David Rydh: Functorial flatification by blow-ups

Abstract:

Raynaud–Gruson (1971) and Hironaka (1975) showed that morphisms and coherent sheaves in algebraic and complex geometry can be flatified via sequences of blow-ups. I will describe a functorial algorithm that accomplishes this, similar in spirit to functorial resolution of singularities. By functoriality the algorithm automatically extends to algebraic stacks. The functorial flatification algorithm leads to a functorial algorithm for weak factorization.

I actually gave a talk  at KTH on this algorithm almost 10 years ago but the result is now properly written down ( arXiv:2501.08394 ). McQuillan has recently proven a similar result ( arXiv:2412.00998 ).