[Paper Review] Lectures on Factorization of Birational Maps

Kenji Matsuki|ArXiv.org|Feb 11, 2000

Algebraic Geometry and Number Theory23 references18 citations

TL;DR

This paper provides a detailed exposition of the weak factorization theorem for birational maps between nonsingular complete varieties over an algebraically closed field of characteristic zero, using the theory of birational cobordism and toroidal structures. It reduces the general factorization problem to the combinatorial factorization of toroidal maps via Morelli’s algorithm, establishing that any such birational map decomposes into blowups and blowdowns along smooth centers disjoint from a common open subset, with projective intermediate varieties when the input varieties are projective.

ABSTRACT

This is an expanded version of the notes for the lectures given by the author at RIMS in the summer of 1999 to give a detailed account of the proof for the (weak) factorization theorem of birational maps by Abramovich-Karu-Matsuki-Włodarczyk.

Motivation & Objective

• To provide a comprehensive, accessible account of the proof of the weak factorization theorem for birational maps between nonsingular varieties.
• To clarify the reduction step from general birational maps to toroidal ones, using the framework of birational cobordism.
• To highlight the role of Morelli’s combinatorial algorithm for toroidal factorization as a black box, and to situate the method within the broader context of resolution of singularities and log geometry.
• To lay the foundation for generalizations, including equivariant factorization and factorization in the logarithmic category.
• To motivate the strong factorization conjecture and the toroidalization conjecture as central open problems in birational geometry.

Proposed method

• The proof relies on the theory of birational cobordism, constructed via Geometric Invariant Theory, to model birational maps as sequences of blowups and blowdowns.
• A key construction is the 'torific ideal', which enables the 'torification' of a variety, transforming it into a toroidal embedding.
• The method uses canonical resolution of singularities and canonical principalization of ideals to recover nonsingularity after torification.
• The reduction from general to toroidal maps is achieved via elimination of indeterminacy points and the use of K*-actions on locally toric structures.
• The factorization of toroidal maps is treated as a black box, relying on Morelli’s algorithm for toric birational maps.
• The framework is extended to handle generalizations such as bimeromorphic maps, group actions, and non-algebraically closed fields.

Experimental results

Research questions

• RQ1 How can the weak factorization theorem for birational maps be proven using birational cobordism and toroidal structures?
• RQ2 What is the role of the torific ideal in reducing general birational maps to toroidal ones?
• RQ3 How does the theory of birational cobordism facilitate the construction of factorizations with smooth centers disjoint from a common open subset?
• RQ4 What are the implications of the weak factorization theorem for the strong factorization conjecture and toroidalization?
• RQ5 Can the methods used in the weak factorization proof be adapted to yield an algorithmic resolution of singularities for morphisms in the logarithmic category?

Key findings

• The weak factorization theorem holds: any birational map between nonsingular complete varieties over a field of characteristic zero factors into blowups and blowdowns along smooth centers disjoint from a common open subset.
• When the source and target are projective, the factorization can be chosen so that all intermediate varieties are projective, with a central stage where maps to both source and target are projective morphisms.
• The reduction from general to toroidal maps is achieved via torification and elimination of indeterminacy, relying on the existence of a torific ideal with specific properties.
• The factorization of toroidal maps is achieved via Morelli’s combinatorial algorithm, which is effective and constructive, though treated as a black box in this work.
• The method provides a framework for generalizations, including equivariant factorization and factorization over non-algebraically closed fields.
• The paper motivates the toroidalization conjecture as a potential path toward proving the strong factorization conjecture, linking it to algorithmic resolution of singularities in the logarithmic category.