{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/families-of-gr-obner-degenerations","title":"Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras","arxiv_id":"2007.14972","date":"2020-07-29","proceeding":null,"authors":["Lara Bossinger","Fatemeh Mohammadi","Alfredo Nájera Chávez"],"abstract":"Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Gr\\\"obner fan of $J$ with $m$ rays. We construct a flat family over $\\mathbb A^m$ that assembles the Gr\\\"obner degenerations of $V$ associated with all faces of $C$. This is a multi-parameter generalization of the classical one-parameter Gr\\\"obner degeneration associated to 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 $X_C$ (the toric variety associated to $C$) along the universal torsor $\\mathbb A^m \\to X_C$. We apply this construction to the Grassmannians ${\\rm Gr}(2,\\mathbb C^n)$ with their Pl\\\"ucker embeddings and the Grassmannian ${\\rm Gr}\\big(3,\\mathbb C^6\\big)$ with its cluster embedding. In each case, there exists a unique maximal Gr\\\"obner 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 to this cone. Further, for ${\\rm Gr}(2,\\mathbb C^n)$ we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation.","url_abs":"https://arxiv.org/abs/2007.14972v3","url_pdf":"https://arxiv.org/pdf/2007.14972v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"families-of-gr-obner-degenerations","repo_url":"https://github.com/lbossing/clusterGr36","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}