### Abstract

We study the modification A → A′ of an affine domain A which produces another affine domain A′ = A[I/f] where I is a nontrivial ideal of A and f is a nonzero element of I. First appeared in passing in the basic paper of O. Zariski [Zar], it was further considered by E. D. Davis [Da]. In [Ka1] its geometric counterpart was applied to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions (more general than those in [Ka1]) which guarantee preservation of the topology under a modification. As an application, we show that the group of biregular automorphisms of the affine hypersurface X ⊂ C^{k+2}, given by the equation uv = p(x_{1}, . . . , x_{k}) where p ∈ C[x_{1}, . . . , x_{k}], k ≥ 2, acts m-transitively on the smooth part regX of X for any m ∈ N. We present examples of such hypersurfaces diffeomorphic to Euclidean spaces. * Partially supported by the NSA grant MDA904-96-01-0012.

Original language | English (US) |
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Pages (from-to) | 53-95 |

Number of pages | 43 |

Journal | Transformation Groups |

Volume | 4 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1999 |

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology

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## Cite this

*Transformation Groups*,

*4*(1), 53-95. https://doi.org/10.1007/BF01236662