A survey on toric degenerations of projective varieties

TL;DR

This survey reviews methods of constructing toric degenerations of projective varieties using valuations and Gr"obner theory, highlighting their equivalences and introducing generalized Newton polytopes called $ ext{B}$-Newton polytopes.

Contribution

It introduces the concept of $ ext{B}$-Newton polytopes and explains their role in unifying various toric degeneration constructions via valuations and Gr"obner theory.

Findings

$ ext{B}$-Newton polytopes generalize classical Newton polytopes.

The survey establishes the equivalence of different toric degeneration methods.

It connects adapted bases with Newton--Okounkov bodies.

Abstract

In this survey I summarize the constructions of toric degenerations obtained from valuations and Gr\"obner theory and describe in which sense they are equivalent. I show how adapted bases can be used to generalize the classical Newton polytope to what is called a $\mathbb B$-Newton polytope. The $\mathbb B$-Newton polytope determines the Newton--Okounkov polytopes of all Khovanskii-finite valuations sharing the adapted standard monomial basis $\mathbb B$.