Rational Complexity-One T-Varieties Are Well-Poised

Nathan Ilten; Christopher Manon

doi:10.1093/imrn/rnx254

Rational Complexity-One T-Varieties Are Well-Poised

Nathan Ilten, Christopher Manon

Mathematics

Research output: Contribution to journal › Review article › peer-review

7 Scopus citations

Abstract

Given an affine rational complexity-one T-variety X, we construct an explicit embedding of X in affine space Aⁿ. 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
Pages (from-to)	4198-4232
Number of pages	35
Journal	International Mathematics Research Notices
Volume	2019
Issue number	13
DOIs	https://doi.org/10.1093/imrn/rnx254
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)

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10.1093/imrn/rnx254

Cite this

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title = "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 An. We show that this embedding is well-poised, that is, every initial ideal of IX is a prime ideal, and we determine the tropicalization Trop(X). We then study valuations of the coordinate ring RX of X which respect the torus action, showing that for full rank valuations, the natural generators of RX 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 S{\"u}{\ss} and Ilten.",

author = "Nathan Ilten and Christopher Manon",

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: {\textcopyright} 2017 The Author(s). Published by Oxford University Press. All rights reserved.",

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AU - Ilten, Nathan

AU - Manon, Christopher

N1 - 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.

PY - 2019/7/1

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N2 - Given an affine rational complexity-one T-variety X, we construct an explicit embedding of X in affine space An. We show that this embedding is well-poised, that is, every initial ideal of IX is a prime ideal, and we determine the tropicalization Trop(X). We then study valuations of the coordinate ring RX of X which respect the torus action, showing that for full rank valuations, the natural generators of RX 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.

AB - Given an affine rational complexity-one T-variety X, we construct an explicit embedding of X in affine space An. We show that this embedding is well-poised, that is, every initial ideal of IX is a prime ideal, and we determine the tropicalization Trop(X). We then study valuations of the coordinate ring RX of X which respect the torus action, showing that for full rank valuations, the natural generators of RX 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.

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DO - 10.1093/imrn/rnx254

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JO - International Mathematics Research Notices

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IS - 13

ER -

Rational Complexity-One T-Varieties Are Well-Poised

Abstract

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