5

Show that a compact Riemann surface admits a branched cover of the sphere with only simple branch points.

I have this problem and it seems to me that the way is to use Riemann Roch theorem, but I do not know how to exactly apply!

Any help is welcome.

Manoel
  • 2,216

1 Answers1

1

Choose a branch set $D$ of size greater than or equal to the genus. Then by Riemann-Roch, the dimension of the space of meromorphic functions branching on $D$ with at worst simple poles is greater than or equal to $1$.

Siran Victor Li
  • 69
  • Can't these functions still have non-simple branched points? Branched points need not to be poles. – flowing Jun 14 '23 at 07:06