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

1998/03/31 by D. Abramovich, K. Matsuki, S. Rashid
Mathematics · #math.AG

paper · pdf

published as Tohoku Math J. 51 (1999), 489-537 · the final and published version with minor mistakes corrected

arxiv created 2000/02/11 · arxiv updated 2009/11/30

Abstract

The recent two proofs for the (weak) factorization theorem for birational maps, one by Włodarczyk and the other by Abramovich-Karu-Matsuki-Włodarczyk rely on the results of Morelli. The former uses the process for π-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.

Related

• On the Strong Factorization of Toric Birational Maps
• Lectures on Factorization of Birational Maps
• Torification and Factorization of Birational Maps
• Birational cobordisms and factorization of birational maps
• Toroidal varieties and the weak Factorization Theorem