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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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/

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