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 𝔸ᵐ 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 with 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 𝑋C (the toric variety associated to 𝐶) along the universal torsor 𝔸ᵐ → 𝑋C. We apply this construction to the Grassmannians Gr(2, ℂⁿ) with their Plücker embeddings and the Grassmannian Gr(3, ℂ⁶) 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 with this cone. Further, for Gr(2, ℂⁿ), we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as a tropicalized cluster mutation.