Geometry of blowups
In this video, we introduce a fundamental construction in algebraic geometry called the blowup. It replaces an algebraic variety X with a binational one Bl X where the blown up point on X is replaced with an exceptional divisor, which is a projective space in the smooth case. We next look at the special case of blowing up a point on a smooth surface. In this case, the blowup has an "extra" curve, the exceptional divisor say E, and it is natural to ask how similar the geometry of the blowup is. We give a good answer to this by showing the intersection theory on Bl X, is essentially that of X, except that you have to add an orthogonal component ZE and, most interestingly, the self-intersection of E is -1!