# Existence of holomorphic coverings having small degree

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?

• 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 $d\geq \frac{g+2}{2}$. – Jason Starr Aug 20 '19 at 10:44