Khovanskii bases, higher rank valuations and tropical geometry

TL;DR

This paper establishes a connection between valuations, tropical geometry, and Khovanskii bases for finitely generated algebras, providing new tools for understanding degenerations and compactifications.

Contribution

It introduces the notion of Khovanskii bases for algebras with valuations, linking tropical geometry with Newton-Okounkov bodies and toric degenerations.

Findings

Provides a necessary and sufficient condition for the existence of finitely generated value semigroups.

Connects Khovanskii bases to tropical geometry and Newton-Okounkov bodies.

Constructs a compactification of the spectrum of the algebra.

Abstract

Given a finitely generated algebra $A$, it is a fundamental question whether $A$ has a full rank discrete (Krull) valuation $\mathfrak{v}$ with finitely generated value semigroup. We give a necessary and sufficient condition for this, in terms of tropical geometry of $A$. In the course of this we introduce the notion of a Khovanskii basis for $(A, \mathfrak{v})$ which provides a framework for far extending Gr\"obner theory on polynomial algebras to general finitely generated algebras. In particular, this makes a direct connection between the theory of Newton-Okounkov bodies and tropical geometry, and toric degenerations arising in both contexts. We also construct an associated compactification of $Spec(A)$. Our approach includes many familiar examples such as the Gel'fand-Zetlin degenerations of coordinate rings of flag varieties as well as wonderful compactifications of reductive groups. We expect that many examples coming from cluster algebras naturally fit into our framework.