In 2001 Van den Bergh defined the notion of blowup for noncommutative surfaces and showed that the blowup of a noncommutative P2 in 6 points is isomorphic to a cubic surface inside a noncommutative P3. This yields a rational map from the moduli space of noncommutative P2s equipped with 6 points to a component of the moduli space of noncommutative P3s. In this talk I will show that the abstractly defined map as such turns out to be the composition of some concrete maps. This has various consequences, including the blowdown theorem. This is a joint work in progress with Ingalls, Sierra, and Van den Bergh.