On the non-very generic intersections in discriminantal arrangements

Simona Settepanella; So Yamagata

doi:10.5802/crmath.360

Combinatoire

On the non-very generic intersections in discriminantal arrangements

Simona Settepanella ¹ ; So Yamagata ²

¹ Department of Economics and Statistics, Torino University, Italy
² Department of Mathematics, Hokkaido University, Japan

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1027-1038.

Résumé

In 1985 Crapo introduced in [3] a new mathematical object that he called geometry of circuits. Four years later, in 1989, Manin and Schechtman defined in [13] the same object and called it discriminantal arrangement, the name by which it is known now a days. Those discriminantal arrangements $ℬ (n, k, 𝒜^{0})$ are builded from an arrangement $𝒜^{0}$ of $n$ hyperplanes in general position in a $k$ -dimensional space and their combinatorics depends on the arrangement $𝒜^{0}$ . On this basis, in 1997 Bayer and Brandt (see [2]) distinguished two different type of arrangements $𝒜^{0}$ calling very generic the ones for which the intersection lattice of $ℬ (n, k, 𝒜^{0})$ has maximum cardinality and non-very generic the others. Results on the combinatorics of $ℬ (n, k, 𝒜^{0})$ in the very generic case already appear in Crapo [3] and in 1997 in Athanasiadis [1] while the first known result on non-very generic case is due to Libgober and the first author in 2018. In their paper [12] they provided a necessary and sufficient condition on $𝒜^{0}$ for which the cardinality of rank 2 intersections in $ℬ (n, k, 𝒜^{0})$ is not maximal anymore. In this paper we further develop their result providing a sufficient condition on $𝒜^{0}$ for which the cardinality of rank r, $r \geq 2$ , intersections in $ℬ (n, k, 𝒜^{0})$ decreases.

Reçu le : 2022-02-12
Révisé le : 2022-03-14
Accepté le : 2022-03-31
Publié le : 2022-09-29

DOI : 10.5802/crmath.360

Classification : 52C35, 05B35, 05C99

Affiliations des auteurs :

Simona Settepanella ¹ ; So Yamagata ²

¹ Department of Economics and Statistics, Torino University, Italy
² Department of Mathematics, Hokkaido University, Japan

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On the non-very generic intersections in discriminantal arrangements},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1027--1038},
     publisher = {Acad\'emie des sciences, Paris},
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Simona Settepanella; So Yamagata. On the non-very generic intersections in discriminantal arrangements. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1027-1038. doi : 10.5802/crmath.360. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.360/

Bibliographie
Cité par

[1] Christos A. Athanasiadis The Largest Intersection Lattice of a Discriminantal Arrangement, Beitr. Algebra Geom., Volume 40 (1999) no. 2, pp. 283-289 | MR | Zbl

[2] Margaret M. Bayer; Keith A. Brandt Discriminantal arrangements, fiber polytopes and formality, J. Algebr. Comb., Volume 6 (1997) no. 3, pp. 229-246 | DOI | MR | Zbl

[3] Henry Crapo The combinatorial theory of structures, Matroid theory (A. Recski; L. Locaśz, eds.) (Colloquia Mathematica Societatis János Bolyai), Volume 40, North-Holland, 1985, pp. 107-213 | MR | Zbl

[4] Henry Crapo; Gian-Carlo Rota The resolving bracket, Invariant methods in discrete and computational geometry (Neil White, ed.), Kluwer Academic Publishers, 1995, pp. 197-222 | DOI | Zbl

[5] Michael Falk A note on discriminantal arrangements, Proc. Am. Math. Soc., Volume 122 (1994) no. 4, pp. 1221-1227 | DOI | MR | Zbl

[6] Stefan Felsner; Günter M. Ziegler Zonotopes associated wuth hugher Bruhat orders, Discrete Math., Volume 241 (2001) no. 1-3, pp. 301-312 | DOI | Zbl

[7] Mikhail M. Kapranov; Vladimir A. Voevodsky Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results), Cah. Topologie Géom. Différ. Catég., Volume 32 (1991) no. 1, pp. 11-27 | Numdam | MR | Zbl

[8] Mikhail M. Kapranov; Vladimir A. Voevodsky Free $n$ -category generated by a cube, oriented matroids, and higher Bruhat orders, Funct. Anal. Appl., Volume 25 (1991) no. 1, pp. 50-52 | MR | Zbl

[9] Mikhail M. Kapranov; Vladimir A. Voevodsky Braided monoidal 2-categories and Manin-Schechtman higher braid groups, J. Pure Appl. Algebra, Volume 92 (1994) no. 3, pp. 241-267 | DOI | MR | Zbl

[10] Toshitake Kohno Integrable connections related to Manin and Schechtman’s higher braid groups, Ill. J. Math., Volume 34 (1990) no. 2, pp. 476-484 | MR | Zbl

[11] Ruth J. Lawrence A presentation for Manin and Schechtman’s higher braid groups (1991) MSRI pre-print (http://www.ma.huji.ac.il/~ruthel/papers/91premsh.html)

[12] Anatoly Libgober; Simona Settepanella Strata of discriminantal arrangements, J. Singul., Volume 18 (2018), pp. 440-454 (Volume in honor of E. Brieskorn) | MR | Zbl

[13] Yurĭ I. Manin; Vadim V. Schechtman Arrangements of Hyperplanes, Higher Braid Groups and Higher Bruhat Orders, Algebraic number theory (Advanced Studies in Pure Mathematics), Volume 17, Academic Press, 1989, pp. 289-308 | DOI | MR | Zbl

[14] Peter Orlik Introduction to Arrangements, Regional Conference Series in Mathematics, 72, American Mathematical Society, 1989 | DOI | Zbl

[15] Peter Orlik; Hiroaki Terao Arrangements of Hyperplanes, Grundlehren der Mathematischen Wissenschaften, 300, Springer, 1992 | DOI | Zbl

[16] Markus Perling Divisorial cohomology vanishing on toric varieties, Doc. Math., Volume 16 (2011), pp. 209-251 | MR | Zbl

[17] Sumire Sawada; Simona Settepanella; So Yamagata Discriminantal arrangement, 3×3 minors of Plücker matrix and hypersurfaces in Grassmannian Gr(3,n), C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 11, pp. 1111-1120 | DOI | Zbl

[18] Sumire Sawada; Simona Settepanella; So Yamagata Pappus’s Theorem in Grassmannian $Gr (3, ℂ^{n})$ , Ars Math. Contemp., Volume 16 (2019) no. 1, pp. 257-276 | MR | Zbl

[19] Günter M. Ziegler Higher Bruhat orders and cyclic hyperplane arrangements, Topology, Volume 32 (1993) no. 2, pp. 259-279 | DOI | MR | Zbl

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