(i) Let $X$ be a curve, and $p \in X$ be a smooth point on $X$. Define what a local parameter at $p$ is.

Now let $f: X \rightarrow Y$ be a rational map to a quasi-projective variety $Y$. Show that if $Y$ is projective, $f$ extends to a morphism defined at $p$.

Give an example where this fails if $Y$ is not projective, and an example of a morphism $f: \mathbb{C}^{2} \backslash\{0\} \rightarrow \mathbf{P}^{1}$ which does not extend to $0 .$

(ii) Let $V=Z\left(X_{0}^{8}+X_{1}^{8}+X_{2}^{8}\right)$ and $W=Z\left(X_{0}^{4}+X_{1}^{4}+X_{2}^{4}\right)$ be curves in $\mathbf{P}^{2}$ over a field of characteristic not equal to 2 . Let $\phi: V \rightarrow W$ be the map $\left[X_{0}: X_{1}: X_{2}\right] \mapsto\left[X_{0}^{2}: X_{1}^{2}: X_{2}^{2}\right]$. Determine the degree of $\phi$, and the ramification $e_{p}$ for all $p \in V$.