Projective space, immersion, embeding

Question

How can I check if the map $F:\Bbb{P}^2\to \Bbb{P}^5$, given as follows $F([x,y,z])=([x^2,y^2,z^2,xy,yz,zx])$, is smooth, an immersion or an embedding?

Answer 1

$\mathbf{P}^n$ has $n+1$ open affine charts $\{x_i\not=0\}$ where I note $[x_0,\ldots,x_n]$ the coordinates on $\mathbf{P}^n$, so that you have to look $F$ through $3$ charts $\varphi_i : U_i \rightarrow \mathbf{R}^3$ at the source and $6$ charts $\psi_j : V_j \rightarrow \mathbf{R}^6$ at the target, that will give you $3\times 6 = 18$ "chart representations" $\psi_j\circ F \circ \varphi_i^{-1}$ of $F$ which will be smooth diffeomorphisms on their open set of definition (as you will check) which will show to you that $F$ is smooth. Looking at the $\psi_j\circ F \circ \varphi_i^{-1}$'s instead of looking at $F$ is called "looking $F$ through charts (at the source and the target)."