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

Figures from this paper

Standard monomial theory and toric degenerations of Richardson varieties in the Grassmannian
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 studyingExpand
Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux
TLDR
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
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.
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 thisExpand
Standard monomial theory and toric degenerations of Richardson varieties inside Grassmannians and flag varieties
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 theExpand
Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux
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 inExpand
Toric degenerations of flag varieties from matching field tableaux
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
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 andExpand

References

SHOWING 1-10 OF 73 REFERENCES
Toric bundles
  • valuations, and tropical geometry over semifield of piecewise linear functions. arXiv preprint arXiv:1907.00543
  • 2019
Toric flat families, valuations, and tropical geometry over the semifield of piecewise linear functions
Using the notion of a valuation into the semifield of piecewise linear functions, we give a classification of torus equivariant flat families of finite type over a toric variety base, by certainExpand
Canonical bases for cluster algebras
In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonicalExpand
Khovanskii bases
  • higher rank valuations, and tropical geometry. SIAM J. Appl. Algebra Geom., 3(2):292–336
  • 2019
Wall-crossing for Newton-Okounkov bodies and the tropical Grassmannian
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 andExpand
Combinatorial mutations and block diagonal polytopes
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
Newton-Okounkov polytopes of Schubert varieties arising from cluster structures
The theory of Newton-Okounkov bodies is a generalization of that of Newton polytopes for toric varieties, and it gives a systematic method of constructing toric degenerations of projective varieties.Expand
On cluster duality for Grassmannians
  • preparation
  • 2020
The Cremona Group and Its Subgroups
— We deal with the CREMONA group via its subgroups. We first describe the isometric action of the CREMONA group on an infinite dimension hyperbolic space. Then we mention the classification of theExpand
Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux
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 inExpand
...
1
2
3
4
5
...