Intersection between $\mathbb{R}P^2$ and $\mathbb{C}P^1$

Question

I would like to compute the number of points in the intersection of $\mathbb{R}P^2$ and $\mathbb{C}P^1$ in $\mathbb{C}P^2$. I choose $$\mathbb{R}P^2=\{ [x_0:y_0:z_0]\mid x_i\in\mathbb{R}\}$$ and $$\mathbb{C}P^2=\{[z_0:z_1:z_2]\mid a_0 z_0+a_1z_1+a_2z_2=0\} $$ where $a_i\in\mathbb{C}$. So I need to compute the number of points satisying $a_0 x_0+a_1x_1 a_2x_2=0$ where $a_i\in\mathbb{C}$ and $x_i\in\mathbb{R}$, but I don't know how to proceed. Any help would be appreciated.

Answer 1

They intersect in a circle = $\mathbb{RP}^{1}$. Just choose the line $\mathbb{CP}^{1} \subset \mathbb{CP}^{2}$, given by the equation $\{z_{2}=0\}$ (corresponding to the choice $a_{0}=a_{1}=0, a_{2}=1$), then the real points satisfying this equation are of the form $[x_{0}:x_{1}:0]$, where $x_{1},x_{2} \in \mathbb{R}$ (not both zero).

This is a copy of 1-dimensional real projective space, which is diffeomorphic to a circle.