On the non-very generic intersections in discriminantal arrangements

Settepanella, Simona; Yamagata, So

Comptes Rendus. Mathématique, Tome 360 (2022) no. G9, pp. 1027-1038.

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, $r\ge 2$, intersections in $ℬ(n,k,{𝒜}^0)$ decreases.

Classification : 52C35, 05B35, 05C99

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