One can show that $\mathbb{P}^n\times \mathbb{P}^m$ is birational to $\mathbb{P}^{n+m}$ by making note of the Zariski topologies and the canonical isomorphism between affine spaces $\mathbb{A}^n\times \mathbb{A}^m\to \mathbb{A}^{n+m}$.
Question: Does it follow that there is isomorphism $\mathbb{P}^n\times \mathbb{P}^m\to \mathbb{P}^{n+m}$? If so, what is an example of such an isomorphism?
Unfortunately I am having a hard time coming up with one, and haven't seen too many examples thus far.
As a user commented above, your question is a bit confusing since you already gave a birational map between $\mathbb{P}^n\times\mathbb{P}^m\to\mathbb{P}^{n+m}$. Now, if you are looking for an isomorphism, that is, a regular map with a regular inverse, this can't exist. One way is to see this is that $\mbox{Pic}(\mathbb{P}^n\times\mathbb{P}^m)\simeq\mathbb{Z}\oplus\mathbb{Z}$ and $\mbox{Pic}(\mathbb{P}^{n+m})\simeq\mathbb{Z}$.