Consider the bijective correspondance between compact Riemann surfaces and smooth irreducible complex projective algebraic curves.
1) Do we take only plane curves or not.
2) If yes then every compact Riemann surface will correspond to a curve $$C=\{[x:y:z]\in P^2(\mathbb C)\, |\, f(x,y,z)=0\}$$ where $f$ is a homogeneous polynomial with coefficients in $\mathbb C$. Is this correct and i'm also asking : when precising complex or real projective curve are we making precise points in $ P^2(\mathbb C)$, in $ P^2(\mathbb R)$ or making precise the coefficients of the polynomial $f$ being in $\mathbb C$ or $\mathbb R$?