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?