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 ${\mathbb{A}}^m$ 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 $\mathrm{Gr}(2,{\mathbb{C}}^n)$ with their Plücker embeddings and the Grassmannian $\mathrm{Gr}(3,{\mathbb{C}}^6)$ 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 $\mathrm{Gr}(2,{\mathbb{C}}^n)$ we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation.

References in corpus (2)

• Minkowski Polynomials and Mutations
• Mutations of Laurent Polynomials and Flat Families with Toric Fibers

Cited by in corpus (2)

• Wall-crossing for Newton-Okounkov bodies and the tropical Grassmannian
• Standard monomial theory and toric degenerations of Richardson varieties inside Grassmannians and flag varieties