Anabelian geometry for Deligne-Mumford curves
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918481188880384 |
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| author | Collas, Benjamin Philip, Séverin Yamaguchi, Naganori |
| author_facet | Collas, Benjamin Philip, Séverin Yamaguchi, Naganori |
| contents | We develop an anabelian framework for general Deligne-Mumford curves, showing that their stack and orbifold structures are encoded in the group-theoretic properties of their étale fundamental groups. After establishing the required properties for profinite F-groups, we prove that fundamental geometric features, including hyperbolicity, affineness, and inertia data, can already be detected from low-level solvable quotients of the associated profinite groups, namely at the optimal 3-step level. As a consequence, we obtain some anabelian reconstruction results for Deligne-Mumford curves, their rigidifications, and their coarsification. While the m-step Grothendieck conjecture doesn't hold for Deligne-Mumford curves, we establish a 5-step anabelian theorem for the rigidification of affine Deligne-Mumford curves, namely affine stacky curves. A certain emphasis is given to the role of stack inertia groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02577 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Anabelian geometry for Deligne-Mumford curves Collas, Benjamin Philip, Séverin Yamaguchi, Naganori Algebraic Geometry 14A20, 14H30, 20E18 (Primary) 14G32, 20F34, 14H37 (Secondary) We develop an anabelian framework for general Deligne-Mumford curves, showing that their stack and orbifold structures are encoded in the group-theoretic properties of their étale fundamental groups. After establishing the required properties for profinite F-groups, we prove that fundamental geometric features, including hyperbolicity, affineness, and inertia data, can already be detected from low-level solvable quotients of the associated profinite groups, namely at the optimal 3-step level. As a consequence, we obtain some anabelian reconstruction results for Deligne-Mumford curves, their rigidifications, and their coarsification. While the m-step Grothendieck conjecture doesn't hold for Deligne-Mumford curves, we establish a 5-step anabelian theorem for the rigidification of affine Deligne-Mumford curves, namely affine stacky curves. A certain emphasis is given to the role of stack inertia groups. |
| title | Anabelian geometry for Deligne-Mumford curves |
| topic | Algebraic Geometry 14A20, 14H30, 20E18 (Primary) 14G32, 20F34, 14H37 (Secondary) |
| url | https://arxiv.org/abs/2605.02577 |