On the family of affine threefolds $a(x)y=F(x,z,t)$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909278727569408 |
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| author | Ghosh, Parnashree Gupta, Neena Pal, Ananya |
| author_facet | Ghosh, Parnashree Gupta, Neena Pal, Ananya |
| contents | In recent decades, linear affine threefolds have enabled researchers to solve some of the challenging problems on affine spaces. Koras-Russell threefolds, especially the Russell Cubic over $\mathbb{C}$ and Asanuma threefolds over a field of positive characteristic, are striking examples of such linear threefolds.In this paper, we apply tools from $K$-theory and theory of $\mathbb{G}_a$-actions to linear threefolds of the form $G:=a(X)Y-F(X,Z,T)\in k[X,Y,Z,T]$, over an arbitrary field $k$. We give some equivalent conditions for $G$ to be a hyperplane (i.e., $k[X,Y,Z,T]/(G)=k^{[3]}$) in the following cases: (i) $k$ is a field of characteristic zero (ii) $k$ is an arbitrary field and $a(X)$ has only multiple roots. We also establish the Abhyankar-Sathaye Conjecture affirmatively in these cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17397 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the family of affine threefolds $a(x)y=F(x,z,t)$ Ghosh, Parnashree Gupta, Neena Pal, Ananya Algebraic Geometry Commutative Algebra Primary: 14R10, Secondary: 13B25, 13A50, 13A02 In recent decades, linear affine threefolds have enabled researchers to solve some of the challenging problems on affine spaces. Koras-Russell threefolds, especially the Russell Cubic over $\mathbb{C}$ and Asanuma threefolds over a field of positive characteristic, are striking examples of such linear threefolds.In this paper, we apply tools from $K$-theory and theory of $\mathbb{G}_a$-actions to linear threefolds of the form $G:=a(X)Y-F(X,Z,T)\in k[X,Y,Z,T]$, over an arbitrary field $k$. We give some equivalent conditions for $G$ to be a hyperplane (i.e., $k[X,Y,Z,T]/(G)=k^{[3]}$) in the following cases: (i) $k$ is a field of characteristic zero (ii) $k$ is an arbitrary field and $a(X)$ has only multiple roots. We also establish the Abhyankar-Sathaye Conjecture affirmatively in these cases. |
| title | On the family of affine threefolds $a(x)y=F(x,z,t)$ |
| topic | Algebraic Geometry Commutative Algebra Primary: 14R10, Secondary: 13B25, 13A50, 13A02 |
| url | https://arxiv.org/abs/2403.17397 |