Families of Gröbner degenerations, Grassmannians, and universal cluster algebras

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 affine $m$-space 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 show that our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pullback of a toric family defined by a Rees algebra with base $X_C$ (the toric variety associated to $C$) along the universal torsor $\mathbb{A}^m \to X_C$. If time permits I will explain how to apply this construction to the Grassmannians $\mathrm{Gr}(2,n)$ (with Plücker embedding) and $\mathrm{Gr}(3,6)$ (with "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 $\mathrm{Gr}(2,n)$ we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation. This is joint work with Fatemeh Mohammadi and Alfredo Nájera Chávez.

algebraic geometrycombinatorics

Audience: researchers in the topic

( slides | video )

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Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html

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