Local monomialization of transcendental extensions

TL;DR

The paper proves that for a dominant morphism between varieties over a characteristic zero field, one can perform blowups to locally monomialize the map near a valuation, extending previous results to arbitrary valuations.

Contribution

It generalizes earlier monomialization results by allowing arbitrary valuations, not just discrete or generically finite cases.

Findings

Existence of sequences of blowups making the morphism locally monomial near the valuation center.

Extension of previous results to arbitrary valuations, not restricted to discrete or finite cases.

Construction of nonsingular models where the morphism is locally monomial.

Abstract

Suppose that f is a dominant morphism from a k-variety X to a k-variety Y, where k is a field of characteristic 0 and v is a valuation of the function field k(X). We allow v to be an arbitary valuation, so it may not be discrete. We prove that there exist sequences of blowups of nonsingular subvarieties from X' to X and from Y' to Y such that X', Y' are nonsingular and X' to Y' is locally a monomial mapping near the center of v. This extends an earlier result of ours (in Asterisque 260) which proves the above result with the restriction that f is generically finite.