Consider $F(x,y,z)$ a non-singular homogeneous polynomial of degree $d$. If I consider the zero locus $X=\{[x:y:z]\in\mathbb{P}^2:F(x,y,z)=0\}$
How can I define a complex structure in $X$ in order to obtain a Riemann Surface? If I do it on the three open subsets $$ U_1=\{[x:y:z]\in X:x\neq 0\} \\ U_2=\{[x:y:z]\in X:y\neq 0\} \\ U_3=\{[x:y:z]\in X:z\neq 0\} $$ then I'm defining them as $$ \begin{array}{ccccc} \phi_i:&U_i&\rightarrow&\mathbb{C}\\ &[x_1:x_2:x_3]&\mapsto &\dfrac{x_j}{x_i},& \text{ for any fixed }j\neq i \end{array} $$ but with this definition how can $\phi_i$ be a local homeomorphism? I cannot guarantee local injectivity.