(i) Let $k$ be an algebraically closed field, and let $I$ be an ideal in $k\left[x_{0}, \ldots, x_{n}\right]$. Define what it means for $I$ to be homogeneous.

Now let $Z \subseteq \mathbb{A}^{n+1}$ be a Zariski closed subvariety invariant under $k^{*}=k-\{0\}$; that is, if $z \in Z$ and $\lambda \in k^{*}$, then $\lambda z \in Z$. Show that $I(Z)$ is a homogeneous ideal.

(ii) Let $f \in k\left[x_{1}, \ldots, x_{n-1}\right]$, and let $\Gamma=\left\{(x, f(x)) \mid x \in \mathbb{A}^{n-1}\right\} \subseteq \mathbb{A}^{n}$ be the graph of $f$.

Let $\bar{\Gamma}$ be the closure of $\Gamma$ in $\mathbb{P}^{n}$.

Write, in terms of $f$, the homogeneous equations defining $\bar{\Gamma}$.

Assume that $k$ is an algebraically closed field of characteristic zero. Now suppose $n=3$ and $f(x, y)=y^{3}-x^{2} \in k[x, y]$. Find the singular points of the projective surface $\bar{\Gamma}$.