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Let $\Sigma$ be a closed Riemann surface of genus $g$. In the book of Farkas and Kra, they prove that there exists a holomorphic covering map $F : \Sigma \to \mathbb{S}^2$ of degree less than or equal to $\left \lfloor{\frac{g+3}{2}}\right \rfloor$ (where $\mathbb{S}^2$ is the unit sphere). Can this bound be improved?

In a similar vein, if we now assume that $\Sigma$ has $r$ boundary components, Gabard showed that there exists a branched holomorphic covering $G : \Sigma \to \mathbb{D}$ satisfying $\deg G \leq g + r$, improving a theorem of Ahlfors (where $\mathbb{D}$ is the unit disk). Do you believe this bound is optimal?

More generally, do you know similar results?

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    $\begingroup$ Denote by $d$ the least such degree for a closed Riemann surface of genus $g$ of general moduli. By the Riemann-Hurwitz formula, the number of branch points equals $b=2g+2d-2$. The number of moduli of $b$ points on a genus $0$ closed Riemann surface equals $b-3 = 2g+2d-5$. Since the number of moduli of genus $g$ curves equals $3g-3$, it follows that $2g+2d-5 \geq 3g-3$, i.e., $2d\geq g+2$. Therefore the least degree for a genus-$g$ closed Riemann surface satisfies $d\geq \frac{g+2}{2}$. $\endgroup$
    – Jason Starr
    Commented Aug 20, 2019 at 10:44

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