We prove that the open subvariety
Accepted:
Published online:
Keywords: Chow quotient, Grassmannian, matroid, tight span
Corey, Daniel 1; Luber, Dante 2

@article{ALCO_2023__6_5_1273_0, author = {Corey, Daniel and Luber, Dante}, title = {The {Grassmannian} of $3$-planes in $\mathbb{C}^{8}$ is sch\"on}, journal = {Algebraic Combinatorics}, pages = {1273--1299}, publisher = {The Combinatorics Consortium}, volume = {6}, number = {5}, year = {2023}, doi = {10.5802/alco.302}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.302/} }
TY - JOUR AU - Corey, Daniel AU - Luber, Dante TI - The Grassmannian of $3$-planes in $\mathbb{C}^{8}$ is schön JO - Algebraic Combinatorics PY - 2023 SP - 1273 EP - 1299 VL - 6 IS - 5 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.302/ DO - 10.5802/alco.302 LA - en ID - ALCO_2023__6_5_1273_0 ER -
%0 Journal Article %A Corey, Daniel %A Luber, Dante %T The Grassmannian of $3$-planes in $\mathbb{C}^{8}$ is schön %J Algebraic Combinatorics %D 2023 %P 1273-1299 %V 6 %N 5 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.302/ %R 10.5802/alco.302 %G en %F ALCO_2023__6_5_1273_0
Corey, Daniel; Luber, Dante. The Grassmannian of $3$-planes in $\mathbb{C}^{8}$ is schön. Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1273-1299. doi : 10.5802/alco.302. https://alco.centre-mersenne.org/articles/10.5802/alco.302/
[1] Category theory, Oxford Logic Guides, 52, Oxford University Press, Oxford, 2010, xvi+311 pages
[2] Massively parallel computation of tropical varieties, their positive part, and tropical Grassmannians, J. Symbolic Comput., Volume 120 (2024), Paper no. 102224, 28 pages | MR | Zbl
[3] Julia: a fresh approach to numerical computing, SIAM Rev., Volume 59 (2017) no. 1, pp. 65-98 | DOI | MR | Zbl
[4] Computational synthetic geometry, Lecture Notes in Mathematics, 1355, Springer-Verlag, Berlin, 1989, vi+168 pages | DOI
[5] Ample families, multihomogeneous spectra, and algebraization of formal schemes, Pacific J. Math., Volume 208 (2003) no. 2, pp. 209-230 | DOI | MR | Zbl
[6] Initial degenerations of Grassmannians, Sel. Math. New Ser., Volume 27 (2021), Paper no. 57, 40 pages (with an appendix by María Angélica Cueto) | MR | Zbl
[7] Initial degenerations of spinor varieties, 2022 | arXiv
[8] Singular matroid realization spaces, 2023 | arXiv
[9] Triangulations. Structures for Algorithms and Applications, Algorithms and Computation in Mathematics, 25, Springer-Verlag, Berlin, 2010, xiv+535 pages
[10] The OSCAR book (Eder, Christian; Decker, Wolfram; Fieker, Claus; Horn, Max; Joswig, Michael, eds.), 2024
[11] polymake: a framework for analyzing convex polytopes, Polytopes—combinatorics and computation (Oberwolfach, 1997) (DMV Sem.), Volume 29, Birkhäuser, Basel, 2000, pp. 43-73 | DOI | MR | Zbl
[12] Combinatorial geometries, convex polyhedra, and Schubert cells, Adv. in Math., Volume 63 (1987) no. 3, pp. 301-316 | DOI | MR | Zbl
[13] Geometry in Grassmannians and a generalization of the dilogarithm, Adv. in Math., Volume 44 (1982) no. 3, pp. 279-312 | DOI | MR | Zbl
[14] Combinatorial geometries and the strata of a torus on homogeneous compact manifolds, Uspekhi Mat. Nauk, Volume 42 (1987) no. 2(254), p. 107-134, 287 | MR
[15] Configurations and their realization, Discrete Math., Volume 174 (1997) no. 1-3, pp. 137-151 Combinatorics (Rome and Montesilvano, 1994) | DOI | MR | Zbl
[16] Configurations of Points and Lines, Graduate Studies in Mathematics, 103, American Mathematical Society, Providence, RI, 2009, 399 pages
[17] Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces, Invent. Math., Volume 178 (2009) no. 1, pp. 173-227 | DOI | MR | Zbl
[18] Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977, xvi+496 pages | DOI
[19] Monodromy filtrations and the topology of tropical varieties, Canad. J. Math., Volume 64 (2012) no. 4, pp. 845-868 | DOI | MR | Zbl
