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Torification and factorization of birational maps. (English) Zbl 1032.14003
The authors prove the “weak factorization theorem” for complete, smooth, algebraic varieties over an algebraically closed field \(\mathbb K\):
First, it states that every birational equivalence between two varieties can be realized by birational maps originating from a third one. Second, every birational map can be split into a sequence consisting of blowing up in smooth centers – or the inverse of those blowing ups. (Without referring to inverse blowing ups, we would obtain the yet unproven strong factorization theorem.)
The proof uses the notion of a birational cobordism \(B\) between birational equivalent varieties, developed by the fourth author [J. Włodarczyk, J. Algebr. Geom. 9, No. 3, 425-449 (2000; Zbl 1010.14002)]. Being a variety of one dimension higher and with a \(\mathbb K^\ast\)-action, this notion is the algebro-geometrical analog of the combinatorial cobordism between fans introduced by Morelli to prove the weak factorization theorem in the toric situation.
Proceeding similar to Morse theory (use the fixed points of the \(\mathbb K\)-action instead of the critical points of a Morse function), one obtains, locally in \(B\), a toric description of the birational transformation going on. However, the embedded torus may vary from point to point. Blowing up so-called torific ideals in \(B\) and using canonical resolution of singularities, the authors eventually reach a more global situation; the birational map becomes toroidal, and they can apply the weak factorization theorem for the toric case.

MSC:
14E05 Rational and birational maps
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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