## Abstract

Given an affine rational complexity-one T-variety X, we construct an explicit embedding of X in affine space 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 Trop(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 integral special fibres of K* × T-equivariant degenerations of rational projective complexity-one T-varieties, generalizing a result of Süß and Ilten.

Original language | English |
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Pages (from-to) | 4198-4232 |

Number of pages | 35 |

Journal | International Mathematics Research Notices |

Volume | 2019 |

Issue number | 13 |

DOIs | |

State | Published - Jul 1 2019 |

### Bibliographical note

Funding Information:supported by NSF grant DMS 1500966 to C.M.

Funding Information:

This work was partially supported by an NSERC Discovery Grant to N.I. and partially

Publisher Copyright:

© 2017 The Author(s). Published by Oxford University Press. All rights reserved.

## ASJC Scopus subject areas

- Mathematics (all)