Rational Complexity-One 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 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 ${\mathbb{K}}^{\ast }\times T$-equivariant degenerations of rational projective complexity-one $T$-varieties, generalizing a results of Süß and the first author.

v2 minor revisions, 25 pages, to appear in IMRN

References in corpus (2)

• K-Stability for Fano Manifolds with Torus Action of Complexity One
• Log terminal singularities, platonic tuples and iteration of Cox rings

Cited by in corpus (3)

• Wall-crossing for Newton-Okounkov bodies and the tropical Grassmannian
• On irregular Sasaki-Einstein metrics in dimension 5
• On the finite generation of valuation semigroups on toric surfaces