A survey on toric degenerations of projective varieties
Lara Bossinger

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 -Newton polytope. The -Newton polytope determines the Newton--Okounkov polytopes of all Khovanskii-finite valuations sharing the adapted standard monomial basis .
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Taxonomy
TopicsCommutative Algebra and Its Applications · Algebraic Geometry and Number Theory · Polynomial and algebraic computation
