I was looking for the $\mathbb{C}-$algebra maps $\varphi:\mathbb{C}[x]\to\mathbb{C}[x,y]$ such that the associated map $\varphi^\star:\text{Spec}\ \mathbb{C}[x,y]\to \text{Spec}\ \mathbb{C}[x]$ are both open and closed.
I know that $\varphi^\star$ is closed if and only if $\varphi$ satisfies the Going Up property, so I was trying to study which maps have this property, but I didn’t conclude anything. I can describe the spectrum of $\mathbb{C}[x]$ (it is a PID, and since $\mathbb{C}$ is algebraically closed the prime ideals are $(0)$ and all of the form $(x-\alpha),$ $\alpha\in\mathbb{C}$) and in a similar way at least a partial description of the spectrum of $\mathbb{C}[x,y]$ (I have to consider the ideals of the form $(p(x,y)),$ where $p(x,y)$ is irreducible), but it seems to me I don’t get anything useful. I don’t know if the geometric point of view can be useful. Any hint?