The Calculus of Blowups on a Ruled Surface
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910258140545024 |
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| author | Birkett, Richard A. P. |
| author_facet | Birkett, Richard A. P. |
| contents | The purposes of this article are threefold. First, to determine numerically when an arbitrary blowup of a smooth surface is smooth. We show the surface is smooth if and only if certain rational parameters involving log discrepancy and multiplicity of the exceptional divisors form a generalised Farey sequence within the dual graph of divisors. Second, in doing the above we provide an exposition of the Berkovich projective line $\mathbb P^1_{\text{an}}(\mathbb K)$ over the Puiseux series as a universal dual graph for divisors on a ruled surface. Third, to explain how non-Archimedean skew products interact with this multiplicity structure of the tree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26598 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Calculus of Blowups on a Ruled Surface Birkett, Richard A. P. Algebraic Geometry Dynamical Systems 14H20, 14J17 (Primary) 14E15, 13A18, 37P50 (Secondary) The purposes of this article are threefold. First, to determine numerically when an arbitrary blowup of a smooth surface is smooth. We show the surface is smooth if and only if certain rational parameters involving log discrepancy and multiplicity of the exceptional divisors form a generalised Farey sequence within the dual graph of divisors. Second, in doing the above we provide an exposition of the Berkovich projective line $\mathbb P^1_{\text{an}}(\mathbb K)$ over the Puiseux series as a universal dual graph for divisors on a ruled surface. Third, to explain how non-Archimedean skew products interact with this multiplicity structure of the tree. |
| title | The Calculus of Blowups on a Ruled Surface |
| topic | Algebraic Geometry Dynamical Systems 14H20, 14J17 (Primary) 14E15, 13A18, 37P50 (Secondary) |
| url | https://arxiv.org/abs/2605.26598 |