Projective blowups: a formal and multicentered proof

Abstract

We give a machine-checked proof, in Lean 4, of the universal property of multicentered blowups: given a finite family $Z$ of closed subschemes of a scheme $X$, presented on each chart by ideals, there is a scheme $\Bl_Z X$ over $X$, terminal among $X$-schemes on which every member of the family becomes an effective Cartier divisor. To the best of our knowledge this is the first formalisation of blowups, single- or multicentered, in any theorem prover. The proof is organised around the universal property of the multicentered dilatation of a ring: the initial algebra in which a finite family of ideals becomes generated by non-zero-divisors. Existence and uniqueness of the dilatation are established locally, on affine charts, and every subsequent step of the construction, the base change along open immersions, the comparison isomorphisms on overlaps, the triple overlap maps, the cocycle identity, and the independence of the chosen presentation, is deduced from that single local property by uniqueness alone. We give the exact theorem statement as formalized, the proof architecture, and the main new mathematical ingredients, together with the corresponding Lean code.