### 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) |
---|---|

Pages (from-to) | 53-95 |

Number of pages | 43 |

Journal | Transformation Groups |

Volume | 4 |

Issue number | 1 |

State | Published - 1999 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Transformation Groups*,

*4*(1), 53-95.

**Affine modifications and affine hypersurfaces with a very transitive automorphism group.** / Kaliman, Shulim; Zaidenberg, M.

Research output: Contribution to journal › Article

*Transformation Groups*, vol. 4, no. 1, pp. 53-95.

}

TY - JOUR

T1 - Affine modifications and affine hypersurfaces with a very transitive automorphism group

AU - Kaliman, Shulim

AU - Zaidenberg, M.

PY - 1999

Y1 - 1999

N2 - 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 ⊂ Ck+2, given by the equation uv = p(x1, . . . , xk) where p ∈ C[x1, . . . , xk], 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.

AB - 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 ⊂ Ck+2, given by the equation uv = p(x1, . . . , xk) where p ∈ C[x1, . . . , xk], 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=0033409410&partnerID=8YFLogxK

M3 - Article

VL - 4

SP - 53

EP - 95

JO - Transformation Groups

JF - Transformation Groups

SN - 1083-4362

IS - 1

ER -