Here $U_{r,n}$ is the disjoint union of the $r$-planes ($r$-dimensional $\mathbb{C}$-linear subspaces) in $\mathbb{C}^n$.
I am confused with how he used $f_i\circ\phi_i^{-1}$ to get $\mathcal{O}(P_1(\mathbb{C}),E^k)$. I see that $f_0(p)=f_0\circ\phi_0^{-1}(z)=\sum_{i=0}^{-k}{(\frac{z_0}{z_1})}^i$ for $p\in U_0\cap U_1$ and by multiplying $z_1^{-k}$ we get a homogeneous polynomial in $\mathbb{C}^2$ with degree $-k$, but then why is it $f\in\mathcal{O}(P_1(\mathbb{C}),E^k)$?
Anyway, I think I don't understand where the multiplication, I described above, happens to transform the $\sum_{i=0}^{-k}{(\frac{z_0}{z_1})}^i$ to a homogeneous polynomial in $\mathbb{C}^2$ with degree $-k$ and how he got $f$ from $f_i(p)$ when $p\in U_1\cap U_0$
The remark(c) is below:
