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I read the following in Fulton's book. Could you help me to solve it? Thanks!

Example: $V = V(XW - YZ) \in A^4(k)$. $\Gamma(V) = k[X, Y, Z, W]/(XW-YZ)$. Let $\overline{X}, \overline{Y}, \overline{Z}, \overline{W}$ be the residue of $X, Y, Z, W$ in $\Gamma{V}$. Then $\overline{X}/\overline{Y} = \overline{Z}/\overline{W} = f \in k(V)$ is defined at $P = (x, y, z, w) \in V$ if $y \neq 0 $ or $w \neq 0$.

Exercise 2-20: Show that it is impossible to write $f = a/b$, where $a, b \in \Gamma(V)$, and $b(P) \neq 0$ for every $P$ where $f$ is defined. Show that the pole set of $f$ is exactly $\{(x, y, z, w) \mid y = w =0\}$.

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Suppose that $f=a/b$ with $a,b\in\Gamma(V)$. Note that $\Gamma(V)$ is an integral domain.

Then $a/b = \bar{X}/\bar{Y}$ thus in $k(V) = \mathrm{Frac}(\Gamma(V))$, we have $a\bar{Y} = b\bar{X}$, i.e. $aY-bX$ vanishes on $V$. Now, taking $P=(1,0,1,0)$, note that $P\in V$ and that $a(P)Y(P)=0$ but $X(P)\neq 0$. Hence $b(P)=0$.

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  • $\begingroup$ Thanks you! I see your argument works for the second assertion. What's about the first one? $\endgroup$ Aug 16 '17 at 16:33
  • $\begingroup$ Here is how to prove the first part: suppose $b(x,y,z,w)\neq 0$ for all $y\neq 0$. It then follows that $b$ depends on $y$ alone (see this post for a simple argument). So if $b(x,y,z,w)\neq 0$ also for all $w\neq 0$, then $b$ is actually a constant. Since $zb-aw\in (xw-yz)$, evaluating at $(0,0,1,0)$ we find $b=0$, that is impossible. $\endgroup$
    – Math101
    Dec 17 '19 at 11:09
  • $\begingroup$ @Math101, Shouldn't it be evaluating at $(0,0,1,1)$? $\endgroup$
    – Steve
    Dec 10 '20 at 8:39

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