Solving some exercises from Miranda's book, I stumble across cyclic covers of the projective line. In Holomorphic maps on the cyclic cover of the projective line I asked for a solution of one of these exercises.
Now, reading the solution and using the notations given there, I have that $Y/G$ is exactly the projective line.
I'm trying to answer to the following questions:
1) In general, given a compact Riemann surface $X$, there always exists an automorphism $\sigma$ such that $X/<\sigma>=\mathbb{P}^1$? May you provide some counterexamples or a complete proof (or some detailed reference)?
2) When it happens, are there conditions on the order of $\sigma$?
3) Assume that $X$ is a compact Riemann surface $X$ admitting an automorphism $\sigma$ such that $X/<\sigma> \simeq \mathbb{P}^1$. Does $\sigma$ fix at least a point? How many points does it fix? Is there some correlation with the genus of $X$?
4) Assume that $X$ is a cyclic cover of the projective line. Is there a birational transform such that the equation of the cover assumes the form
$$ y^n=\prod\limits_{i=1}^s (x-\alpha_i)^{m_i} $$
with $1 \le m_i \le n-1$ (and this is easy!) and with $\sum\limits_{i=1}^s m_i \equiv 0 \pmod{n}$ (this is the part I'm interested in!)?
1) A "generic" compact Riemann surfaces of genus $g >2 $ do not have automorphisms. See here.
Example: $\{ x^3y+y^3z+z^3(x-y)=0 \} \subset \mathbb{P}^2$ does not have nontrivial automorphisms.
2) Riemann-Hurwitz formula gives the restrictions. (Yes, anything can happen.)
3) You fall exactly in the situation discussed in page 73 and you have fixed points. I believe you can derive this in general from Lefschetz fixed-point theorem.
4) This is equivalent to removing the ramification from $x=\infty$. Maybe it is well-known, as in here it is stated without proof, but I don't know a proof for this "fact".