Let X be a quasi-compact and quasi-separated scheme. Let U, V be quasi-compact disjoint open subschemes of X. Then there exist a (U ∪ V)-admissible blowup b : X′ → X such that X′ is a disjoint union of open subschemes X′ = X′1 ⨿ X′2 with b^{−1}(U) ⊂ X′1 and b^{−1}(V) ⊂ X′2. See Lemma Tag 080P.

Slogan: separate irreducible components by blowing up.