The question is, in how many points does a line in $\mathbb{CP}^n$ intersect $\mathbb{CP}^2$?
By a line in $\mathbb{CP}^n$, I mean a copy of $\mathbb{CP}^1$. I have tried with a system of equations, (because a line in $\mathbb{CP}^n$ is the zero locus of a polynomial in $n$ variables of degree $1$) and see if that line passes through one point or two.
Thank you.
It depends on $n$, and on the line. $\def\C{\mathbb C}\def\P{\mathbb P}\def\CP{\C\P}$We consider the embedding: $$ \CP^2 = \{[x_0:x_1:x_2: 0 :\cdots : 0] \mid [x_0:x_1:x_2] \in \CP^2\} \subseteq \CP^n $$ Lifting this to $\C^{n+1}$, we have $\C^3 \subseteq \C^{n+1}$ embedded as the first three coordinates. A line in $\CP^n$ corresponds to a 2-dimensional subspace $A\subseteq \C^{n+1}$. This subspace intersects $\C^3$ in $A$, or (possible for $n \ge 3$) in a one-dimensional subspace $L$, or only in $\{0\}$ (possible for $n \ge 4$).
Hence a line $[A]$ in $\CP^n$ intersects $\CP^2$ in the line $[A]$, a point $[L]$ (possible for $n \ge 3$) or in no point (possible for $n \ge 4$).