A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension. -- Correction by Kenji Matsuki
The strong form of the factorization conjecture by \textit{K. Miyake} and \textit{T. Oda} asserts that every proper equivariant birational map between two smooth toric varieties can be factored into a sequence of blow-ups followed by a sequence of blow-downs centered on smooth orbit closures. In the weak form of the conjecture the factorization is allowed to be a sequence of blow-ups and blow-downs in any order [see \textit{T. Oda}, ``Lectures on torus embeddings and applications''. Lect. Math. Phys. 57, Tata Inst. Fund. Res. 57 (1978; Zbl 0417.14043); p. 60]. It turned out that Morelli's proof adressed in the title [\textit{R. Morelli}, J. Algebr. Geom. 5, No. 4, 751-782 (1996; Zbl 0871.14041)] contained a number of gaps and errors that were partly corrected by himself on his homepage. The purpose of the article under review is to clarify Morelli's arguments and to remove the remaining discrepancies, that essentially occurred in two parts namely in the process of \(\pi\)-desingularization and when deriving strong factorization from weak factorization. Moreover, the article under review also contains a section presenting a generalization of the factorization results to the toroidal case. Whereas the authors succeeded with the process of \(\pi\)-desingularization and thus completed Morelli's proof of weak factorization, there is a serious problem with the proof of the strong factorization conjecture. This was first noticed by K. Karu and then pointed out by \textit{K. Matsuki} in his correction to this article [Tohoku Math. J. 52, 629-631 (2000)]. In fact, the strong factorization conjecture for toric birational maps is still an open problem. Nevertheless the central idea of Morelli to consider a combinatorial cobordism was a breakthrough, and it inspired \textit{J. Włodarczyk} to define a more general notion of birational cobordism [J. Algebr. Geom. 9, No. 3, 425-449 (2000; Zbl 1010.14002)]. Using this notion and the algorithm of \(\pi\)-desingularization he could prove the analogous weak factorization result for birational maps between general complete smooth varieties [see \textit{J. Włodarczyk}, ``Combinatorial structures on toroidal varieties and a proof of the weak factorization theorem'', math.AG/9904076 (1999)]. An independent proof of the same result based on the theory of birational cobordism and a refined version of the aforementioned weak factorization of toroidal birational maps was given by \textit{J. Włodarczyk} together with \textit{D. Abramovich, K. Karu, K. Matsuki} and \textit{J. Włodarczyk} [see ``Torification and factorization of birational maps'', J. Am. Math. Soc. 15, No. 3, 531-572 (2002; Zbl 1032.14003)].
factorization of toroidal birational maps
scientific article; zbMATH DE number 1445640
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