Pole set of rational function on $V(WZ-XY)$ Let $V = V(WZ - XY)\subset \mathbb{A}(k)^4$ (k is algebraically closed). This is an irreducible algebraic set so the coordinate ring is an integral domain which allows us to form a field of fractions, $k(V).$ Let $\overline{W}, \overline{X}$ denote the image of $W$ and $X$ in the coordinate ring. Let $f=\dfrac{\overline{W}}{\overline{X}}\in k(V).$ I want to find the pole set of $f.$ I have a feeling I am over complicating this. Essentially I should just view $f$ as the function $W/X$ restricted to $V.$ I know that points on $V$ where $X=0$ and $W\neq 0$ are in the pole set, and this occurs when $Z=0.$ But what happens if both $W$ and $X$ are $0$ ?
 A: Indeed, the pole set of $f=\frac{\overline{W}}{\overline{X}}$ is precisely the set 
$$
S(f)=\{(w,0,y,0) \in \Bbb{A}^4 \; \vert \; w,y \in k\}=V(X,Z)
$$
It is clear that $f$ is defined at every point not in $S(f)$, because 
$$
f=\frac{\overline{W}}{\overline{X}}=\frac{\overline{Y}}{\overline{Z}}
$$
and from this you see that $f$ is defined at $(w,x,y,z)$ if either $x \neq 0$ or $z \neq 0$.
In order to see that $f$ is not defined anywhere else, let $P=(a,0,b,0) \in \Bbb{A}^4$, and assume you could write
$$
\frac{\overline{W}}{\overline{X}}=\frac{\overline{F}}{\overline{G}}
$$
with $F,G \in k[X,Y,Z,W]$ and $G(P) \neq 0$.
This implies that $G \cdot W-F \cdot X \in (WZ-XY)$, which means that $G \cdot W-F \cdot X=H \cdot (WZ-XY)$ for some $H \in k[X,Y,Z,W]$. Now you get
$$
X \cdot (F-HY)=W \cdot (G-HZ)
$$
from which it follows that $X$ must divide $G-HZ$, i.e. $G-HZ \in (X)$. But then evaluating at $P=(a,0,b,0)$ yields
$$
G(P)=0
$$
which contradicts our choice of $G$.
A: By definition of pole set (e.g. Fulton's 'Algebraic Curves', sec 2.4, I'm guessing you have this book :) , we want to analyze: points $P\in V$ where $f$ is not defined, so these $P$ correspond to where $X=0$ and $Z=0$ which is $(X=0, Z=0)$ on $V$; note that we include $Z=0$ precisely because $\overline{W}/\overline{X} = \overline{Y}/\overline{Z} = f$ in $k(V)$. Thus, $$pole(f)_V = (X=0, Z=0) = \{(x,y,z,w): x=0, z=0\}.$$
What happens at $W=0$ or $Y=0$ is uninteresting here since we are interested in the poles, not zeroes of $f$
It's also helpful to think in terms of valuations, as in the Lemma following Prop 2 in sec 2.4 of Fulton.
As additional exercises, calculate the codimension of $pole(f)_V$ in $V$ and what is (geometrically and algebraically speaking) the intersection $pole(f)_V\cap pole(1/f)_V$?
