Toroidal varieties and the weak Factorization Theorem

TL;DR

This paper extends the theory of toroidal embeddings to stratified toroidal varieties and provides a new proof of the weak factorization theorem, showing that birational maps can be decomposed into blow ups and downs with smooth centers.

Contribution

It generalizes the theory of toroidal embeddings to stratified varieties and offers an alternative proof of the weak factorization theorem.

Findings

Extended toroidal embedding theory to stratified toroidal varieties.

Provided a new proof of the weak factorization theorem.

Confirmed that birational maps decompose into smooth blow ups and downs.

Abstract

The main goal of the present paper is two-fold. First we extend the theory of toroidal embeddings introduced by Kempf, Knudsen, Mumford and Saint-Donat to the class of toroidal varieties with stratifications (which is the main body of the paper). Second we give a proof of the following weak factorization theorem as an application and illustration of the theory: A birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blow ups and blow downs with smooth centers. Another proof of the weak factorization theorem appeared in a joint paper with Abramovich, Karu and Matsuki (math.AG/9904135) In that paper the theorem is stated and proven in general for proper algebraic and analytic spaces.