The Stable Equivalence and Cancellation Problems

Let $K$ be an arbitrary field of characteristic 0, and ${\text{Aff}}^n$ the $n$-dimensional affine space over $K$. A well-known cancellation problem asks, given two algebraic varieties $V_1,V_2\subseteq {\text{Aff}}^n$ with isomorphic cylinders $V_1\times {\text{Aff}}^1$ and $V_2\times {\text{Aff}}^1$, 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 ${\text{Aff}}^{n+1}$) cylinders $V_1 \times \Aff^1$and$V_2 \times \Aff^1$,are$V_1$and$V_2$ equivalent under an automorphism of ${\text{Aff}}^n$? We call this stable equivalence problem. We show that the answer is positive for any two curves $V_1,V_2\subseteq {\text{Aff}}^2$. 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.