classically, blowups were defined extrinsically, by first defining the blowup on spaces such as projective space using an explicit construction in coordinates and then defining blowups on other spaces in terms of an embedding. this is reflected in some of the terminology, such as the classical term "monoidal transformation". contemporary algebraic geometry treats blowing up as an intrinsic operation on an algebraic variety. from this perspective, a blowup is the universal (in the sense of category theory) way to turn a subvariety into a cartier divisor.