# equivalence of compact riemann surfaces and smooth projective curves

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 $\mathbb R$?

• The answer to (1) is no. The equivalence is for all compact Riemann surfaces and all smooth projective curves.
– user18119
Commented Nov 26, 2012 at 12:53

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 $(d-1)(d-2)/2$.