a) Let $ X $ projective variety and $ S (X) $ your homogeneous coordinate ring
Let $ \phi: \mathbb{P}^1 \rightarrow \mathbb{P}^2 $ given by $ \phi([a: b]) = [a^2: ab: b^2] $ be $ C = Im \phi $. How can I show that $ C \simeq \mathbb{P}^1 $ but $ S(C) \not \cong S (\mathbb{P}^1) $, specifically, how can I see that $ S(C) \not \cong S (\mathbb{P}^1) $
I know $ S (\mathbb{P}^1) = K[x,y] $, how do I calculate $ S (C) $?
b) How can I prove that $ Z(xy-zw) \subset \mathbb{P}^3 $ is birational to $ \mathbb{P}^2 $, but not isomorphic.