Toroidalization of Locally Toroidal Morphisms from N-folds to Surfaces

Abstract

The toroidalization conjecture of D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk asks whether any given morphism of nonsingular varieties over an algebraically closed field of characteristic zero can be modified into a toroidal morphism. Following a suggestion by Dale Cutkosky, we define the notion of \emph{locally toroidal} morphisms and ask whether any locally toroidal morphism can be modified into a toroidal morphism. In this paper, we answer the question in the affirmative when the morphism is between any arbitrary variety and a surface.Comment: 22 pages. Final version (added an outline of the proof to the introduction, fixed some errors of notation); to appear in J of Pure and Applied Algebr