Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
Let be the weighted projective variety defined by a weighted homogeneous ideal and a maximal cone in the Gröbner fan of with rays. We construct a flat family over ᵐ that assembles the Gröbner degenerations of associated with all faces of . This is a multi-parameter generalization of the classi...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2021 |
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Інститут математики НАН України
2021
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211364 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras. Lara Bossinger, Fatemeh Mohammadi and Alfredo Nájera Chávez. SIGMA 17 (2021), 059, 46 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862532627095879680 |
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| author | Bossinger, Lara Mohammadi, Fatemeh Nájera Chávez, Alfredo |
| author_facet | Bossinger, Lara Mohammadi, Fatemeh Nájera Chávez, Alfredo |
| citation_txt | Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras. Lara Bossinger, Fatemeh Mohammadi and Alfredo Nájera Chávez. SIGMA 17 (2021), 059, 46 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Let be the weighted projective variety defined by a weighted homogeneous ideal and a maximal cone in the Gröbner fan of with rays. We construct a flat family over ᵐ that assembles the Gröbner degenerations of associated with all faces of . This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated with a weight. We explain how our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algebra with base C (the toric variety associated to ) along the universal torsor ᵐ → C. We apply this construction to the Grassmannians Gr(2, ℂⁿ) with their Plücker embeddings and the Grassmannian Gr(3, ℂ⁶) with its cluster embedding. In each case, there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated with this cone. Further, for Gr(2, ℂⁿ), we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as a tropicalized cluster mutation.
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| first_indexed | 2026-03-12T14:01:37Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211364 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T14:01:37Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
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| spelling | Bossinger, Lara Mohammadi, Fatemeh Nájera Chávez, Alfredo 2025-12-30T15:56:47Z 2021 Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras. Lara Bossinger, Fatemeh Mohammadi and Alfredo Nájera Chávez. SIGMA 17 (2021), 059, 46 pages 1815-0659 2020 Mathematics Subject Classification: 13F60; 14D06; 14M25; 14M15; 13P10 arXiv:2007.14972 https://nasplib.isofts.kiev.ua/handle/123456789/211364 https://doi.org/10.3842/SIGMA.2021.059 Let be the weighted projective variety defined by a weighted homogeneous ideal and a maximal cone in the Gröbner fan of with rays. We construct a flat family over ᵐ that assembles the Gröbner degenerations of associated with all faces of . This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated with a weight. We explain how our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algebra with base C (the toric variety associated to ) along the universal torsor ᵐ → C. We apply this construction to the Grassmannians Gr(2, ℂⁿ) with their Plücker embeddings and the Grassmannian Gr(3, ℂ⁶) with its cluster embedding. In each case, there exists a unique maximal Gröbner cone whose associated initial ideal is the Stanley-Reisner ideal of the cluster complex. We show that the corresponding cluster algebra with universal coefficients arises as the algebra defining the flat family associated with this cone. Further, for Gr(2, ℂⁿ), we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as a tropicalized cluster mutation. This work was partially supported by CONACyT grant CB2016 no. 284621. L.B. was supported by Programa de Becas Posdoctorales en la UNAM 2018, Instituto de Matemáticas, Universidad Nacional Autonoma de México. F.M. thanks the Instituto de Matemáticas, UNAM Unidad Oaxaca, for their hospitality during this project and also acknowledges partial support by the EPSRC Early Career Fellowship EP/R023379/1, the Starting Grant of Ghent University BOF/STA/201909/038, and the FWO grants (G023721N and G0F5921N). L.B. would like to thank Kiumars Kaveh for providing an opportunity to present this work, and further Nathan Ilten for his insightful comments. A special thanks goes to Christopher Manon, who patiently explained his joint work with Kaveh to us. The connection between cluster mutation and Escobar-Harada's flip map was first observed in a discussion with Megumi Harada during the MFO-Workshop on Toric geometry in 2019, see [10]. We would like to thank the organizers of the meeting and, in particular, Megumi Harada for explaining to us the results of [19]. We are grateful to Brenda Policarpo Sibaja and Nathan Ilten for pointing out misprints in an earlier version of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras Article published earlier |
| spellingShingle | Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras Bossinger, Lara Mohammadi, Fatemeh Nájera Chávez, Alfredo |
| title | Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras |
| title_full | Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras |
| title_fullStr | Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras |
| title_full_unstemmed | Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras |
| title_short | Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras |
| title_sort | families of gröbner degenerations, grassmannians and universal cluster algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211364 |
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