Affine modifications and affine hypersurfaces with a very transitive automorphism group

Sh Kaliman, M. Zaidenberg

Research output: Contribution to journalArticle

48 Scopus citations

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 ⊂ 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.

Original languageEnglish (US)
Pages (from-to)53-95
Number of pages43
JournalTransformation Groups
Volume4
Issue number1
DOIs
StatePublished - Jan 1 1999

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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