The stable equivalence and cancellation problems

Abstract

Let K be an arbitrary field of characteristic 0, and $\mathbf{A}^n$ the n-dimensional affine space over K. A well-known cancellation problem asks, given two algebraic varieties $V_1, V_2 \subseteq \mathbf{A}^n$ with isomorphic cylinders $V_1 \times \mathbf{A}^1$ and $V_2 \times \mathbf{A}^1$, whether $V_1$ and $V_2$ themselves are isomorphic. In this paper, we focus on a related problem: given two varieties with equivalent (under an automorphism of $\mathbf{A}^{n+1}$) cylinders $V_1 \times \mathbf{A}^1$ and $V_2 \times \mathbf{A}^1$, are $V_1$ and $V_2$ equivalent under an automorphism of $\mathbf{A}^n$? We call this stable equivalence problem. We show that the answer is positive for any two curves $V_1, V_2 \subseteq \mathbf{A}^2$. For an arbitrary $n \ge 2$, 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.