Take the 2-minute tour ×  I have the following questions:    1.
Suppose the compact Riemann surface $C$ is an n-fold branched
cover of $\mathbb{P}^1$ branched at exactly four points
$x_1,x_2,x_3$ and $x_4$. I believe that $C$ is a smooth genus
$(n-1)$ Riemann surface. Is this true?    2. In case 1 is true,
what is the algebraic equation for this Riemann surface?
[Naively, I thought $y^n=(x-x_1)(x-x_2)(x-x_3)(x-x_4)$ would do
the job, but this surface is singular when it is given as a sub-
variety of $\mathbb{P}^2$. Do I have to embed the initial
$\mathbb{P}^1$ in a more complicated weighted projective space?
How do I resolve the singularity to get a smooth genus $(n-1)$
surface?]    3. Given the algebraic equation for the Riemann
surface $C$ of question 2, what are the $(n-1)$ holomorphic
differentials?    4. I have a basic knowledge of Riemann
surfaces obtained through undergraduate/early graduate level
courses, but would like to learn more. In particular I would
like to be address problems similar to questio