Affine modifications and affine hypersurfaces with a very transitive automorphism group

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(x₁, . . . , x_k) where p ∈ C[x₁, . . . , 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.