A survey on toric degenerations of projective varieties

Abstract

In this survey I summarize the constructions of toric degenerations obtained from valuations and Gröbner 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 $\mathbb{B}$.

I wrote this survey for the Proceedings of the Nottingham Algebraic Geometry Seminar in 2021. I hope it serves as an overview or introduction to Toric Degenerations, it is especially addressed to students and non experts. Comments are very welcome