##
**Normally flat deformations of rational and minimally elliptic singularities.**
*(English)*
Zbl 0531.14003

Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 1, 619-639 (1983).

[For the entire collection see Zbl 0509.00008.]

Normally flat deformations of an isolated singularity may be characterized by the property that they admit a singular section such that the Hilbert Samuel function is constant along this section. Since such deformations cannot be smoothings, it turns out to be a very interesting problem to describe the singularities which appear generically in normally flat deformations. Good candidates should be cone singularities. The aim of this paper is to give an affirmative answer in the important special case of rational and minimally elliptic singularities. As an application, one obtains a necessary condition to smooth minimally elliptic singularities. - It is not difficult to show that rational and elliptic cone singularities are generic singularities in the sense above. The crucial point is to construct non-trivial normally flat deformations of a given rational or minimally elliptic singularity (V,p): Let M be the minimal resolution of (V,p). Then a deformation of M to which the fundamental cycle Z lifts blows down to a normally flat deformation of (V,p). So one has to check smoothability of Z via deformations of M. Using a method introduced by the author in Math. Ann. 247, 43-65 (1980; Zbl 0407.14014)], it remains to verify the existence of a reduced subcycle of Z which satisfies certain simple numerical conditions. As a first step, the author describes a general strategy how to construct such nice subcycle. However, the verification of this algorithmic procedure turns out to be no at all trivial and needs a lot of tedious and delicate combinatorial arguments.

Normally flat deformations of an isolated singularity may be characterized by the property that they admit a singular section such that the Hilbert Samuel function is constant along this section. Since such deformations cannot be smoothings, it turns out to be a very interesting problem to describe the singularities which appear generically in normally flat deformations. Good candidates should be cone singularities. The aim of this paper is to give an affirmative answer in the important special case of rational and minimally elliptic singularities. As an application, one obtains a necessary condition to smooth minimally elliptic singularities. - It is not difficult to show that rational and elliptic cone singularities are generic singularities in the sense above. The crucial point is to construct non-trivial normally flat deformations of a given rational or minimally elliptic singularity (V,p): Let M be the minimal resolution of (V,p). Then a deformation of M to which the fundamental cycle Z lifts blows down to a normally flat deformation of (V,p). So one has to check smoothability of Z via deformations of M. Using a method introduced by the author in Math. Ann. 247, 43-65 (1980; Zbl 0407.14014)], it remains to verify the existence of a reduced subcycle of Z which satisfies certain simple numerical conditions. As a first step, the author describes a general strategy how to construct such nice subcycle. However, the verification of this algorithmic procedure turns out to be no at all trivial and needs a lot of tedious and delicate combinatorial arguments.

### MSC:

14B07 | Deformations of singularities |

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

32S30 | Deformations of complex singularities; vanishing cycles |

32Sxx | Complex singularities |

14B05 | Singularities in algebraic geometry |

14C20 | Divisors, linear systems, invertible sheaves |

14D15 | Formal methods and deformations in algebraic geometry |

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |