The interplay between projective and affine varieties.

Question

I'm studying Algebraic Geometry first course from Harris and I didn't understand this equality:

In another words, I'm having troubles to understand the interplay between $f_{\alpha}$ and $F_{\alpha}$.

In order to grasp this intuitively I'm working in $\mathbb P^2$ where $U_0\cong \mathbb A^2$ is a plane $Z_0=1$.

Thanks in advance

Answer 1

It is the change from a homogeneous polynomial for the projective variety to a polynomial for the affine one.

Answer 2

For $X=V_+(F)$ and $f(z_1,\ldots,z_n)=F(1,z_1,\ldots,z_n)$ we have $X\cap U_0=\{(x_0:x_1:\ldots:x_n)\in k^{n+1}:x_1\neq 0, F(x_0,\ldots,x_n)=0\}=\{(1:z_1:\ldots:z_n):(z_1,\ldots,z_n)\in k^n, F(1,z_1,\ldots,z_n)=0\}\cong\{(z_1,\ldots,z_n):f(z_1,\ldots,z_n)=0\}$