# Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras

@article{Bossinger2020FamiliesOG, title={Families of Gr{\"o}bner Degenerations, Grassmannians and Universal Cluster Algebras}, author={Lara Bossinger and Fatemeh Mohammadi and Alfredo N'ajera Ch'avez}, journal={arXiv: Algebraic Geometry}, year={2020} }

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… Expand

#### 8 Citations

Standard monomial theory and toric degenerations of Richardson varieties in the Grassmannian

- Mathematics
- 2021

Richardson varieties are obtained as intersections of Schubert and opposite Schubert varieties. We provide a new family of toric degenerations of Richardson varieties inside Grassmannians by studying… Expand

Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux

- Mathematics, Computer Science
- J. Symb. Comput.
- 2021

An analogue of matching field ideals for Schubert varieties inside the flag variety and a complete characterization of toric ideals among them are described to show that block diagonal matching fields give rise to toric degenerations. Expand

Combinatorial mutations and block diagonal polytopes

- Mathematics
- 2020

Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes and more recently have been shown to give rise to toric degenerations of various families of varieties.… Expand

Quadratic Gr\"obner bases of block diagonal matching field ideals and toric degenerations of Grassmannians.

- Mathematics
- 2020

In the present paper, we prove that the toric ideals of certain $s$-block diagonal matching fields have quadratic Grobner bases. Thus, in particular, those are quadratically generated. By using this… Expand

Standard monomial theory and toric degenerations of Richardson varieties inside Grassmannians and flag varieties

- Mathematics
- 2020

We study toric degenerations of opposite Schubert and Richardson varieties inside degenerations of Grassmannians and flag varieties. These degenerations are parametrized by matching fields in the… Expand

Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux

- Mathematics
- 2020

Abstract We study Grobner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by matching field tableaux in… Expand

Toric degenerations of flag varieties from matching field tableaux

- Mathematics
- 2019

We present families of tableaux which interpolate between the classical semi-standard Young tableaux and matching field tableaux. Algebraically, this corresponds to SAGBI bases of Plucker algebras.… Expand

Wall-Crossing for Newton–Okounkov Bodies and the Tropical Grassmannian

- Mathematics
- 2019

Let $X$ be an irreducible complex projective variety. Tropical geometry and the theory of Newton-Okounkov bodies are two methods which can produce toric degenerations of $X$, and in 2016, Kaveh and… Expand

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Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes and more recently have been shown to give rise to toric degenerations of various families of varieties.… Expand

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Abstract We study Grobner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by matching field tableaux in… Expand