Related papers: On the Strong Factorization of Toric Birational Ma…

This paper has been withdrawn by the author due to a crucial error.

This is an erratum to math.AG/9803126, Tohoku 51 (1999) 489-537. This erratum describes: 1. the failure of the algorithm in [AMR] and [Morelli1] for the strong factorization pointed out by Kalle Karu, 2. the statement of a refined weak…

The recent two proofs for the (weak) factorization theorem for birational maps, one by W{\l}odarczyk and the other by Abramovich-Karu-Matsuki-W{\l}odarczyk rely on the results of Morelli. The former uses the process for…

This paper has been withdrawn by the author due to the presented idea is wrong.

The paper has been withdrawn by the author, due to it being fundamentally flawed. The author apologizes for any inconvenience it may have caused.

This paper has been withdrawn because of serious errors.

The paper has been withdrawn.

Although the results are correct, it was pointed out that the results follow from some previously known results. Accordingly, this version of the paper is withdrawn by the authors.

This paper has been withdrawn by the author, due to possible counter-examples.

This paper has been withdrawn by the author, due an error in the proof of Proposion 2.13.

This paper has been withdrawn by the author(s), due to a crucial error in eq. 6.

This paper has been withdrawn by the author.

This paper has been withdrawn by the author due to its publication

This paper has been withdrawn by the author due to a crucial argument error at p.10.

This paper has been withdrawn

This paper has been withdrawn by the author because there are some typos in proofs.

The article has been withdrawn by the author due to the existence of counterexamples.

This paper has been withdrawn.

This paper has been withdrawn.

The paper is withdrawn due to mistakes in the proofs for Proposition 1.2 and Theorem 2.2.