The Artin-Castelnuovo contraction theorem
The blowup is a natural and well understood method of constructing a more complicated surface from a simpler one. It replaces a point with a projective with line with self-intersection -1. It is thus natural to ask, when is a smooth surface the blowup of a simpler one? More generally, one can ask when you can contract a curve that's isomorphic to the projective line. This is given by the Castelnuovo contraction theorem, which was generalised by Artin. It turns out, the contraction exists when the self-intersection is negative, though the contraction is smooth precisely when that self-intersection is -1. We go through the proof of this contraction theorem. It is a wonderful application of cohomology theory and the theory of linear systems.