This is exercise 5.6.10 in Gathmann's note: Show that $\mathbb{A}^2_\mathbb{C}\not\simeq \mathbb{A}^1_\mathbb{C} \times_{\operatorname{Spec}(\mathbb{Z})} \mathbb{A}^1_\mathbb{C}$.
The problem is equivalent as proving that $\mathbb{C}[x]\otimes_{\mathbb{Z}}\mathbb{C}[x]\not\simeq \mathbb{C}[x,y]$. I know one way to solve this is to show that $\mathbb{C}[x]\otimes_{\mathbb{Z}}\mathbb{C}[x]$ is not an integral domain by proving that $(1\otimes 1 + i\otimes i)(1\otimes 1 - i\otimes i)=0$. The problem is also answered here.
Since those methods are all algebraic, I am wandering if we have any geometric way to solve this problem? How should I understand $\mathbb{A}^1_\mathbb{C} \times_{\operatorname{Spec}(\mathbb{Z})} \mathbb{A}^1_\mathbb{C}$ geometrically?
