A note on the Hurwitz problem and cone spherical metrics

TL;DR

This paper proves the existence of branched covers with specific branch data on compact Riemann surfaces of positive genus, motivated by cone spherical metrics, extending previous results from genus zero.

Contribution

It establishes a new existence theorem for branched covers with prescribed branch data on higher genus surfaces, generalizing earlier genus-zero results.

Findings

Existence of branched covers with given branch data on genus g surfaces.

Extension of genus-zero results to positive genus cases.

Motivated by cone spherical metrics on Riemann surfaces.

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 $\Lambda$ 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 $\Lambda$. 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.