Monomialization of strongly prepared morphisms from nonsingular n-folds to surfaces

Local monomialization of transcendental extensions

“…A generalization of this result to prove monomialization of strongly prepared morphisms from N -folds to surfaces appears in the paper [17] with Olga Kashcheyeva.…”

Resolution of Singularities

“…A generalized multiplicity is defined in this paper, and it can go up, causing a very high complexity in the proof. An extension of this result to strongly prepared morphisms from n-folds to surfaces is proven in [33]. It is not known if monomialization is true even for birational morphisms of varieties of dimension ≥ 3, although it is true locally along a valuation, from the following Theorem 2.18.…”

A simpler proof of toroidalization of morphisms from 3-folds to surfaces

“…This second step is generalized in [16] to prove that a strongly prepared morphism from an n-fold to a surface can be toroidalized. Thus to prove toroidalization of a morphism from an n-fold to a surface, it suffices to proof strong preparation.…”