The scheme-theoreic definition of projective plane is formed by several steps:
1.Define the line through $(a_1,...,a_n)$ by the closed subscheme of $\Bbb A^{n+1}_C$ defined by $a_jx_i=a_ix_j$.
2.A closed subscheme $L\subset \Bbb A_R^{n+1}$ is a principle $R$-line if it is the line through some point $a\in (\Bbb A^{n+1}\setminus \{0\})(R)$
3.A closed subscheme $L\subset \Bbb A_R^{n+1}$ is an $R$-line if it is principle locally on $R$. (i.e. exists $f_1,...,f_n\in R$ generates $\langle 1\rangle$ such that $L_{R[1/f_i]}$ is a principle $R_{[1/f_i]}$-line in $\Bbb A^{m+1}_{R[1/f_i]}$
4.$\Bbb P^n_R(C)=\{\text{C-lines in $\Bbb A^{n+1}_C$}\}$
It seems the scheme theoreic definition of the projective plane is much more complicated. There are someone pointed out that this is related to the topological definition of projective space, but they are different.
I am confused. I cannot see the explicit connection and difference between this definition of projective space and the one we use in topology, which is "identifying lines through the origin in the same direction".(See here: https://en.wikipedia.org/wiki/Projective_space)
May I please ask which are the relations and main difference between these two definiton? And how to visualize the above scheme-theoric definition of the projective space?
Thanks!