# Rational Complexity-One T-Varieties are Well-Poised

## TL;DR

This paper constructs explicit well-poised embeddings for affine rational complexity-one T-varieties, analyzes their tropicalizations, and studies valuations to determine Newton-Okounkov bodies, extending previous results in the field.

## Contribution

It provides explicit embeddings and tropicalizations for rational complexity-one T-varieties and establishes conditions under which generators form a Khovanskii basis, generalizing prior work.

## Key findings

- Embeddings are well-poised with prime initial ideals.
- Generators form a Khovanskii basis for full rank valuations.
- Newton-Okounkov bodies are determined for rational projective complexity-one T-varieties.

## Abstract

Given an affine rational complexity-one $T$-variety $X$, we construct an explicit embedding of $X$ in affine space $\mathbb{A}^n$. We show that this embedding is well-poised, that is, every initial ideal of $I_X$ is a prime ideal, and determine the tropicalization of $X$. We then study valuations of the coordinate ring $R_X$ of $X$ which respect the torus action, showing that for full rank valuations, the natural generators of $R_X$ form a Khovanskii basis. This allows us to determine Newton-Okounkov bodies of rational projective complexity-one $T$-varieties, partially recovering (and generalizing) results of Petersen. We apply our results to describe all irreducible special fibers of $\mathbb{K}^*\times T$-equivariant degenerations of rational projective complexity-one $T$-varieties, generalizing a results of S\"u\ss{} and the first author.