This is a continuation of this question On the flatness of a particular ring map on two variable polynomial ring . Given the $\mathbb C$-algebra map $f: \mathbb C[x,y]\to \mathbb C[x,z]$ defined by $f(x)=x$ and $f(y)=xz$ , we have an induced map on the prime spectrum $f^*:\operatorname {Spec}(\mathbb C[x,z]) \to \operatorname {Spec}(\mathbb C[x,y]) $ (given by $f^*(P) :=f^{-1}(P), \forall P \in \operatorname {Spec}(\mathbb C[x,z]) $) . How to describe this map $f^*$ ? Is there some nice description or special properties of this map $f^*$ ?
I know that the only non-zero, non-maximal prime ideals of $\mathbb C[x,y]$ are of the form $(f(x,y))$ for some irreducible polynomial $f(x,y)\in \mathbb C[x,y]$ but am not sure if that would be helpful or not.
(Notice that I'm taking the prime spectrum and not the maximal spectrum ... describing the induced map on maximal spectrum is easy and some properties of the induced map on the maximal spectrum are already there in the answers of the linked question)
Please help
Once you put Spec, you may want to think of your example as an example of a blow-up. Below $k$ is a field.
Let $X = \operatorname{Spec} k[x,y]$ and $\pi: X'\to X$ be the blow of $X$ at the origin. Then $X'$ is covered by two affine charts. To be concrete, let $X' = \operatorname{Proj} R[xt,yt]$, where $R = k[x,y]$ and $U = D(xt)$ be an affine chart.
It is well-known that $U = \operatorname{Spec} R[yt/xt]$. Notice that $$ R[yt/xt] = R[y/x] = k[x,y,y/x] = k[x,y/x]. $$ Once you set $z = y/x$, $U = \operatorname{Spec} k[x,z]$, and the morphism in question is the restriction of the blow up map $\pi$ to $U$. That is, $\pi|_U$ is $f^*$ in your post. Now, you may look at the properties of blow-ups. (Also look at the blow-up figure in Sec I.4 of Hartshorne's book on Algebraic Geometry.) Remember, blow-ups are not flat in general.