Embeddings of hypersurfaces in affine spaces

In this paper, we address the following two general problems: given two alg ebraic varieties in C-n, find out whether or not they are (1) isomorphic an d (2) equivalent under an automorphism of C-n. Although a complete solution of either of those problems is out of the question at this time, we give h ere some handy and useful invariants of isomorphic as well as of equivalent varieties. Furthermore, and more importantly, we give a universal procedur e for obtaining all possible algebraic varieties isomorphic to a given one and use it to construct numerous examples of isomorphic but inequivalent al gebraic varieties in C-n. Among other things, we establish the following in teresting fact: for isomorphic hypersurfaces {p(x(1),...,x(n)) = 0} and {q( x(1),..., x(n)) = 0}, the number of zeros of gad(p) might be different from that of grad(q). This implies, in particular, that, although the fibers {p = 0} and {q = 0} are isomorphic, there are some other fibers {p = c} and { q = c} which are not. We construct examples like this for any n greater tha n or equal to 2. (C) 2001 Academic Press.