Weakly special varieties, Campana stacks, and Remarks on Orbifold Mordell
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arXiv
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| Format: | Preprint |
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2026
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| author | Bartsch, Finn Javanpeykar, Ariyan |
| author_facet | Bartsch, Finn Javanpeykar, Ariyan |
| contents | We construct the first weakly special surfaces that are not Campana-special, including the complement of the plane curve $x^2y^3 = 1$ in $\mathbb{A}^2$. We prove that the set of $\mathcal{O}_{K,S}$-integral points on this surface is non-dense for every number field $K$ and finite set $S$ of finite places of $K$ if and only if Campana's Orbifold Mordell conjecture holds for $(\mathbb{G}_m, \tfrac{1}{2}[1])$. This basic example carries a natural $\mathbb{G}_m$-action, and the quotient stack is an Artin stack parametrizing points on a C-pair. This leads to the introduction of ``Campana stacks'', which encode morphisms of C-pairs in a manner analogous to the role of root stacks for integral points satisfying prescribed divisibility conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28745 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weakly special varieties, Campana stacks, and Remarks on Orbifold Mordell Bartsch, Finn Javanpeykar, Ariyan Algebraic Geometry Number Theory We construct the first weakly special surfaces that are not Campana-special, including the complement of the plane curve $x^2y^3 = 1$ in $\mathbb{A}^2$. We prove that the set of $\mathcal{O}_{K,S}$-integral points on this surface is non-dense for every number field $K$ and finite set $S$ of finite places of $K$ if and only if Campana's Orbifold Mordell conjecture holds for $(\mathbb{G}_m, \tfrac{1}{2}[1])$. This basic example carries a natural $\mathbb{G}_m$-action, and the quotient stack is an Artin stack parametrizing points on a C-pair. This leads to the introduction of ``Campana stacks'', which encode morphisms of C-pairs in a manner analogous to the role of root stacks for integral points satisfying prescribed divisibility conditions. |
| title | Weakly special varieties, Campana stacks, and Remarks on Orbifold Mordell |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2603.28745 |