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 Grobner fan of $J$ with $m$ rays. We construct a flat family over $\mathbb A^m$ that assembles the Grobner degenerations of $V$ associated with all faces of $C$. This is a multi-parameter generalization of the classical one-parameter Grobner 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 $\mathbb A^m \to X_C$. We apply this construction to the Grassmannians ${\rm Gr}(2,\mathbb C^n)$ with their Plucker embeddings and the Grassmannian ${\rm Gr}(3,\mathbb C^6)$ with its cluster embedding. In each case there exists a unique maximal Grobner 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.