Strong Toroidalization of Birational Morphisms of 3-Folds

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''.