I have a problem understanding blowing up a subvariety. I've had some experience blowing up singular points on curves. I suppose the best way to address the question is to pose an example. Take the equation
$f(x,y,z,w)=(x-y)(z-w)$, and $\nabla f=0$ on the subvariety $x-y=0$, $z-w=0$.
My question is how do I blow up these singular lines? If you could possibly direct me to a resource or give an explanation that would be much appreciated.
