Birational between affine and projective

Question

I'm asked to prove that $\mathbb{A}^n$ is birational to $\mathbb{P}^n$. I know I'm suppose to find a rational function whose inverse is also a rational function. But I have no idea where to start. Maybe it's because I'm having trouble believing this is result is even true. This is because I know that $\mathbb{A}^n$ is birational to $\mathbb{P}^{n}$ with one of the coordinates fixed as $1$.

Answer 1

You mention that $\mathbb{A}^n$ is birational to any of the subsets $U$ of $\mathbb{P}^n$ obtained by fixing a coordinate to be 1, e.g. $$U = \{[x_0:x_1:\cdots:x_{n-1}:1]: x_i\in k\}]\subset \mathbb{P}^n$$ where $k$ is the field over which we're working. In fact, they are isomorphic, not just birational: the usual map $\mathbb{A}^n\to U$ given by $(x_0,\ldots,x_{n-1})\mapsto [x_0:\cdots:x_{n-1}:1]$ is an isomorphism. By the definition you gave, this defines a rational map $\phi:\mathbb{A}^n\dashrightarrow\mathbb{P}^n$ (by composing with the inclusion map $U\hookrightarrow \mathbb{P}^n$). See if you can write down a rational map from $\mathbb{P}^n$ to $\mathbb{A}^n$ which is an inverse for $\phi$.