Questions on a Certain Branched Cover of the Two-sphere I have the following questions:


*

*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?

*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?]

*Given the algebraic equation for the Riemann surface $C$ of question 2, what are the $(n-1)$ holomorphic differentials?

*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 questions 1, 2, 3 on my own with relative ease. Do you have any references(on/offline) that I can learn from?
Thank you in advance!
 A: *

*The formula is true if the cover is branching is "total", that is, if all sheets of the cover join into a single point. This is a straightforward application of the Riemann-Hurwitz formula (algebraic geometry). However, as Jason Starr pointed out, if only some of the sheets join, the genus will be lower - another straightforward application of this formula.

*One way to get the correct surface is to blow up the singular point, which involves looking at $\mathbb P^2 \times \mathbb P^1$. Are you familiar with this process? Since your formula gives the correct birational equivalence class, blowing-up will give the correct surface.

*If you get one holomorphic differential, the other ones will be rational function multiples of it. You could figure out where it has zeroes and try to find enough rational functions that have poles there. But this probably isn't the best way. Another, somewhat silly, way is that if you know an embedding into a high-enough dimensional space, you can get all the holomorphic differentials as restrictions of holomorphic differentials on $\mathbb P^n$. You can get bigger-dimensional embeddings out of smaller ones by using the Veronese maps.

*This is fairly standard algebraic geometry, so I would suggest a standard algebraic geometry text like Hartshorne. However learning algebraic geometry is quite a bit of work just to answer these questions. I don't know good sources on it from a complex analytic perspective.
A: Look at the Riemann-Hurwitz [RH] formula carefully.
I recommend Plane Algebraic Curves by G. Fisher, or Complex Algebraic Curves
by Kirwan. All kinds of interesting things can happen, and in symmetric ways,
if 4 divides n. For example, you might try to have the inverse image of each
of branch point be exactly 3 points, and look for the local branching  at each branch point to be 2: as in $z \to z^2$.  The total degree will then be $6 = 2*3$. 
Does the Riemann surface exist? If so, its genus must be 3, by RH.
