# Projection of Curve into $\mathbb{P}^{1}$.

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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$p \in \mathbb{P}^{1}$, and if suppose. $\rho^{-1}(a:b) = (c:d:e)$ how would $\rho^{-1}p$ look like. – Maka Vijay May 3 '12 at 20:58
If you're given $x$ and $y$ in $(x:y:z)$, how many choices for $z$ do you have? – Jonathan May 3 '12 at 21:02