Khovanskii-finite valuations, rational curves, and torus actions

Ilten, Nathan; Wrobel, Milena

doi:10.4171/JCA/41

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Journal of Combinatorial Algebra

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Volume 4, Issue 2, 2020, pp. 141–166

DOI: 10.4171/JCA/41

Published online: 2020-06-25

Khovanskii-finite valuations, rational curves, and torus actions

Nathan Ilten^[1] and Milena Wrobel^[2]

(1) Simon Fraser University, Burnaby, Canada
(2) Simon Fraser University, Burnaby, Canada

We study full rank homogeneous valuations on (multi)-graded domains and ask when they have finite Khovanskii bases. We show that there is a natural reduction from multigraded to simply graded domains. As special cases, we consider projective coordinate rings of rational curves, and almost toric varieties. Our results relate to several problems posed by Kaveh and Manon, and imply that the procedure of Bossinger–Lamboglia–Mincheva–Mohammadi for producing tropical prime cones will not terminate in general.

Keywords: Khovanskii bases, valuations, degenerations, torus actions

Ilten Nathan, Wrobel Milena: Khovanskii-finite valuations, rational curves, and torus actions. J. Comb. Algebra 4 (2020), 141-166. doi: 10.4171/JCA/41