On $\mathbb{A}^1$-contractibility of certain simple birational extensions of affine spaces

Abstract

Over a field of characteristic zero, upto isomorphism of varieties, affine spaces are the only smooth $\mathbb{A}^1$-contractible affine varieties in dimensions $\leqslant 2$. However, in dimensions $\geqslant 3$, examples of smooth $\mathbb{A}^1$-contractible affine varieties, not isomorphic to affine spaces are constructed by Dubouloz, Fasel [2018] and Dubouloz, Ghosh [2026]. In this paper, we consider a generalized class of smooth affine varieties containing the examples of $\mathbb{A}^1$-contractible varieties by Dubouloz, Fasel [2018] and Dubouloz, Ghosh [2026], given as $$ a(x_m)b(x_1,\ldots,x_{m-1})y+f(z,t)+x_m=0, $$ and investigate when such a variety is actually isomorphic to an affine space. We establish that, for a large subfamilies of these varieties to be affine spaces, it is necessary and sufficient that the embedding of the corresponding hyperplane in $\mathbb{A}^{m+3}$ must be rectifiable, in the sense that, there exists an automorphism of the ambient space that takes it to a coordinate hyperplane. Thus our result naturally connects $\mathbb A^1$-contractibility of these varieties with the classical embedding problem for affine spaces in codimension one, and provide new evidences towards the conjecture of Abhyankar and Sathaye. A key ingredient in our approach is to study singular $\mathbb A^1$-contractible affine curves over perfect fields. We describe the possible singularities of such curves and obtain a generalization of the classical result of Lin-Zaidenberg for topologically contractible affine plane curves. We further construct new examples of $\mathbb{A}^1$-contractible affine varieties in every dimension $\geqslant 4$, in the sense that they are not isomorphic to the existing examples.