Just curious, is it true that $\mathbb{A}^1\times\mathbb{P}^1\cong\mathbb{P}^1\times\mathbb{P}^1$?
Here I'm writing $\mathbb{A}^1$ is affine space, and $\mathbb{P}^1$ projective space, both over an algebraically closed field, for simplicity.
I know that affine space "behaves well" under taking products, in the sense that $\mathbb{A}^n\times\mathbb{A}^m\cong\mathbb{A}^{n+m}$, but the same is not true for projective space in the sense that $\mathbb{P}^n\times\mathbb{P}^m\not\cong\mathbb{P}^{n+m}$. So my gut feeling is that the two are in fact not isomorphic. Is my hunch correct, or maybe I am wrong?
A connected projective variety has no non-constant regular functions. But $A^1\times P^1 \to A^1$ is a non-constant regular function.