Let $C$ be an algebraic curve in $\mathbb P^2( \mathbb C)$ with singular points $p_i : \{1 \le i \le n \}$ . Then there exists a holomorphic map $\Phi : S \to C$ , where $S$ is a Riemann surface.
How to i construct such a map , considering the fact that its a map from a surface to a curve , it doesn't look obvious since the open sets are different on a surface than on a curve . Will the constructed surface have Hausdroff and compactness property ? Your help is appreciated . Thanks
The normalization $S$ of $C$ is a smooth projective algebraic curve over $\mathbb C$, so it is a Riemann surface. The normalization map $S\to C$ is holomorphic because it is even algebraic.