Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Families of Gr\"obner Degenerations, Grassmannians and Universal Cluster
  Algebras   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 $\mathbb A^m$ 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 $\mathbb A^m \to X_C$. We apply this construction to the
Grassmannians ${\rm Gr}(2,\mathbb C^n)$ with their Pl\"ucker embeddings and the
Grassmannian ${\rm Gr}\big(3,\mathbb C^6\big)$ 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 ${\rm
Gr}(2,\mathbb C^n)$ we show how Escobar-Harada's mutation of Newton-Okounkov
bodies can be recovered as tropicalized cluster mutation.

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The categories of this paper are: econ
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The categories of this paper are: gr-qc
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The categories of this paper are: astro-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: physics
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: cs
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The categories of this paper are: eess
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The categories of this paper are: math-ph