Let $k$ be an algebraically closed field. Let $\phi : \mathbb A^n (k) \to \mathbb P^n(k) $ be the map defined as $\phi_i((a_1,...,a_n))=[1: a_1 : ... : a_n]$. Let $Y \subseteq \mathbb A^n(k)$ be an affine variety (irreducible algebraic set). Then is it true that $Y$ and $\overline {\phi(Y)}$ are birational ?
I know that we are done if we can show that $Y$ has a non-empty open subset which is isomorphic with some open subset of $\overline {\phi(Y)}$. Now we know that $\phi :\mathbb A^n(k) \to \phi(\mathbb A^n(k))$ is an isomorphism. So we are done if we can show that there is a non-empty open subset $U$ of $\overline {\phi (Y)}$ such that $U \subseteq \phi (Y)$, because then we can take $\phi^{-1} (U)$ to be our open subset of $Y$. Am I on right track ? Is there any alternative or easier approach (may be considering the rational function field of $Y$ and $\overline {\phi(Y)}$ ) ?