David Rydh: Weak factorization of Deligne–Mumford stacks

The weak factorization theorem relates two birational smooth varieties in characteristic zero via a sequence of blow-ups and blow-downs in smooth centers. I will present a variant of the weak factorization theorem for Deligne–Mumford stacks in characteristic zero where blow-ups are replaced with stacky blow-ups.

As for weak factorization for schemes, the main idea is to use cobordisms (Morelli, Włodarczyk) which are analogous to cobordisms in Morse theory. I will outline the proof for Deligne–Mumford stacks which actually is simpler than for schemes. Using functorial flatification, one also obtains functorial weak factorization with respect to smooth morphisms. As a corollary, we recover the weak factorization theorem for simplicial toric varieties (the weak Oda conjecture, proven by Włodarczyk).