# Branched covers of the sphere branched over few points

Let $$X$$ be a compact Riemann surface of genus $$g\geq 2$$. By the Riemann-Roch theorem, $$X$$ is a branched cover of the sphere, branched over finitely many branched values. What is the smallest number of such values, for general $$X$$ of genus $$g$$?

Since moduli space has complex dimension $$3g-3$$, and branched covers branched over $$B$$ points are contained in the union of countably many varieties of dimension at most $$B-3$$, for general $$X$$ we need at least $$3g$$ branched values.

Riemann-Roch shows that there is a holomorphic $$X\to\hat{\mathbb{C}}$$ of degree $$g+1$$ having a single pole, of degree $$g+1$$. By Riemann-Hurwitz, this function has $$4g$$ critical points, of which $$g$$ are over infinity. So in total there are at most $$3g+1$$ critical values. So my question becomes:

Question. Let $$X$$ be a compact Riemann surface of genus $$g\geq 2$$. Is there a holomorphic function $$f\colon X\to \hat{\mathbb{C}}$$ which has at most $$3g$$ critical values?

It is plausible that the answer is positive, but in either case the answer is surely known. Does anyone know a reference?

I have been told that Brill-Noether theory shows the existence of a meromorphic function of degree $$\lfloor(g+3)/2\rfloor$$ on $$X$$. For even $$g$$, applying Riemann-Hurwitz then shows that there are $$3g$$ critical points for this function. That answers the question in the positive for even $$g$$. But for odd $$g$$, we get the same number $$3g+1$$ as via Riemann-Roch.

• You just need to apply your argument with a Weierstrass point — that is a point $p$ for which there exists a meromorphic function with a pole of order $k\leq g$ at $p$ (they always exist). Then the number of critical values is at most $2g+k\leq 3g$.
– abx
Commented Apr 2, 2021 at 16:31
• @abx Thanks! Alex Eremenko just pointed out the same thing to me. :D Do you want to post this as an answer? Commented Apr 2, 2021 at 17:50

Let me post my comment as an answer. Take a Weierstrass point on $$X$$, that is, a point $$P$$ for which there exists a meromorphic function $$f$$ with a pole of order $$k\leq g$$ at $$P$$ (there always exists such a point). Then apply the argument in the post: by Riemann-Hurwitz the number of critical points of $$f$$, counted with multiplicity, is $$2g-2+2k$$. But $$P$$ appears with multiplicity $$k-1$$ in this count, so the number of critical values outside $$\infty$$ is $$\leq 2g-2+2k-(k-1)=2g+k-1$$, and the total number of critical values is $$\leq 2g+k\leq 3g$$.
• Do you think that for each $g$ and each $0<k<3g-3$ (i.e. avoiding this case and Belyi case) it's open whether every curve of genus $g$ defined over an algebraically closed field of transcendence degree $k$ has a map to $\mathbb P^1$ branched over at most $k+3$ points? Commented Apr 3, 2021 at 0:53