Equations for formal toric degenerations

TL;DR

This paper develops equations for formal toric degenerations of certain local domains, enabling the representation of valuations as limits of finitely generated semivaluations, thus advancing the understanding of degenerations in algebraic geometry.

Contribution

It introduces equations in a generalized power series ring for degenerations of local domains to toric graded algebras, and shows how to approximate valuations by finitely generated semivaluations.

Findings

Produced equations for algebraic degenerations to toric algebras.

Represented valuations as limits of finitely generated semivaluations.

Extended the understanding of semigroup structures in valuation theory.

Abstract

Let $R$ be a complete equicharacteristic noetherian local domain and $\nu$ a valuation of its field of fractions whose valuation ring dominates $R$ with trivial residue field extension. The semigroup of values of $\nu$ on $R\setminus \{0\}$ is not finitely generated in general. We produce equations in an appropriate generalized power series ring for the algebra encoding the degeneration of $R$ to the toric graded algebra ${\rm gr}_\nu R$ associated to the filtration defined by $\nu$. We apply this to represent $\nu$ as the limit of a sequence of Abhyankar semivaluations (valuations on quotients) of $R$ with finitely generated semigroups.