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Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
Lara Bossinger, Fatemeh Mohammadi, Alfredo Nájera Chávez
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Let V be the weighted projective variety defined by a weighted homogeneous ideal J and C a maximal cone in the Gr\"obner fan of J with m rays. We construct a flat family over 𝔸ᵐ that assembles the Gr\"obner degenerations of V associated with all faces of C. This is a multi-parameter generalization of the classical one-parameter Gr\"obner 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 𝔸ᵐ →X_C. We apply this construction to the Grassmannians Gr(2,ℂⁿ) with their Pl\"ucker embeddings and the Grassmannian Gr(3,ℂ⁶) with its cluster embedding. In each case, there exists a unique maximal Gr\"obner 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 Gr(2,ℂⁿ) we show how Escobar-Harada's mutation of Newton-Okounkov bodies can be recovered as tropicalized cluster mutation.
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