A deformation theory for non-isolated singularities. (English) Zbl 0721.32014
With this paper, we get enough knowledge of the deformation theory for non-isolated singularities. The contents are the following: § 1 The functor of admissible deformations (functors for hypersurfaces), § 2 Infinitesimal theory (deformations of \(\Sigma\), the complex D(\(\Sigma\),f)). § 3 Special conditions on \(\Sigma\) (the complex H(\(\sigma\),f), examples and applications).
§ 3 deals with the
Conjecture: Let f: \({\mathbb{C}}^ 3\to {\mathbb{C}}\) be a germ of a function with a one-dimensional reduced singular locus \(\Sigma\). The \(T^ 1(\Sigma,f)\) is only zero if f is right equivalent to the \(A_{\infty}\), \(D_{\infty}\) or \(T_{\infty,\infty,\infty}\)-singularity.
And the Question: Is it true that for space curve singularities the inequality \(\dim_{{\mathbb{C}}}(N^*/I)\geq 3\cdot \dim_{{\mathbb{C}}}(\int I/I^ 2)\) holds?
Huneke has shown that for a space curve singularity the following holds: \[ \dim_{{\mathbb{C}}}(\int I/I^ 2)\geq \left( \begin{matrix} t-1\\ 2\end{matrix} \right), \] where t denotes the Gorenstein type of \(\Sigma\) i.e. the number of generators of the dualizing module \(\omega_{\Sigma}\). In particular, \(\int I/I^ 2\) is never zero if \(\sigma\) is not a complete intersection. This implies that for \(f\in I^ 2\), \(\Sigma\) not a complete intersection, the base space of the semi-universal admissible deformation has at least two components.
[For special notations see the paper itself.]
§ 3 deals with the
Conjecture: Let f: \({\mathbb{C}}^ 3\to {\mathbb{C}}\) be a germ of a function with a one-dimensional reduced singular locus \(\Sigma\). The \(T^ 1(\Sigma,f)\) is only zero if f is right equivalent to the \(A_{\infty}\), \(D_{\infty}\) or \(T_{\infty,\infty,\infty}\)-singularity.
And the Question: Is it true that for space curve singularities the inequality \(\dim_{{\mathbb{C}}}(N^*/I)\geq 3\cdot \dim_{{\mathbb{C}}}(\int I/I^ 2)\) holds?
Huneke has shown that for a space curve singularity the following holds: \[ \dim_{{\mathbb{C}}}(\int I/I^ 2)\geq \left( \begin{matrix} t-1\\ 2\end{matrix} \right), \] where t denotes the Gorenstein type of \(\Sigma\) i.e. the number of generators of the dualizing module \(\omega_{\Sigma}\). In particular, \(\int I/I^ 2\) is never zero if \(\sigma\) is not a complete intersection. This implies that for \(f\in I^ 2\), \(\Sigma\) not a complete intersection, the base space of the semi-universal admissible deformation has at least two components.
[For special notations see the paper itself.]
Reviewer: Sh.Ohyanagi (Tokyo)
MSC:
| 32S30 | Deformations of complex singularities; vanishing cycles |
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