equivalence of compact riemann surfaces and smooth projective curves

Question

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 $\mathbb R$?

Answer 1

1) I don't understand. Not all smooth projective curves are plane curves. In fact,the genus of a smooth projective plane curve of degree d is $(d-1)(d-2)/2$.

Moreover, there is not just a bijection of sets, but an equivalence of categories and even more....