In $\mathbb{A}^3$ we can transform the surface $x^2=y^2z$ to a smooth surface by blowing up the z-axis.
What if we want to resolve this surface everywhere in $\mathbb{P}^3$? If we take projective coordinates $$[x;y;z;1]=[X;Y;Z;W],$$ so that we have $$X^2\,W=Y^2\,Z,$$ is it strictly necessary to consider this in separate affine charts, or can we perform a blow-up at the subvarieties $X=W=0$, $X=Y=0$ etc as if we were in $\mathbb{A}^4$ with coordinates $(X,Y,Z,W)$?
Thanks