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):

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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?

  • 1
    $\begingroup$ 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 $\mathbb{P}^1$. $\endgroup$ May 12 '19 at 22:13
  • $\begingroup$ @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: $\endgroup$
    – KarlPeter
    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?

  • $\begingroup$ Yes. That's correct. $\endgroup$ May 12 '19 at 23:11

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