# Blow Up of a Surface

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?

• Because he's realized $E$ as the cofibered product of $\mathbb{A}^1$ with itself along $\mathbb{G}_m$ where the gluing map is inversion. This cofibered product is just $\mathbb{P}^1$. May 12 '19 at 22:13
• @AlexYoucis: I'm not sure if I understood you completely so please correct me if I wrong. Since it's not allowed to draw diagrams in comments I outsourced it as an answer: May 12 '19 at 23:08

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?

• Yes. That's correct. May 12 '19 at 23:11