I am struggling with part (a) of Exercise 17.5.B in Vakil's Foundations of Algebraic Geometry. The statement of the problem goes as follows:
Suppose $C = V(f(x, y, z)) \subseteq \mathbb{P}_k^2$ is a curve, and $p$ is a smooth $k$-valued point of $C$. Recall that the lines in $\mathbb{P}^2$ through $p$ correspond to points of $\mathbb{P}^1$. Explain how we get a map from $C - p$ to $\mathbb{P}^1$, given by mapping a point $q$ to the line $pq$ through $p$.
There is a part (b) as well, which involves extending this map over $p$, but I am stuck on part (a) because I am unable to construct a map $C - p \to \mathbb{P}^1$.
Intuitively, I want to take a point $q \in C$ and find the equation for the line which connects it to $p = [p_0: p_1: p_2] \in C$, and use this data to find two global sections of some line bundle on $V(f(x, y, z))$ which don't vanish simultaneously anywhere on $C - p$ in order to define a map to $\mathbb{P}^1$. If $q = [x_0: x_1: x_2]$ is a point on $C - p$, the line $pq$ is given by the equation $$\det \begin{pmatrix} p_0 & p_1 & p_2 \\ x_0 & x_1 & x_2 \\ X & Y & Z \end{pmatrix} = 0,$$ or in other words $X(p_1x_2 - p_2x_1) - Y(p_0x_2 - p_2x_0) + Z(p_0x_1 - p_1x_0) = 0$. I thus have three linear forms associated with the line $pq$: $p_1x_2 - p_2x_1$, $p_0x_2 - p_2x_0$, and $p_1x_1 - p_1x_0$, which we can denote by $f_0, f_1, f_2$.
At this point, I'm not sure how to proceed; I have three sections of $\mathcal{O}_{V(f(x, y, z))}(1)$, which seems like "too much" information to describe a map to $\mathbb{P}^1$. How should I go about doing this?
Some advice: solve the problem first, then worry about translating it in to the right language.
Up to an automorphism of $\Bbb P^2_k$, we may assume that $p$ is $[0:0:1]$. The lines through $p$ are parametrized by the line at infinity $V(z)$: given a line through $p$, it intersects $V(z)$ in one point, whereas given a point on $V(z)$ we get a unique line through that point and $p$. So given $q=[a:b:c]$ not equal to $p$, the line $\overline{pq}$ has intersection with $V(z)$ given by $[a:b:0]$, giving a map $\Bbb P^2\setminus p \to \Bbb P^1$ by $[x:y:z]\mapsto [x:y]$. We can then compose this with the immersion $C\setminus p \to \Bbb P^2\setminus p$ to get a map $C\setminus p \to \Bbb P^1$.
In more scheme-theoretic terms, the map $\Bbb P^2\setminus p \to \Bbb P^1$ is given by picking a basis of the two-dimensional vector subspace of global sections of $\mathcal{O}_{\Bbb P^2}(1)$ which vanish at $p$.