Local factorization and monomialization of morphisms

TL;DR

This paper proves that for a generically finite map between nonsingular varieties over a characteristic zero field, one can perform transformations to make the map monomial at a valuation, enabling factorization of birational morphisms.

Contribution

It introduces a method to monomialize morphisms via monoidal transforms, facilitating valuation-based factorization of birational maps.

Findings

Achieved monomialization of morphisms at a valuation

Facilitated factorization of birational morphisms

Provided a sequence of blowups and blowdowns with nonsingular centers

Abstract

Suppose that X to Y is a generically finite map of nonsingular varieties over a field of characteristic zero, and v is a valuation of the function field of X. We prove that it is possible to perform a sequence of monoidal transforms X' to X and Y' to Y so that X' to Y' is a monomial mapping at the center of v. We deduce from this that a birational morphism of nonsingular varieties can be factored along a valuation by a sequence of blowups and blowdowns with nonsingular centers.