A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension

Details A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension

1998-03-25

arxiv.org/abs/math/9803126v2

Tohoku Math J. 51 (1999), 489-537

Mathematics

Algebraic Geometry

the final and published version with minor mistakes corrected

Scientific paper

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 $\pi$-desingularization (the most subtle part of Morelli's combinatorial algorithm), while the latter uses the strong factorization of toric (toroidal) birational maps directly. This paper provides a coherent account of Morelli's work (and its toroidal extension) clarifying some discrepancies in the original argument.

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