Let $f: \mathbb{P}^{2} \rightarrow \mathbb{P}^{2}$ be the rational map given by $f\left(X_{0}: X_{1}: X_{2}\right)=\left(X_{1} X_{2}: X_{0} X_{2}\right.$ : $\left.X_{0} X_{1}\right)$. Determine whether $f$ is defined at the following points: $(1: 1: 1),(0: 1: 1),(0:$ $0: 1)$.

Let $C \subset \mathbb{P}^{2}$ be the curve defined by $X_{1}^{2} X_{2}-X_{0}^{3}=0$. Define a bijective morphism $\alpha: \mathbb{P}^{1} \rightarrow C$. Prove that $\alpha$ is not an isomorphism.