On the discriminant of the Artin component. (English) Zbl 0596.14002

Let (V,p) be a normal surface singularity, and \(\pi : M\to V\) be the minimal resolution. We assume that (V,p) is rational, i.e. \(R^ 1\pi_*{\mathcal O}_ M=0\). Let \(\omega : {\mathcal M}\to R\) be a 1-convex flat representative of the semi-universal deformation of the germ of M at the exceptional set, where R is a complex manifold. Then \(\omega\) can be simultaneously blown down to a deformation of the singularity (V,p). There is an induced holomorphic map germ \(\phi : (R,0)\to (S,0)\) to the base space of the semi-universal deformation \(\vartheta : ({\mathcal V},p)\to (S,0)\) of (V,p). It was proved by M. Artin that \(\phi\) is finite onto its image, and that the germ of its image \(S_ a=\phi (R)\) is an irreducible component of the base space (S,0). This component is called the Artin component. - If (V,p) is a rational double point (RDP), then \((S_ a,0)\) is the whole base space (S,0). It was shown by E. Brieskorn that in this case the map germ \(\phi\) can be represented by a Galois covering with group W. Here the weighted dual graph associated to the minimal resolution of such a singularity is a Dynkin diagram of type \(A_ k, D_ k, E_ 6, E_ 7\) or \(E_ 8\), and W is the corresponding Weyl group. Furthermore the discriminant \(\Delta\) \(\subset S\) of the semi- universal deformation \(\vartheta\) is an irreducible hypersurface such that the fiber over a generic point of \(\Delta\) has only one singularity which is an ordinary double point.
The main results of the paper are generalizations of these results to arbitrary rational singularities. The author shows that in the general case \(\phi\) can be represented by a Galois covering with group W which is isomorphic to a direct product of Weyl groups corresponding to the maximal RDP-configurations on the minimal resolution of the given rational singularity. He describes the components of the discriminant \(\Delta_ a=\Delta \cap S_ a\) of the Artin component and the singularities of the fibers corresponding to generic points of \(\Delta_ a.\)
In the formal category the same results are valid and were obtained earlier under an additional assumption by J. Wahl [Compos. Math. 38, 43-54 (1979; Zbl 0412.14008) and Duke Math. J. 46, 341-375 (1979; Zbl 0472.14002)].
Reviewer: W.Ebeling


14B07 Deformations of singularities
14J17 Singularities of surfaces or higher-dimensional varieties
32S30 Deformations of complex singularities; vanishing cycles
14B05 Singularities in algebraic geometry
Full Text: Numdam EuDML


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