In $\mathbb{P}^3$ define the following sets: $$X=\{w_0w_1^2=w_2^2w_3-w_3^3\}\\Y_1=\{y_3=0\}\\Y_2=\{\sum_{i=0}^3 w_i=0\}\\Y_3=\{w_0+w_1+w_2+2w_3=0\}$$ Does the set $Z=X\cap Y_3\setminus((X\cap Y_2)\cup(X\cap Y_1))$ is affine variey ?
I think that it is affine since $X\cap Y_j$ is an hyper surface so one may look at Z as a union of two affine subsets $Z=(X\cap Y_3)\setminus(X\cap Y_2)\cup(X\cap Y_3)\setminus(X\cap Y_1)$ but this is not enough to show it. How can I show that $Z$ is affine (or if not: how can I show it's projective?)