Étale Extensions of Unipotent Torsors
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866915982070513664 |
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| author | Bassan, Gabriel |
| author_facet | Bassan, Gabriel |
| contents | In this paper we study extension problems for torsors in positive characteristic. Let $F$ be a field of characteristic $p>0$ and $U/F$ be a unipotent algebraic group. As our first main result, we prove that every $U$-torsor defined over the generic point of a discrete valuation ring $\mathcal{O}_{K}$, containing a field $F$, extends to the normalization of $\mathcal{O}_{K}$ in some finite separable extension of its fraction field. We then globalize this result and prove that for $X/F$ a normal integral curve over an algebraically closed field $F$, every $U$-torsor over an open set $X^{\circ}\subseteq X$ extends to some ramified cover of $X$ which is étale over $X^{\circ}$. As an application, we are able to find isomorphisms between certain unipotent variants of Nori's fundamental group scheme for curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04182 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Étale Extensions of Unipotent Torsors Bassan, Gabriel Algebraic Geometry 14L15, 14H30 In this paper we study extension problems for torsors in positive characteristic. Let $F$ be a field of characteristic $p>0$ and $U/F$ be a unipotent algebraic group. As our first main result, we prove that every $U$-torsor defined over the generic point of a discrete valuation ring $\mathcal{O}_{K}$, containing a field $F$, extends to the normalization of $\mathcal{O}_{K}$ in some finite separable extension of its fraction field. We then globalize this result and prove that for $X/F$ a normal integral curve over an algebraically closed field $F$, every $U$-torsor over an open set $X^{\circ}\subseteq X$ extends to some ramified cover of $X$ which is étale over $X^{\circ}$. As an application, we are able to find isomorphisms between certain unipotent variants of Nori's fundamental group scheme for curves. |
| title | Étale Extensions of Unipotent Torsors |
| topic | Algebraic Geometry 14L15, 14H30 |
| url | https://arxiv.org/abs/2605.04182 |