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 $⩽2$. However, in dimensions $⩾3$, 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 $$a(x_m)b(x_1,\dots ,x_{m-1})y+f(z,t)+x_m=0,$$ 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 $⩾4$, in the sense that they are not isomorphic to the existing examples.

Some results are strengthened, and a new section with new results are added