Let $C$ be a smooth projective curve over an algebraic field $k$. I suppose the Kodaira dimension of $C$ is $0$. Why the genus must be $1$?
If $C$ is a smooth projective surface, and if I suppose the Kodaira dimension is $0$ why $dim_{k}(H^{0}(C, nK_{C})) \leq 1$ for all $n$ and is equal to $1$ for some $n$? Here $K_{C}$ is the canonical sheaf.