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

Abstract

Let $K$ be a field. We study $\A^1$-contractibility of Koras--Russell type varieties defined by \[ \frac{K[x_1,\ldots,x_m,y,z,t]} {\langle x_m^2a(x_m)b(x_1,\ldots,x_{m-1})y+f(z,t)+x_m\rangle}. \] We prove that if such a variety is $\A^1$-contractible, then the plane curve $\Gamma=\mathrm{Spec}(K[z,t]/(f))$ has only unibranched singularities. Over a perfect field, we show moreover that the normalization of $\Gamma$ is $\A_K^1$ and that $\Gamma$ and $\A_K^1$ represent isomorphic Nisnevich sheaves on $Sm_K$; over an arbitrary field, the corresponding statement holds after base change to an algebraic closure. We also prove that, in characteristic zero, singular $\A^1$-contractible affine curves are rational and can have at most unibranched singularities. Using this criterion for $\A^1$-contractible curves, over algebraically closed fields of characteristic zero, we give sufficient conditions for stable $\A^1$-contractibility of the Koras-Russell type varieties in terms of $\A^1$-contractibility of the associated plane curves $\{f(z,t)=\lambda\}$ appearing in the fiber of the morphism $\mathrm{Spec}\,A \to \Spec(K[x_m])$. 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.