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



About maps induced for a divisor D in P^1 
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]. 


