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Working out some questions from Ravi Vakil's notes. Here is a question:

Question: Suppose $\operatorname{char}\bar{k} \neq 2$ and let $C$ be the curve defined by $x^{2}+y^{2} = z^{2}$. Let $\rho$ be the projection $C \to \mathbb{P}^{1}$ given by $(x:y:z) \to (x:y)$. If $p$ is a point in $\mathbb{P}^{1}$, how many points does $\rho^{-1}p$ have?

Would be grateful if you help.

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    If you're given $x$ and $y$ in $(x:y:z)$, how many choices for $z$ do you have?2012-05-03

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