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A
note on rational maps with three branching points on the
Riemann sphere
Zhiqiang Wei, Yingyi Wu and Bin Xu |
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Algebraic & Geometric Topology 25 (2025) 5607–5617
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Abstract |
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Studying the existence of rational maps with given branching datum is a classical problem in the field of complex analysis and algebraic geometry. This problem dates back to Hurwitz and remains open to this day. We utilize complex analysis to establish a property of rational maps with 3 branching points on the Riemann sphere. Given two compact Riemann surfaces and , a pair of an integer and a collection of nontrivial partitions of is called a candidate branching datum if it satisfies the Riemann–Hurwitz formula. A candidate branching datum is considered exceptional when there is no holomorphic map realization for it. As applications of our main theorem, we present some new types of exceptional branching datum which cover some previously known results. We also deduce the realizability of a certain type of candidate branching datum on the Riemann sphere. |
Keywords
branched cover, Hurwitz existence problem, rational map
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Mathematical Subject Classification
Primary: 57M12
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References |
Publication
Received: 23 January 2024
Revised: 3 September 2024
Accepted: 4 November 2024
Published: 18 December 2025
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Authors |
| Zhiqiang Wei | |
| School of Mathematics and
Statistics Henan University Kaifeng China |
Center for Applied Mathematics of
Henan Province Henan University Zhengzhou China |
| Yingyi Wu | |
| School of Mathematical
Sciences University of Chinese Academy of Sciences Beijing China |
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| Bin Xu | |
| CAS Wu Wen-Tsun Key Laboratory of
Mathematics and School of Mathematical Sciences University of Science and Technology of China Hefei China |
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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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