Strong Toroidalization of Birational Morphisms of 3-Folds

TL;DR

This paper proves that any birational morphism between nonsingular complete 3-folds can be modified through specific blow-ups to become a toroidal morphism, generalizing previous results in the field.

Contribution

It establishes strong toroidalization for birational morphisms of 3-folds, extending prior work to a more general setting with explicit blow-up procedures.

Findings

Existence of modifications making morphisms toroidal

Use of blow-ups supported in preimages of divisors

Generalization of previous toroidalization theorems

Abstract

In this paper we prove strong toroidalization of birational morphisms of 3-folds. Suppose that f:X\to Y is a birational morphism of nonsingular complete 3-folds, and D_Y, D_X are simple normal crossings divisors on Y and X such that f^{-1}(D_Y)=D_X and D_X contains the singular locus of the morphism f. We prove that there exist morphisms \Phi:X_1\to X and \Psi:Y_1\to Y which are products of blow ups of points and nonsingular curves which are supported in the preimage of D_Y and make simple normal crossings with this preimage, such that f_1=\Psi_1^{-1}\circ f\circ \Phi_1 is a toroidal morphism. This theorem generalizes the toroidalization theorem which we prove in ``Toroidalization of birational morphisms of 3-folds''.