Non-extendable isomorphisms between affine varieties

In this paper, we report several large classes of affine varieties (over an arbitrary field K of characteristic 0) with the following property: each variety in these classes has an isomorphic copy such that the corresponding isomorphism cannot be extended to an automorphism of the ambient affine space K-n. This implies, in particular. that each of these varieties has at least two inequivalent embeddings in K-n. The following application of our results seems interesting: we show that lines in K-2 are distinguished among irreducible algebraic retracts by the property of having a unique embedding in K-2. (C) 2001 Elsevier Science B.V. All rights reserved.