duke Math. J. 52, 263-272 (1985; Zbl 0587.32038)]. Next the case of hypersurface singularities is investigated in detail (some modifications of the general setting of local moduli suites are here necessary). Here one is interested in the dimensions of the loci \(M_{\mu \tau}\) of all points in \(M_{\tau}\) corresponding to singularities with a given Milnor number \(\mu\). Quite precise results are obtained in the case of weighted homogeneous plane curve singularities; in particular a coarse moduli space is proved to exist for all plane curve singularities with given semigroup \(\Gamma =<a_ 1,a_ 2>\), \((a_ 1,a_ 2)=1\) and minimal Tjurina number \(\tau\) and a computation is given for its dimension. Part of these results are joint work of the authors with \textit{B. Martin}; they are a remarkable contribution to Zariski's program on the moduli problem for curve singularities.
reviewed by: Alexandru Buium
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14D22
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14H10
zbMATH DE Number: 192997
zbMATH Keywords: local moduli suite
zbMATH Keywords: formal versal family
zbMATH Keywords: hypersurface singularities
zbMATH Keywords: homogeneous plane curve singularities
zbMATH Keywords: coarse moduli space
zbMATH Keywords: Tjurina number
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 192997