Monomialization of Morphisms From 3 Folds to Surfaces

Abstract

Suppose that $\Phi:X\to Y$ is a morphism from a 3 fold to a surface (over an algebraically closed field of characteristic zero). We prove that there exist sequences of blowups of nonsingular subvarieties $X_1\to X$ and $Y_1\to Y$ such that the morphism $\Phi_1:X_1\to Y_1$ is a ``Monomial Morphism''. As a corollary, we show that we can make $\Phi_1$ a toroidal morphism.