Necessary and sufficient conditions for $\A^1$-contractibility of Koras-Russell type varieties
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915984924737536 |
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| author | Ghosh, Parnashree |
| author_facet | Ghosh, Parnashree |
| contents | 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
$Γ=\mathrm{Spec}(K[z,t]/(f))$ has only unibranched singularities. Over a perfect field, we show moreover that the normalization of $Γ$ is $\A_K^1$ and that $Γ$ 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 $\{f(z,t)=λ\}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09166 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Necessary and sufficient conditions for $\A^1$-contractibility of Koras-Russell type varieties Ghosh, Parnashree Commutative Algebra Primary: 14R10, 14F42, Secondary: 14F45, 14R20, 14H20, 13A50 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 $Γ=\mathrm{Spec}(K[z,t]/(f))$ has only unibranched singularities. Over a perfect field, we show moreover that the normalization of $Γ$ is $\A_K^1$ and that $Γ$ 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 $\{f(z,t)=λ\}$ 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. |
| title | Necessary and sufficient conditions for $\A^1$-contractibility of Koras-Russell type varieties |
| topic | Commutative Algebra Primary: 14R10, 14F42, Secondary: 14F45, 14R20, 14H20, 13A50 |
| url | https://arxiv.org/abs/2506.09166 |