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 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 ally arbitrary variety and a surface. (C) 2008 Elsevier B.V. All rights reserved.
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