# Projective and Affine varieties

by hlfmanhoffdor
Tags: affine, projective, varieties
No, of course not. The entire set $\mathbb{P}^n\setminus U_0$ would also be in the zero locus of the $F_\alpha$ since it is easy to see that if $Z_0=0$, then $F_\alpha(0,Z_1,...,Z_n)=0$.
So you obtain $X\cup (\mathbb{P}^n\setminus U_0)$. If you want to obtain just X, then you need to perform the same procedure on the $U_1,...,U_n$. This will give you extra polynomials. The zero locus of everything will then be X.