Where can I find (book, notes, ...) the following result:
Theorem: A hyperelliptic Riemann surface $S$ of genus $g$ with two to one map $f: \longrightarrow \mathbb{P}^1$. Then $S$ is determined completely by the $2g + 2$ branch points of $f$.
Thank you!
Every compact Riemann surface with a 2 to 1 holomorphic map to the projective line $\mathbb P^1$ is in fact an Hyperelliptic Riemann surface i.e. a curve of the type $y^2=(z-b_1)(z-b_2)...(z-b_{2g+2})$. Thanks to the Riemann existence theorem one can prove that given the map f and given the 2g+2 branch points the surface is completely determined up to isomorphism. You can find everything on Rick Miranda's "algebraic curves and riemann surfaces".