Affine modifications and affine hypersurfaces with a very transitive automorphism group

We study the modification A bar right arrow 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]. I n [Ka1] its geometric counterpart was applied to construct contractible smo oth affine varieties non-isomorphic to Euclidean spaces. Here we provide ce rtain conditions (more general than those in [Ka1]) which guarantee preserv ation of the topology under a modification. As an application, we show that the group of biregular automorphisms of the affine hypersurface X subset of Ck+2, gi,,, by the equation uv = p(x(1),.. .,x(k)) where p is an element of C[x(1),...,x(k)], k greater than or equal to 2, acts m-transitively on the smooth part regX of X for any m is an elem ent of N. We present examples of such hypersurfaces diffeomorphic to Euclid ean spaces.