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 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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- Mathematics(all)
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Affine modifications and affine hypersurfaces with a very transitive automorphism group. / Kaliman, Shulim; Zaidenberg, M.
In: Transformation Groups, Vol. 4, No. 1, 1999, p. 53-95.Research output: Contribution to journal › Article
}
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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M3 - Article
VL - 4
SP - 53
EP - 95
JO - Transformation Groups
JF - Transformation Groups
SN - 1083-4362
IS - 1
ER -