Toric degenerations and Newton-Okounkov bodies

TL;DR

This paper extends the wall-crossing results linking Newton-Okounkov bodies and tropical geometry to non-prime cones by developing a method to convert them into prime cones, demonstrated through the example of Gr(3,6).

Contribution

It introduces a novel procedure to transform non-prime cones into prime cones, enabling the application of existing wall-crossing results in broader contexts.

Findings

Successfully applied the method to Gr(3,6)

Computed new embeddings and tropicalizations

Analyzed the relation between original and new cones

Abstract

In recent times, a wide variety of combinatorics has been introduced in order to solve problems from algebraic geometry. Newton-Okounkov bodies and tropical geometry are two such combinatorial theories. As shown by Kaveh and Manon, there is a certain correspondence between these two. Building on this correspondence, and exploiting the link of both theories to toric degenerations, Harada and Escobar obtained their wall-crossing result for prime cones. This result states that moving between two adjacent prime maximal cones in a tropical variety corresponds to a mutation between the associated Newton-Okounkov bodies of these cones. In this thesis, we provide a method for applying the wall-crossing result to non-prime cones. Our approach uses a procedure developed by Bossinger, Lamboglia, Mincheva and Mohammadi in order to compute an embedding which changes a tropical variety in such a way that a non-prime cone becomes prime. Assuming that adjacent cones stay adjacent, the wall-crossing result can then be applied in this new embedding. Building on computations by Clarke, Mohammadi and Zaffalon, we show that for Gr(3,6), this approach works. We compute the new embedding and its tropicalization in this case, and study the relation between cones in the new embedding and the original embedding.