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arXiv · 2605.26598

The Calculus of Blowups on a Ruled Surface

Richard A. P. Birkett
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Abstract

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.

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Richard A. P. Birkett. 2026-05-26. The Calculus of Blowups on a Ruled Surface. https://arxiv.org/abs/2605.26598

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