About maps induced for a divisor D in P^1  Let $D$ be an effective $ \mathbb{Z}$-divisor on $ \mathbb{P}^1$. Is there a form to associate a curve $C$, and morphism $C \to P^1$ to the divisor $D$ ?  For example, let $Y$ be a singular plane curve, sometimes, we can built a cover of $ \mathbb{P}^2$ that branches along $Y$.  Is there a similar construction for covers of $\mathbb{P}^1$ ?  I will appreciate any reference in this direction, or anything related? 
Thanks
 A: Yes, this is the famous Riemann Existence Theorem.
In its general form, it can be stated as follows.
Theorem. Let $Y$ be a compact Riemann surface, and $D \subset Y$ be an effective reduced divisor. Then there is a $1$-$1$ correspondence between the following sets:
$\mathbf{1)}$  finite covers $f \colon X \to Y$ of degree $d$ whose branch locus lies in $D$, up to isomorphism;
$\mathbf{2)}$  group homomorphism $\rho \colon \pi_1(Y - D) \to S_d$ with transitive image, up to conjugacy in $S_d$.
Now, the fundamental group of $\mathbb{P}^1 -\{b_1, \ldots, b_k\}$ is the group generated by $k$ generators $\gamma_1, \cdots, \gamma_k$ with the unique relation $\gamma_1 \gamma_2 \cdots \gamma_k=1$, where each $\gamma_i$ is the homotopy class of a small loop on $\mathbb{P}^1$ around $b_i$. Then we obtain the following corollary:
Corollary. Let $D=\{b_1, \ldots, b_k\} \subset \mathbb{P}^1$ be a finite set of points.Then there is a $1$-$1$ correspondence between the following sets:
$\mathbf{1)}$  finite covers $f \colon X \to \mathbb{P}^1$ of degree $d$ whose branch locus lies in $D$, up to isomorphism;
$\mathbf{2)}$ conjugacy classes of $k$-tuples $(\sigma_1, \ldots, \sigma_k)$ of permutations in $S_d$ such that $\sigma_1 \sigma_2 \cdots \sigma_k=1$ and the subgroup generated by the $\sigma_i$ is transitive. 
Notice that neither $X$ nor the degree  $d$ of the cover are uniquely determined by $D$. 
For more details, see Miranda's book [Algebraic curves and Riemann surfaces, p. 90-92]. 
