Tropical geometry and Newton-Okounkov cones for Grassmannian of planes from compactifications

TL;DR

This paper develops a unified framework linking tropical geometry, Newton-Okounkov bodies, and compactifications to study toric degenerations of Grassmannians of planes, enhancing understanding of their geometric and combinatorial structures.

Contribution

It introduces a new family of compactifications that connect tropical varieties and Newton-Okounkov bodies for Grassmannians, unifying different approaches to toric degenerations.

Findings

Tropical variety of the Plücker ideal is recovered from the compactifications.

Valuations associated with Newton-Okounkov bodies are derived from these compactifications.

The framework unifies multiple perspectives on constructing toric degenerations.

Abstract

We construct a family of compactifications of the affine cone of the Grassmannian variety of 2-planes. We show that both the tropical variety of the Pl\"ucker ideal and familiar valuations associated to the construction of Newton-Okounkov bodies for the Grassmannian variety can be recovered from these compactifications. In this way, we unite various perspectives for constructing toric degenerations of flag varieties.