I have a question a step in the example demonstating the blowing up in Liu's "Algebraic Geometry and Arithmetic Curves" in the excerpt below (or look up at page 320):
We blow up the surface $X= Spec k[S,T,W]/(ST-W^2)= Spec k[s,t,w)$ in the point $x_0$ corresponding to max ideal $m=(s,t,w)$.
The point of my interest is the fiber $E:=\pi^{-1}(x_)$
I don't understand why the observation that
$$E \cap A_1= V(s)=Spec k[ws^{-1}]$$ $$E \cap A_2= V(t)=Spec k[wt^{-1}]$$ $$E \cap A_3= V(w)=Spec k[tw^{-1},sw^{-1}]/((sw^{-1}) \cdot (tw^{-1})-1)$$
leads to conclusion that $E= \mathbb{P}^1_k$?
Indeed, by definition $\mathbb{P}^1_k$ glues together as $Spec k[x/y] \cup Spec k[y/x]$ on the intersection via $k[x/y,y/x] \to k[x/y,y/x]$
by $ x/y \mapsto y/x$.
Why the calculation in the excerpt imply that $E= \mathbb{P}^1_k$ hold?
your cofibered product along $\mathbb{G}_m$ seems to be
$$ \require{AMScd} \begin{CD} Spec( k[tw^{-1},sw^{-1}]/((sw^{-1}) \cdot (tw^{-1})-1)) @>{g} >> Spec( k[ws^{-1}]) \\ @VVfV @VVV \\ Spec( k[wt^{-1}]) @>{}>> Spec (k[ws^{-1}]) \coprod Spec( k[wt^{-1}]) \end{CD} $$
where we identified $E \cap A_1$ and $E \cap A_2$ with $\mathbb{A}^1$ and $E \cap A_3$ with $\mathbb{G}_m$.
Futhermore $g$ is induced by ring map $k[ws^{-1}] \to k[tw^{-1},sw^{-1}]/((sw^{-1}) \cdot (tw^{-1})-1), ws^{-1} \mapsto tw^{-1}$ and
$f$ by $k[wt^{-1}] \to k[tw^{-1},sw^{-1}]/((sw^{-1}) \cdot (tw^{-1})-1), wt^{-1} \mapsto sw^{-1}$
Is this the construction which you mean?