# Blow-up as polar coordinates?

While doing some explicit calculations involving a blow-up of the plane in a point, I realised what I was doing was basically writing things in polar coordinates. Somewhat astonished that I hadn't made the connections

tangent lines through point $$\leftrightarrow$$ lines $$\theta=const$$ in polar coordinates

exceptional divisor $$\leftrightarrow$$ the line $$r=0$$ in polar coordinates

before, I mentioned it to some other algebraic geometry people, and none of them had thought of it either. It's quite obvious when you see it, but somehow it's never mentioned anywhere, which may be because a) the geometric picture as usually presented is what you really want, who cares about coordinates anyway; or b) this isn't the actual motivation behind the construction, as originally conceived.

So, I propose the following conjectural origin story of the blow-up:

Look at picture of singular curve "Hmm, for no apparent reason I wonder what that looks like in polar coordinates." draw the picture "Hey, the curve isn't singular anymore!" work out how to express this in terms of polynomials, like a good algebraist -- and then you recover the usual text-book presentation of the blow-up.

Question: Is this story complete rubbish?

or if you will, I suppose I could just have asked

Question': what is the historical origin of the blow-up construction?

• I wouldn't say it is complete rubish, but your analogy only works for surfaces. Here is something more general to think about: "When you blow up at a subvariety, you replace the subvariety with its projectivized normal cone." Jun 12, 2015 at 12:37
• I think it's reasonable that the idea of blowing up a point came before higher dimensions, and, whether it was explicitly viewed as polar co-ordinates or not, it's likely that it was somewhere in the back of the mind of the person(s) who discovered the idea of blowing up. Jun 12, 2015 at 13:05
• As an evidence for the story not being completely rubbish: Pierre Milman (my advisor in grad school) always (at least from 2005 - that's when I started talking with him) used to introduce blow ups as algebraic geometric version of polar coordinates. Aug 23, 2018 at 1:57

The starting point to construct such analytic triangle (that nowdays is called Newton's polygon) is (in Coolidge's words): Suppose that the point in which we are interested is the origen. We put $y=vx^{\mu}$ and see out those terms ...