2003/10/05 by Leonid Makar-Limanov, Peter van Rossum, Makar-Limanov, Leonid +5
Mathematics · #14E09 #14E25 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.math/0310060
openalex publication_date 2003/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be an arbitrary field of characteristic 0, and \Affn the n-dimensional affine space over K. A well-known cancellation problem asks, given two algebraic varieties V1, V2 ⊆ \Affn with isomorphic cylinders V1 × \Aff1 and V2 × \Aff1, 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 \Affn+1) cylinders V1 × \Aff1 and V2 × \Aff1, are V1 and V2 equivalent under an automorphism of \Affn? We call this stable equivalence problem. We show that the answer is positive for any two curves V1, V2 ⊆ \Aff2. 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.