##
**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)].

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

### MSC:

14B07 | Deformations of singularities |

14J17 | Singularities of surfaces or higher-dimensional varieties |

32S30 | Deformations of complex singularities; vanishing cycles |

14B05 | Singularities in algebraic geometry |

### Keywords:

simultaneous resolution; discriminant of semi-universal; deformation; normal surface singularity; Artin component; rational singularities; Weyl groups### References:

[1] | M. Artin : Algebraic construction of Briesknorn’s resolutions . J. of Algebra 29 (1974) 330-348. · Zbl 0292.14013 |

[2] | E. Brieskorn : Singular elements of semi simple algebraic groups . In: Actes Congress Intern. Math.(1970) t.2 279-284. · Zbl 0223.22012 |

[3] | D. Burns and M. Rapaport : On the Torelli problem for Kählerian K3-surfaces . Ann. Sci. Ecole Norm. Sup. 8 (1975) 235-274. · Zbl 0324.14008 |

[4] | D. Burns and J. Wahl : Local contributions to global deformations of surfaces . Invent. math. 26 (1974) 67-88. · Zbl 0288.14010 |

[5] | A. Fujuki and S. Nakano : Supplement to on the inverse of monoidal transformations . Publ. Res. Inst. Math. Sci. Kyoto Univ. 7 (1972) 637-644. · Zbl 0234.32019 |

[6] | E. Horikawa : Algebraic surfaces of general type with small c21 . II. Invent. math. 37 (1976) 121-155. · Zbl 0339.14025 |

[7] | U. Karras : Methoden zur Berechnung von algebraischen Invarianten und zur Konstruktion von Deformationen normaler Flächensingularitäten. Habilitationsschrift Dortmund (1981). |

[8] | U. Karras : Normally flat deformations of rational and minimally elliptic singularities . Proc. of Symp. in Pure Math. 40(I) (1983) 619-639. · Zbl 0531.14003 |

[9] | U. Karras : On pencils of elliptic curves and deformations of minimally elliptic singularities . Math. Ann. 247 (1980) 43-65. · Zbl 0407.14014 |

[10] | H. Laufer : Deformations of resolutions of two-dimensional singularities . Rice Univ. · Zbl 0281.32009 |

[11] | H. Laufer : Ambient deformations for exceptional sets in two-manifolds . Invent. math. 55 (1979) 1-36. · Zbl 0509.32010 |

[12] | H. Laufer : Versal deformations for two-dimensional pseudoconvex manifolds. Annali d . Scuola Norm. Sup. di Pisa 7 (1980) 511-521. · Zbl 0512.32016 |

[13] | H. Laufer : Lifting cycles to deformations of two-dimensional pseudoconvex manifolds . Trans. of the A.M.S. 296 (1981) 183-202. · Zbl 0512.32017 |

[14] | L. Lipman : Double point resolutions of deformations of rational singularities . Comp. Math. 38 (1979) 37-42. · Zbl 0405.14010 |

[15] | O. Riemenschneider : Familien komplexer Räume mit streng pseudoknovexer spezieller Faser . Comm. Math. Helv. 51 (1977) 547-565. · Zbl 0338.32013 |

[16] | J. Wahl : Vanishing theorems for resolutions of surface singularities . Invent. math. 31 (1975) 17-41. · Zbl 0314.14010 |

[17] | J. Wahl : Equisingular deformations of normal surface singularities . Ann. of Math. 104 (1976), 325-356. · Zbl 0358.14007 |

[18] | J. Wahl : Simultaneous resolution of rational singularities . Comp. Math. 38 (1979) 43-54. · Zbl 0412.14008 |

[19] | J. Wahl : Simultaneous resolution and discriminantal loci . Duke Math. J. 46 (1979) 341-375. · Zbl 0472.14002 |

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