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
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.
scientific article; zbMATH DE number 192997
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