Rational Complexity-One $\boldsymbol{T}$-Varieties Are Well-Poised

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 we determine the tropicalization ${\mathrm{Trop}}(X^\circ)$. 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 integral special fibres of ${\mathbb{K}}^*\times T$-equivariant degenerations of rational projective complexity-one $T$-varieties, generalizing a result of Süß and Ilten.