Embeddings of Hypersurfaces in Affine Spaces

In this paper, we address the following two general problems: given two algebraic varieties in Cn, find out whether or not they are (1) isomorphic and (2) equivalent under an automorphism of Cn. Although a complete solution of either of those problems is out of the question at this time, we give here some handy and useful invariants of isomorphic as well as of equivalent varieties. Furthermore, and more importantly, we give a universal procedure for obtaining all possible algebraic varieties isomorphic to a given one and use it to construct numerous examples of isomorphic but inequivalent algebraic varieties in Cn. Among other things, we establish the following interesting fact: for isomorphic hypersurfaces {p(x1,…,xn) = 0} and {q(x1,…,xn) = 0}, the number of zeros of grad(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 ≥ 2.