Structure and realizability for rational maps

2025/11/10 by Wei, Zhiqiang
#57M12 #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2511.06784

Abstract

We prove that if D is a collection of k nontrivial partitions of a positive integer d satisfying the Riemann-Hurwitz formula, and if k is greater than one plus the minimum length among the partitions in D, then D is realizable as the branch data of a rational map f \colon S² → S². As an application, we confirm a conjecture of Zheng [Topology Appl. 153 (2006), no.~12, 2124--2134].

Citations

• Hurwitz existence problem and fiber products
• A note on Rational Maps with three branching points on the Riemann sphere
• Solution of the Hurwitz problem with a length-2 partition
• Metrics of constant positive curvature with conic singularities. A survey
• Spherical metrics with conical singularities on a 2-sphere: angle\n constraints
• Branched covers of the sphere and the prime-degree conjecture
• Solution of the Hurwitz problem for Laurent polynomials
• On the existence of branched coverings between surfaces with prescribed branch data, II
• Gromov-Witten theory, Hurwitz theory, and completed cycles

Related

• A realizability criterion for rational maps with three branch points
• A note on Rational Maps with three branching points on the Riemann sphere
• Solution of the Hurwitz problem for Laurent polynomials
• Hurwitz existence problem and fiber products
• Rational maps with rational multipliers