Suppose $C$ is a projective rational irreducible dimension $1$ Noetherian scheme over field $k$ and I would like to collect some of most usual common preferably as elementary as possible ways to prove that the $k$-space $\Gamma(C, O_C)$ of global sections of $C$ is one dimensional, that is equals $k$.
One appoach I learned from Ravi Vakil's FOAG notes is to use the extension Theorem 16.5.1 for nonsingular curves. It states that if we have a rational map $X \dashedarrow Y$ where $Y$ is projective scheme and $X$ a nonsingular curve then the rational map extends to regular map $X \to Y$.
In our case $C$ is rational, so $k(C)=k(t)=K(\mathbb{P}^1_k)$ and we obtain a birational map $\mathbb{P}^1 \to C$ which is welldefined on dense open sets of both schemes and by extension Theorem this map etends to a regular map. $C$ is irreducible and the image of $C$ is dense therefore we get inclusion $\Gamma(C, O_C) \to \Gamma(\mathbb{P}^1, O_{\mathbb{P}^1}) =k$. Since $\Gamma(C, O_C) \neq 0$, we conclude $\Gamma(C, O_C)=k$.
This proof uses extension Theorem as technical ingredient and I think that there are much easier ways to show $\Gamma(C, O_C)=k$. Any suggestions?