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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.

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I'm confused about which lines you mean. But anyway, a great resource for learning about blowups (in scheme language) is "Geometry of Schemes" by Eisenbud and Harris. –  Andrew Jul 18 '12 at 4:59
Your question is related to this older question: math.stackexchange.com/questions/240301/… and I think the answer there will be useful for you as well. –  Qiao Jul 22 at 21:39

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