I feel a bit stupid about this question, but I feel an urge to get to know more about this!
Let $R = k[t_0,\ldots,t_n]/I$ where $I \trianglelefteq k[t_0,\ldots,t_n]$ is a homogeneous ideal and such that $X = \operatorname{Proj}(R)$ is a projective $k$-scheme of pure dimension 1.
Given a graded finite ring map $\phi:k[x_0,x_1] \hookrightarrow R$, say from the Noether normalization lemma:
- is it possible to find an (graded) isomorphism $R \to k[\phi(x_0),\phi(x_1),s_2\ldots,s_n]/J$?
- does a finite ring map $\phi': k[\phi(x_0),\phi(x_1)] \hookrightarrow k[\phi(x_0),\phi(x_1),s_2\ldots,s_n]/J$ correspond geometrically to the projection onto the first two coordinates $$X \to \mathbb{P}_k^1 ,\quad (a_0:a_1:\ldots:a_n) \mapsto (a_0:a_1)?$$
Thanks in advance for any helpful comment on this!