This book contains a proof that a dominant morphism from a 3-fold \(X\) to a variety \(Y\) can be made toroidal by blowing up in the target and domain. We give applications to factorization of birational morphisms of 3-folds.

Table of Contents

• Introduction
• An outline of the proof
• Notation
• Toroidal morphisms and prepared morphisms
• Toroidal ideals
• Toroidalization of morphisms from 3-folds to surfaces
• Preparation above 2 and 3-points
• Preparation
• The \(\tau\) invariant
• Super parameters
• Good and perfect points
• Relations
• Well prepared morphisms
• Construction of \(\tau\)-well prepared diagrams
• Construction of a \(\tau\)-very well prepared morphism
• Toroidalization
• Proofs of the main results
• List of technical terms
• Bibliography