Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 059, 46 pages      arXiv:2007.14972      https://doi.org/10.3842/SIGMA.2021.059

Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras

Lara Bossinger a, Fatemeh Mohammadi bc and Alfredo Nájera Chávez ad
a) Instituto de Matemáticas UNAM Unidad Oaxaca, León 2, altos, Oaxaca de Juárez, Centro Histórico, 68000 Oaxaca, Mexico
b) Department of Mathematics: Algebra and Geometry, Ghent University, 9000 Gent, Belgium
c) Department of Mathematics and Statistics, UiT - The Arctic University of Norway, 9037 Tromsø, Norway
d) Consejo Nacional de Ciencia y Tecnología, Insurgentes Sur 1582, Alcaldía Benito Juárez, 03940 CDMX, Mexico

Received October 21, 2020, in final form May 29, 2021; Published online June 10, 2021

Abstract
Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Gröbner fan of $J$ with $m$ rays. We construct a flat family over $\mathbb A^m$ that assembles the Gröbner degenerations of $V$ associated with all faces of $C$. This is a multi-parameter generalization of the classical one-parameter Gröbner 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 Plücker embeddings and the Grassmannian ${\rm Gr}\big(3,\mathbb C^6\big)$ 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 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.

Key words: cluster algebras; Gröbner basis; Gröbner fan; Grassmannians; flat degenerations; Newton-Okounkov bodies.

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