[20] On the facets of the secondary polytope, J. Combin. Theory Ser. A, Volume 118 (2011) no. 2, pp. 425-447 | DOI | MR | Zbl
[21] How to Draw Tropical Planes, Electron. J. Combin., Volume 16 (2009) no. 2, Paper no. R6, 26 pages | MR | Zbl
[22] Splitting polytopes, Münster J. Math., Volume 1 (2008), pp. 109-141 | MR | Zbl
[23] Polymake.jl: A New Interface to polymake, Mathematical Software – ICMS 2020 (Bigatti, Anna Maria; Carette, Jacques; Davenport, James H.; Joswig, Michael; de Wolff, Timo, eds.), Springer International Publishing, Cham (2020), pp. 377-385 | DOI | Zbl
[24] Quotients of toric varieties, Math. Ann., Volume 290 (1991) no. 4, pp. 643-655 | DOI | MR | Zbl
[25] Chow quotients of Grassmannians. I, I. M. Gelfand Seminar (Adv. Soviet Math.), Volume 16, Amer. Math. Soc., Providence, RI, 1993, pp. 29-110 | DOI | MR | Zbl
[26] Realization spaces for tropical fans, Combinatorial aspects of commutative algebra and algebraic geometry (Abel Symp.), Volume 6, Springer, Berlin, 2011, pp. 73-88 | DOI | MR | Zbl
[27] Contractible extremal rays on
[28] Geometry of Chow quotients of Grassmannians, Duke Math. J., Volume 134 (2006) no. 2, pp. 259-311 | MR | Zbl
[29] Positroid varieties: juggling and geometry, Compos. Math., Volume 149 (2013) no. 10, pp. 1710-1752 | DOI | MR | Zbl
[30] Chirurgie des grassmanniennes, CRM Monograph Series, 19, American Mathematical Society, Providence, RI, 2003, xx+170 pages | DOI | MR
[31] Mnëv-Sturmfels universality for schemes, A celebration of algebraic geometry (Clay Math. Proc.), Volume 18, Amer. Math. Soc., Providence, RI, 2013, pp. 457-468 | Zbl
[32] The log canonical compactification of the moduli space of six lines in
[33] Some results on tropical compactifications, Trans. Amer. Math. Soc., Volume 363 (2011) no. 9, pp. 4853-4876 | DOI | MR | Zbl
[34] Introduction to Tropical Geometry, Graduate Studies in Mathematics, 161, American Mathematical Society, Providence, RI, 2015, vii+359 pages | DOI
[35] Matroid enumeration for incidence geometry, Discrete Comput. Geom., Volume 47 (2012) no. 1, pp. 17-43 | DOI | MR | Zbl
[36] On the connectivity of the realization spaces of line arrangements, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 11 (2012) no. 4, pp. 921-937 | Numdam | MR | Zbl
[37] On local Dressians of matroids, Algebraic and geometric combinatorics on lattice polytopes, World Sci. Publ., Hackensack, NJ, 2019, pp. 309-329 | DOI | Zbl
[38] OSCAR – Open Source Computer Algebra Research system, Version 0.10.1, 2022 https://oscar.computeralgebra.de
[39] Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992, xii+532 pages
[40] Fibers of tropicalization, Math. Z., Volume 262 (2009) no. 2, pp. 301-311 | DOI | MR | Zbl
[41] Quasilinear tropical compactifications, 2021 | arXiv
[42] Tropical linear spaces, SIAM J. Discrete Math., Volume 22 (2008) no. 4, pp. 1527-1558 | DOI | MR | Zbl
[43] The tropical Grassmannian, Adv. Geom., Volume 4 (2004) no. 3, pp. 389-411 | DOI | MR | Zbl
[44] The Stacks project, https://stacks.math.columbia.edu, 2022
[45] Compactifications of subvarieties of tori, Amer. J. Math., Volume 129 (2007) no. 4, pp. 1087-1104 | DOI | MR | Zbl
[46] Murphy’s law in algebraic geometry: badly-behaved deformation spaces, Invent. Math., Volume 164 (2006) no. 3, pp. 569-590 | DOI | MR | Zbl
[47] Newton polyhedra and estimates of oscillatory integrals, Funkcional. Anal. i Priložen., Volume 10 (1976) no. 3, pp. 13-38 | MR
[48] Zeta-function of monodromy and Newton’s diagram, Invent. Math., Volume 37 (1976) no. 3, pp. 253-262 | DOI | MR | Zbl
- Quasilinear tropical compactifications, Advances in Mathematics, Volume 461 (2025), p. 110037 | DOI:10.1016/j.aim.2024.110037
- Matroids, The Computer Algebra System OSCAR, Volume 32 (2025), p. 351 | DOI:10.1007/978-3-031-62127-7_14
Cited by 2 documents. Sources: Crossref