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A note on the Hurwitz problem and cone spherical metrics

Jijian Song, Bin Xu, Yu Ye

Abstract

We are motivated by cone spherical metrics on compact Riemann surfaces of positive genus to solve a special case of the Hurwitz problem. Precisely speaking, letting $d,\,g$ and $\ell$ be three positive integers and $Λ$ be the following collection of $(\ell+2)$ partitions of a positive integer $d$: \[(a_1,\cdots, a_p),\,(b_1,\cdots, b_q),\,(m_1+1,1,\cdots,1),\cdots, (m_{\ell}+1,1,\cdots,1),\] where $(m_1,\cdots, m_{\ell})$ is a partition of $p+q-2+2g$, we prove that there exists a branched cover from some compact Riemann surface of genus $g$ to the Riemann sphere ${\Bbb P}^1$ with branch data $Λ$. An analogue for the genus-zero case was found by the first two authors ({\it Algebra Colloq.} {\bf 27} (2020), no. 2, 231-246), who were stimulated by such metrics on ${\Bbb P}^1$ and conjectured the veracity of the above statement there.

A note on the Hurwitz problem and cone spherical metrics

Abstract

We are motivated by cone spherical metrics on compact Riemann surfaces of positive genus to solve a special case of the Hurwitz problem. Precisely speaking, letting and be three positive integers and be the following collection of partitions of a positive integer : where is a partition of , we prove that there exists a branched cover from some compact Riemann surface of genus to the Riemann sphere with branch data . An analogue for the genus-zero case was found by the first two authors ({\it Algebra Colloq.} {\bf 27} (2020), no. 2, 231-246), who were stimulated by such metrics on and conjectured the veracity of the above statement there.
Paper Structure (5 sections, 8 theorems, 18 equations)

This paper contains 5 sections, 8 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Proof of Propostion \ref{['prop:subgroup']}
  3. Case $\ell=1$: three partitions
  4. Case $\ell>1$: more than three partitions
  5. Hurwitz numbers

Key Result

Theorem 1.1

Let $d$ and $\ell$ be positive integers. Suppose that a collection $\Lambda$ comprises $\ell + 2$ partitions of $d$ as follows: We define the total branch number $v(\Lambda)$ of $\Lambda$ to be $m_1+\cdots+m_\ell+(d-p)+(d-q)$. Then $\Lambda$ represents the branch data of a branched cover from a compact Riemann surface $X$ to ${\Bbb P}^1$ if and only if it satisfies one of the following two condit

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Proposition 1.4
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 5 more