Overweight deformations of affine toric varieties and local uniformization

TL;DR

This paper proves local uniformization for Abhyankar valuations in excellent equicharacteristic local domains with algebraically closed residue fields, using properties of finitely generated semigroups and birational maps.

Contribution

It establishes a new uniformization result for Abhyankar valuations by linking finitely generated semigroups to monomialization via birational maps.

Findings

Uniformization for zero-dimensional valuations with finitely generated semigroups.

Characterization of Abhyankar valuations through finitely generated semigroups after modifications.

Extension of uniformization results to all Abhyankar valuations in the setting.

Abstract

We study Abhyankar valuations of excellent equicharacteristic local domains with an algebraically closed residue field. For zero dimensional valuations we prove that whenever the ring is complete and the semigroup of values taken by the valuation is finitely generated (which implies that the valuation is Abhyankar) the valuation can be uniformized in an embedded way by a birational map which is monomial with respect to a suitable system of generators of the maximal ideal. We prove that conversely if a valuation is Abhyankar after a birational modification and localization at the point picked by the valuation one obtains a ring whose semigroup of values is finitely generated. Combining the two results and using the good behavior of Abhyankar valuations with respect to composition and completion gives local uniformization for all Abhyankar valuations of excellent equicharacteristic local domains with an algebraically closed residue field. Some general results on Abhyankar valuations are by-products of the method of proof.