Toroidalization of locally toroidal morphisms
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Let X and Y be nonsingular varieties over an algebraically closed field [kappa] of characteristic zero. A toroidal structure on X is a simple normal crossing divisor DX on X. Suppose that DX and DY are toroidal structures on X and Y respectively. A dominant morphism [function] : X [arrow] Y is toroidal (with respect to the toroidal structures DX and DY ) if for all closed points [rho] 2 X, [function] is isomorphic to a toric morphism of toric varieties specified by the toric charts at [rho] and [function] [rho]. A dominant morphism [function] : X [arrow] Y of nonsingular varieties is toroidalizable if there exist sequences of blow ups with nonsingular centers [pi] : Y₁ [arrow] Y and [pi]₁ : X₁ [arrow] X so that the induced map [function]₁ : X1 [arrow] Y1 is toroidal. Let [function] : X [arrow] Y be a dominant morphism. Suppose that there exist finite open covers (Ui) and (Vi) of X and Y respectively such that [function] (Ui) [arrow] Vi and the restricted morphisms [function] : (Ui) [arrow] Vi are toroidal for all i. [function] is then called locally toroidal. It is proved that a locally toroidal morphism from an arbitrary variety to a surface is toroidalizable.