Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras
نویسندگان
چکیده
Let $V$ be the weighted projective variety defined by a homogeneous ideal $J$ and $C$ maximal cone in Gr\"obner fan of with $m$ rays. We construct flat family over $\mathbb A^m$ that assembles degenerations associated all faces $C$. This is multi-parameter generalization classical one-parameter degeneration to weight. explain how our can constructed from Kaveh-Manon's recent work on classification toric families varieties: it pull-back Rees algebra base $X_C$ (the $C$) along universal torsor A^m \to X_C$. apply this construction Grassmannians ${\rm Gr}(2,\mathbb C^n)$ their Pl\"ucker embeddings Grassmannian Gr}\big(3,\mathbb C^6\big)$ its cluster embedding. In each case, there exists unique whose initial Stanley-Reisner complex. show corresponding coefficients arises as defining cone. Further, for we Escobar-Harada's mutation Newton-Okounkov bodies recovered tropicalized mutation.
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ژورنال
عنوان ژورنال: Symmetry Integrability and Geometry-methods and Applications
سال: 2021
ISSN: ['1815-0659']
DOI: https://doi.org/10.3842/sigma.2021.059