I have a line $L\subset \mathbb CP^2$ and a point $R\in \mathbb CP^2-L$. I need to prove that the map \begin{align*} \varphi:\mathbb CP^2&-\{R\} \to L\\ &P\quad\mapsto L\cap L_{RP} \end{align*} where $L_{RP}$ is the line between $R$ and $P$, is a holomorphic map between complex manifolds.
I'm having some troubles finding an explicit description of $\varphi$, so that I can make sense of the composition with the charts. Any help? Are there other ways to prove this?
We'll use points in the projective plane whose coordinates are ratios $(x:y:z)$Let $L$ bethe line $$\ell x+my+nz=0.$$ Let $R$ be the point $(r:s:t)$. Let $P$ be the point $(x_1:y_1:z_1).$ Then $RP$ is the line $$\ell'x+m'y+n'z=0$$ where $$[\ell',m',n']=[x_1,y_1,z_1] \mathbf x [r,s,t]$$ and $L \cap RP$ is the point whose projective coordinates are $$[\ell',m',n'] \mathbf x[\ell,m,n] $$ $$=([x_1,y_1,z_1] \mathbf x [r,s,t]) \mathbf x [\ell,m,n].$$