  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.

Algebraic Geometry