scholarly journals Tropical geometry and Newton–Okounkov cones for Grassmannian of planes from compactifications

2020 ◽  
pp. 1-33
Author(s):  
Christopher Manon ◽  
Jihyeon Jessie Yang
Keyword(s):  
Tropical Geometry ◽  
Flag Varieties ◽  
Tropical Variety ◽  
Affine Cone ◽  
Toric Degenerations

Abstract We construct a family of compactifications of the affine cone of the Grassmannian variety of $2$ -planes. We show that both the tropical variety of the Plücker ideal and familiar valuations associated to the construction of Newton–Okounkov bodies for the Grassmannian variety can be recovered from these compactifications. In this way, we unite various perspectives for constructing toric degenerations of flag varieties.

scholarly journals Tropical ideals do not realise all Bergman fans

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Jan Draisma ◽  
Felipe Rincón
Keyword(s):  
Tropical Geometry ◽  
Tensor Products ◽  
Tropical Variety ◽  
Polyhedral Complex ◽  
Direct Sum ◽  
Weight One ◽  
Polyhedral Complexes

AbstractEvery tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.

scholarly journals Newton–Okounkov Bodies of Flag Varieties and Combinatorial Mutations

2020 ◽  
Author(s):  
Naoki Fujita ◽  
Akihiro Higashitani
Keyword(s):  
Mirror Symmetry ◽  
Projective Variety ◽  
Fundamental Problem ◽  
Flag Varieties ◽  
Systematic Method ◽  
Fano Manifolds ◽  
Projective Varieties ◽  
Combinatorial Properties ◽  
Toric Degenerations ◽  
Higher Rank

Abstract A Newton–Okounkov body is a convex body constructed from a projective variety with a globally generated line bundle and with a higher rank valuation on the function field, which gives a systematic method of constructing toric degenerations of projective varieties. Its combinatorial properties heavily depend on the choice of a valuation, and it is a fundamental problem to relate Newton–Okounkov bodies associated with different kinds of valuations. In this paper, we address this problem for flag varieties using the framework of combinatorial mutations, which was introduced in the context of mirror symmetry for Fano manifolds. By applying iterated combinatorial mutations, we connect specific Newton–Okounkov bodies of flag varieties including string polytopes, Nakashima–Zelevinsky polytopes, and Feigin–Fourier–Littelmann–Vinberg polytopes.

scholarly journals On toric degenerations of flag varieties

2016 ◽  
pp. 187-232 ◽  
Author(s):  
Xin Fang ◽  
Ghislain Fourier ◽  
Peter Littelmann
Keyword(s):  
Flag Varieties ◽  
Toric Degenerations

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

2020 ◽  
Author(s):  
Laura Escobar ◽  
Megumi Harada
Keyword(s):  
Projective Variety ◽  
Tropical Geometry ◽  
Technical Condition ◽  
Complex Projective Variety ◽  
Wall Crossing ◽  
Toric Degenerations ◽  
Complex Projective

Abstract Tropical geometry and the theory of Newton–Okounkov bodies are two methods that produce toric degenerations of an irreducible complex projective variety. Kaveh and Manon showed that the two are related. We give geometric maps between the Newton–Okounkov bodies corresponding to two adjacent maximal-dimensional prime cones in the tropicalization of $X$. Under a technical condition, we produce a natural “algebraic wall-crossing” map on the underlying value semigroups (of the corresponding valuations). In the case of the tropical Grassmannian $Gr(2,m)$, we prove that the algebraic wall-crossing map is the restriction of a geometric map. In an appendix by Nathan Ilten, he explains how the geometric wall-crossing phenomenon can also be derived from the perspective of complexity-one $T$-varieties; Ilten also explains the connection to the “combinatorial mutations” studied by Akhtar–Coates–Galkin–Kasprzyk.

scholarly journals COMPLETIONS, REVERSALS, AND DUALITY FOR TROPICAL VARIETIES

2011 ◽  
Vol 10 (06) ◽  
pp. 1141-1163
Author(s):  
ZUR IZHAKIAN ◽  
LOUIS ROWEN
Keyword(s):  
Algebraic Structure ◽  
Tropical Geometry ◽  
Tropical Variety ◽  
Single Face ◽  
Polyhedral Complexes ◽  
Tropical Varieties

The object of this paper is to present two algebraic results with straightforward proofs, which have interesting consequences in tropical geometry. We start with an identity for polynomials over the max-plus algebra, which shows that any polynomial divides a product of binomials. Interpreted in tropical geometry, any tropical variety W can be completed to a union of tropical primitives, i.e. single-face polyhedral complexes. In certain situations, a tropical variety W has a "reversal" variety, which together with W already yields the union of primitives; this phenomenon is explained in terms of a map defined on the algebraic structure, and yields a duality on tropical hypersurfaces.

scholarly journals Counting curves on Hirzebruch surfaces: tropical geometry and the Fock space

2021 ◽  
pp. 1-41
Author(s):  
RENZO CAVALIERI ◽  
PAUL JOHNSON ◽  
HANNAH MARKWIG ◽  
DHRUV RANGANATHAN
Keyword(s):  
Fock Space ◽  
Tropical Geometry ◽  
Geometric Interpretation ◽  
Toric Surfaces ◽  
Tropical Curves ◽  
Hirzebruch Surfaces ◽  
Witten Theory ◽  
Toric Degenerations ◽  
Correspondence Theorem ◽  
Floor Diagrams

We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are showno coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttsche, Cooper–Pandharipande and Block–Gathmann–Markwig.

scholarly journals Phylogenetic trees and the tropical geometry of flag varieties

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Christopher Manon
Keyword(s):  
Phylogenetic Trees ◽  
Tropical Geometry ◽  
Type A ◽  
Hyperplane Arrangements ◽  
Nous Allons ◽  
Flag Varieties ◽  
Dynkin Diagrams ◽  
The Family ◽  
International Audience ◽  
Tropical Varieties

International audience We will discuss some recent theorems relating the space of weighted phylogenetic trees to the tropical varieties of each flag variety of type A. We will also discuss the tropicalizations of the functions corresponding to semi-standard tableaux, in particular we relate them to familiar functions from phylogenetics. We close with some remarks on the generalization of these results to the tropical geometry of arbitrary flag varieties. This involves the family of Bergman complexes derived from the hyperplane arrangements associated to simple Dynkin diagrams. Nous allons discuter de quelques théorèmes récents concernant l'espace des arbres phylogénétiques aux variétés Tropicales de chaque variété de drapeaux de type A. Nous allons également discuter des tropicalisations des fonctions correspondant à tableaux semi-standard, en particulier, nous les rapporter à des fonctions familières de la phylogénétique. Nous terminerons avec quelques remarques sur la généralisation de ces résultats à la géométrie tropicale de variétés de drapeaux arbitraires. Il s'agit de la famille de complexes Bergman provenant des arrangements d'hyperplans associés à des diagrammes de Dynkin simples.

scholarly journals Computing Toric Degenerations of Flag Varieties

2017 ◽  
pp. 247-281 ◽  
Author(s):  
Lara Bossinger ◽  
Sara Lamboglia ◽  
Kalina Mincheva ◽  
Fatemeh Mohammadi
Keyword(s):  
Flag Varieties ◽  
Toric Degenerations
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