let $k$ be an algebraic closed field. For $I\subseteq k[x_1,\ldots,x_n]$, we donote $I^*$ to be the ideal generated by the set $\{F^*|F\in I\}$, here $F^*=x_{n+1}^{deg F}F(x_1/x_{n+1},\ldots,x_n/x_{n+1})\in k[x_1,\ldots,x_{n+1}]$.
Now, $V=Z(\mathfrak{p})$ is an irreducible algebraic set of $\mathbb{A}^n(k)$. $V^*=Z(\mathfrak{p}^*)$ is a irreducible algebraic set of $\mathbb{P}^n$, called the projective closure of $V$. It is the smallest algebraic set in $\mathbb{P}^n$ containing $V$.
My question is: When is $V^*$ really bigger than $V$ ? That is to say $V^*$ has infinities.
Trivial case: $V$ is a point, then $V^*$ is a point too, thus $V^*$ do not increase points.
So I want to know if this will be the only case that $V^*$ equals $V$?