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Toric degenerations of \(\mathrm{Gr}(2, n)\) and \(\mathrm{Gr}(3, 6)\) via plabic graphs. (English) Zbl 06948249
Summary: We establish an explicit bijection between the toric degenerations of the Grassmannian \(\mathrm{Gr}(2, n)\) arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphs corresponding to \(\mathrm{Gr}(2, n)\). We show that a similar statement does not hold for \(\mathrm{Gr}(3, 6)\).

14M15 Grassmannians, Schubert varieties, flag manifolds
14T05 Tropical geometry (MSC2010)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
05C21 Flows in graphs
Full Text: DOI
[1] Anderson, D., Okounkov bodies and toric degenerations, Math. Ann., 356, 1183-1202, (2013) · Zbl 1273.14104
[2] Brodsky, S.; Ceballos, C.; Labbé, J., Cluster algebras of type D4, tropical planes, and the positive tropical Grassmannian, Beitr. Algebra Geom., 58, 25-46, (2017) · Zbl 1401.13063
[3] Buczyńska, W.; Wiśniewski, J., On geometry of binary symmetric models of phylogenetic trees, J. Eur. Math. Soc., 9, 609-635, (2007) · Zbl 1147.14027
[4] Eisenbud, D.: Commutative Algebra. Grad. Texts in Math., Vol. 150. Springer-Verlag, New York (1995) · Zbl 0819.13001
[5] Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
[6] Gross, M.; Hacking, P.; Keel, S., Birational geometry of cluster algebras, Algebr. Geom., 2, 137-175, (2015) · Zbl 1322.14032
[7] Gross, M.; Hacking, P.; Keel, S.; Kontsevich, M., Canonical bases for cluster algebras, J. Amer. Math. Soc., 31, 497-608, (2018) · Zbl 1446.13015
[8] Hering, M.: Macaulay 2 computation comparing toric degenerations of Gr(3, 6) arising from plabic graphs with the tropical Grassmannian trop Gr(3, 6). Available at http://www.maths.ed.ac.uk/ mhering/Papers/PlabicGraphDegens/PlabicGraphDegensGr36.m2
[9] Kaveh, K.; Khovanskii, A. G., Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2), 176, 925-978, (2012) · Zbl 1270.14022
[10] Kaveh, K., Manon, C.: Khovanskii bases, higher rank valuations and tropical geometry. Preprint (2016)
[11] Kodama, Y.; Williams, L., KP solitons and total positivity for the Grassmannian, Invent. Math., 198, 637-699, (2014) · Zbl 1306.35109
[12] Lakshmibai, V., Brown, J.: The Grassmannian Variety. Geometric and Representation-Theoretic Aspects. Dev. Math., Vol. 42. Springer, New York (2015) · Zbl 1343.14001
[13] Lazarsfeld, R.; Mustaţă, M., Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4), 42, 783-835, (2009) · Zbl 1182.14004
[14] Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. Grad. Stud. Math., Vol. 161. American Mathematical Society, Providence, RI (2015) · Zbl 1321.14048
[15] Postnikov, A.: Total positivity, Grassmannians, and networks. Preprint (2006)
[16] Rietsch, K.,Williams, L.: Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians. Preprint (2017)
[17] Scott, J. S., Grassmannians and cluster algebras, Proc. London Math. Soc. (3), 92, 345-380, (2006) · Zbl 1088.22009
[18] Speyer, D.; Sturmfels, B., The tropical Grassmannian, Adv. Geom., 4, 389-411, (2004) · Zbl 1065.14071
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