Toroidalization of birational morphisms of 3-folds

TL;DR

This paper proves that any birational morphism between projective 3-folds over a characteristic zero field can be transformed into a toroidal morphism through a sequence of blow ups of points and nonsingular curves.

Contribution

It establishes a method to toroidalize birational morphisms of 3-folds, extending the understanding of their structure in algebraic geometry.

Findings

Birational morphisms can be made toroidal via blow ups.

The process applies to projective 3-folds over characteristic zero fields.

Provides a constructive approach to toroidalization.

Abstract

We prove that a birational morphism of projective 3-folds, over a field of characteristic zero, can be made toroidal by performing a sequence of blow ups of points and nonsingular curves above the domain and target.