Toroidalization of locally toroidal morphisms from \(N\)-folds to surfaces (Q2378547)

If \(X\) is a nonsingular variety over a field of characteristic 0, a toroidal structure on \(X\) is a simple normal crossing (SNC) divisor \(D\) on \(X\). A SNC divisor \(D\) on \(X\) defines a toric chart at any closed point of \(X\); and a dominant morphism \(f: X \rightarrow Y\) of varieties \(X\) and \(Y\) with todoidal structures \(D_X\) and \(D_Y\) is toroidal w. r. to \(D_X\) and \(D_Y\) if it is locally toric for the toric charts of \(D_X\) and \(D_Y\). The toroidalization conjecture asks whether any morphism of nonsingular varieties \(X\) and \(Y\) can be modified to a toroidal morphism [see \textit{D. Abramovich ,K. Karu, K. Matsuki, J. Wlodarczyk}, J. Am. Math. Soc. 15, No. 3, 531--572 (2002; Zbl 1032.14003)]. The case when \(\dim(X) = 3\) has been solved by [\textit{S. D. Cutkosky}, Monomialization of morphisms from 3-folds to Surfaces. Lecture Notes in Mathematics 1786. (Berlin): Springer. (2002; Zbl 1057.14009)]. A special case of dim(X) arbitrary and \(\dim(Y) = 2\) is studied in [\textit{S. D. Cutkosky, O. Kascheyeva}, J. Algebra 275, 275--320 (2004; Zbl 1057.14008)]. In this paper the author gives an affirmative answer to the above conjecture for \(\dim(X)\) is arbitrary and \(\dim(Y) = 2\).