Spherical tropical curves are balanced

TL;DR

This paper extends the concept of balancing conditions from tropicalizations of algebraic tori to spherical homogeneous spaces, establishing a generalized balancing condition for spherical tropicalizations of curves.

Contribution

It proves a new balancing condition for spherical tropicalizations of curves in spherical homogeneous spaces, generalizing the classical toric case.

Findings

Established a balancing condition for spherical tropicalizations

Connected spherical tropicalization with Gross's balancing condition

Provided examples illustrating the generalized balancing condition

Abstract

One of the characterizing features of tropicalizations of curves in an algebraic torus is that they are balanced. Tevelev and Vogiannou introduced a spherical tropicalization map for spherical homogeneous spaces $G/H$, where $G$ is a reductive group. This map generalizes the tropicalization map for algebraic tori. We prove a balancing condition for spherical tropicalizations of curves in $G/H$ that generalizes the balancing condition for tropicalizations of curves contained in an algebraic torus. We give examples and describe the relationship with Andreas Gross's balancing condition for tropicalizations of subvarieties of toroidal embeddings.