Necessary and sufficient conditions for ${\text{A}}^1$-contractibility of Koras-Russell type varieties

Let $K$ be a field. We study ${\text{A}}^1$-contractibility of Koras–Russell type varieties defined by $$\frac{K[x_1,\dots ,x_m,y,z,t]}{⟨x_m^{2}a(x_m)b(x_1,\dots ,x_{m-1})y+f(z,t)+x_m⟩}\mathrm{.}$$ We prove that if such a variety is ${\text{A}}^1$-contractible, then the plane curve $\mathrm{\Gamma }=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(K[z,t]\mathrm{/}(f))$ has only unibranched singularities. Over a perfect field, we show moreover that the normalization of $\mathrm{\Gamma }$ is ${\text{A}}_K^1$ and that $\mathrm{\Gamma }$ and ${\text{A}}_K^1$ represent isomorphic Nisnevich sheaves on $S{m}_K$; over an arbitrary field, the corresponding statement holds after base change to an algebraic closure. We also prove that, in characteristic zero, singular ${\text{A}}^1$-contractible affine curves are rational and can have at most unibranched singularities. Using this criterion for ${\text{A}}^1$-contractible curves, over algebraically closed fields of characteristic zero, we give sufficient conditions for stable ${\text{A}}^1$-contractibility of the Koras-Russell type varieties in terms of ${\text{A}}^1$-contractibility of the associated plane curves $\{f(z,t)=\lambda \}$ appearing in the fiber of the morphism $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}A\to \text{Spec}(K[x_m])$. Further we show that, these results have application, to prove rectifiability of a family of embeddings between affine spaces, giving an evidence towards the Abhyankar–Sathaye embedding conjecture.