The stable equivalence and cancellation problems

Abstract

Let K be an arbitrary field of characteristic 0, and An the n-dimensional affine space over K. A well-known cancellation problem asks, given two algebraic varieties V1, V2 ⊆ An with isomorphic cylinders V1 × A1 and V2 × A1, whether V1 and V2 themselves are isomorphic. In this paper, we focus on a related problem: given two varieties with equivalent (under an automorphism of An+1) cylinders V 1 × A1 and V2 × A1, are V1 and V2 equivalent under an automorphism of A n ? We call this stable equivalence problem. We show that the answer is positive for any two curves V1, V2 ⊆ A 2. For an arbitrary n ≥ 2, we consider a special, arguably the most important, case of both problems, where one of the varieties is a hyperplane. We show that a positive solution of the stable equivalence problem in this case implies a positive solution of the cancellation problem.