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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.20593 |
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| _version_ | 1866917227575377920 |
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| author | Balwe, Chetan Gupta, Nidhi |
| author_facet | Balwe, Chetan Gupta, Nidhi |
| contents | For any smooth quadratic hypersurface $X$ in $\mathbb A^n_k$, we use the iterations of the functor of naive $\mathbb{A}^1$-connected components $\mathcal{S}$ to study the field-valued sections of the sheaf of $\mathbb{A}^1$-connected components $π_0^{\mathbb{A}^1}(X)$ of $X$. We prove that for any field $F/k$, the canonical isomorphism $π_0^{\mathbb{A}^1}(X)(F) \xrightarrow{\sim} \lim_{n} \mathcal{S}^n(X)(F)$ stabilizes at $n=2$, meaning that $π_0^{\mathbb{A}^1}(X)(F)=\mathcal{S}^2(X)(F)$. Furthermore, by combining this result with Morel's characterization of $\mathbb{A}^1$-connected spaces in terms of the triviality of field-valued sections of $π_0^{\mathbb{A}^1}$, we provide a complete characterization of $\mathbb{A}^1$-connected smooth quadratic hypersurfaces in $\mathbb{A}^n_k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20593 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $\mathbb A^1$-connected components of affine quadrics Balwe, Chetan Gupta, Nidhi Algebraic Geometry 14F42 For any smooth quadratic hypersurface $X$ in $\mathbb A^n_k$, we use the iterations of the functor of naive $\mathbb{A}^1$-connected components $\mathcal{S}$ to study the field-valued sections of the sheaf of $\mathbb{A}^1$-connected components $π_0^{\mathbb{A}^1}(X)$ of $X$. We prove that for any field $F/k$, the canonical isomorphism $π_0^{\mathbb{A}^1}(X)(F) \xrightarrow{\sim} \lim_{n} \mathcal{S}^n(X)(F)$ stabilizes at $n=2$, meaning that $π_0^{\mathbb{A}^1}(X)(F)=\mathcal{S}^2(X)(F)$. Furthermore, by combining this result with Morel's characterization of $\mathbb{A}^1$-connected spaces in terms of the triviality of field-valued sections of $π_0^{\mathbb{A}^1}$, we provide a complete characterization of $\mathbb{A}^1$-connected smooth quadratic hypersurfaces in $\mathbb{A}^n_k$. |
| title | $\mathbb A^1$-connected components of affine quadrics |
| topic | Algebraic Geometry 14F42 |
| url | https://arxiv.org/abs/2601.20593 |