Abstract
The Oda's Strong Factorization Conjecture states that a proper birational map between smooth toric varieties can be decomposed as a sequence of smooth toric blowups followed by a sequence of smooth toric blowdowns. This article describes an algorithm that conjecturally constructs such a decomposition. Several reductions and simplifications of the algorithm are presented and some special cases of the conjecture are proved. 1. Introduction. The general strong factorization problem asks if a proper birational map between nonsingular varieties (in characteristic zero) can be factored into a sequence of blowups with nonsingular centers followed by a sequence of inverses of such maps. Oda (5) posed the same problem for toric varieties and toric birational maps. Since toric varieties are defined by combinatorial data, the conjecture for toric varieties also takes a combinatorial form. A nonsingular toric variety is determined by a nonsingular fan and a smooth toric blowup corresponds to a smooth star subdivision of the fan. The conjecture then is:
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