Let $P$ be a point in $\mathbb{P}^3$. Show that the collection of lines through $P$ has one-to-one correspondence with $\mathbb{P}^2$.
I'm a bit confused here. I thought projective lines can be determined by two points in $\mathbb{P}^3$. By picking a point $Q \neq P \in \mathbb{P^3}$, I'm giving the line passing through $P$ and $Q$, so wouldn't the collection of lines through $P$ be viewed as $\mathbb{P}^3$ itself?
Best Answer
Lines in $\mathbb{P}^3$ are the images of planes (2 dimensional) in $\mathbb{A}^4$. Given the point $P\in \mathbb{A}^4$, the planes containing $P$ are determined by $P$ and a point $Q\in\mathbb{A}^4$ linearly independent of $P$. Moreover, $P, Q$ span the same plane as $\lambda P, \mu Q$ for nonzero $\lambda, \mu$.
So, roughly speaking, you have a three dimensional space $\mathbb{A}^3$ from which to choose $Q$ (so it's independent of $P$), and then you have to quotient by the nonzero scalars, so there would be a correspondence with points in $\mathbb{P}^2$. I have left out some details, hope this helps.