Abstract
We study the modificationA→A′ of an affine domainA which produces another affine domainA′=A[I/f] whereI is a nontrivial ideal ofA andf is a nonzero element ofI. 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 hypersurfaceX⊂C
k+2, given by the equationuv=(p(x
1,...,xk) wherep∈C[x
1,...,x
k
],k≥2, actsm-transitively on the smooth part regX ofX for anym∈N. We present examples of such hypersurfaces diffeomorphic to Euclidean spaces.
