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If X is a projective variety, say its the locus of the homogeneous polynomials F_α(Z_0, ..., Z_n). Say we define on A^n the polynomials f_α(z_1, ..., z_n) = F_α(1,z_1, ...,z_n) = F_α(Z_0, ..., Z_n) / Z

^{d}_0 where d is the degree of F_α and z_i are the local coords (z_i = Z_i/Z_0). Then he claims the zero locus of the f_α is X intersect U_0. Now since there's a bijection between U_0 and A^n, are we just identifying X intersect U_0 with its image via the local coordinates, meaning its an affine variety too?

For the other direction, I don't really see this. If for example X intersect U_0 is an affine variety, say its the locus of f_α(z_1, ..., z_n),then we can define homogeneous polynomials F_α(Z_0, ... Z_n) = Z

^{d}_0 f_α(Z_1/Z_0,..., Z_n/Z_0) where d = deg(f_α). But then is the zero locus of the F_α just X?

Any help would be appreciated!