The question is In how many points a line in CP^n intersects CP^2?.
By a line in CP, I mean a copy from CP^1. I have tried with a sytem of equations, (Because a line in CP^n is the zero locus of a polynomial in n-variables of degree 1) and watch if that line passes throug one point or two.
I have this:
Let $p_{1}, p_{2},p_{3} \in \mathbb{C}P^{n}$ points whose coordinates are:
$p_{1} = [1:\frac{x_{1}}{x_{0}}: \ldots:\frac{x_{n}}{x_{0}}], p_{2} = [1:\frac{z_{1}}{z_{0}}:\frac{z_{2}}{z_{0}} \ldots:0], p_{3} = [1:\frac{c_{1}}{c_{0}}:\frac{c_{2}}{c_{0}} \ldots:0] $
Notice that althought the points are in $\mathbb{C}P^{n}$, $p{2},p{3} \in \mathbb{C}P^{2}$.
Then, we want to know if there exist a line in the projective space such that passes throught these three points.
Since a line in $\mathbb{C}P^{n}$ is the zero locus of a polynomial in n-variables of degree 1 then we have the next system of equations:
$$x_{0} + x_{1}(\frac{x_{1}}{x_{0}}) + \ldots + x_{n}(\frac{x_{n}}{x_{0}}) = 0 \\ x_{0} + x_{1}(\frac{z_{1}}{z_{0}}) + x_{2}(\frac{z_{2}}{z_{0}}) = 0 \\ x_{0} + x_{1}(\frac{c_{1}}{c_{0}}) + x_{2}(\frac{c_{2}}{c_{0}}) = 0 $$
Then I did a matrix with this system and I proved that the linear tranformation associated to the matrix is not inyective, but my result is that given 3 points there is only on line that passes through them and that can´nt be. What did I do wrong?
Thank you.
Remember that $\Bbb P^n(\Bbb C)$ is the quotient of $\Bbb C^{n+1}\setminus\{0\}$ under the parallelism equivalence of vectors in $\Bbb C^{n+1}$ and there's a quotient map $$ \pi:\Bbb C^{n+1}\setminus\{0\}\longrightarrow\Bbb P^n(\Bbb C) $$ under which the linear subspaces in the target correspond under inverse image to vector subspaces in the source space with $0$ taken out of diension raised by $1$.
Thus $V=\pi^{-1}(\Bbb P^2)\cup\{0\}$ is a subspace of dimension $3$ and $W=\pi^{-1}(\text{line})\cup\{0\}$ is a subspace of dimension $2$.
If $W\subset V$, the line sits inside the plane.
If $dim(V\cap W)=1$ the line and the plane intersect in one point.
If $n\geq4$ it may happen that $V\cap W=\{0\}$ and in this case there is no intersection between line and plane.