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Degeneration in discriminantal arrangements

Takuya Saito

TL;DR

This work studies how discriminantal arrangements $\mathcal{B}(\mathcal{A})$ depend on the underlying geometry beyond combinatorial type by introducing non-very generic varieties and the $(\mathbb{T},r)$-singularity framework. It constructs infinite families of non-very generic examples via wheel configurations and a degeneration mechanism, and provides explicit equations (e.g., wheel and ladder varieties) to describe these strata. The results yield a stratification of the arrangement realization spaces and practical tools for identifying non-very generic cases, with complete lists and equations for small-line configurations (notably eight lines) to guide further analysis. These findings deepen the understanding of the sensitivity of discriminantal combinatorics to geometric data and connect to broader matroid and geometric-lattice theory contexts.

Abstract

Discriminantal arrangements are hyperplane arrangements, which are generalized braid ones. They are constructed from given hyperplane arrangements, but their combinatorics are not invariant under combinatorial equivalence. However, it is known that the combinatorics of the discriminantal arrangement are constant on a Zariski open set of the space of hyperplane arrangements. In the present paper, we introduce non-very generic varieties in the space of hyperplane arrangements to classify discriminantal arrangements and show that the Zariski open set is the complement of non-very generic varieties. We study their basic properties and construction and provide examples, including infinite families of non-very generic varieties. In particular, the construction we call degeneration is a powerful tool for constructing non-very generic varieties. As an application, we provide lists of non-very generic varieties for spaces of small line arrangements.

Degeneration in discriminantal arrangements

TL;DR

This work studies how discriminantal arrangements depend on the underlying geometry beyond combinatorial type by introducing non-very generic varieties and the -singularity framework. It constructs infinite families of non-very generic examples via wheel configurations and a degeneration mechanism, and provides explicit equations (e.g., wheel and ladder varieties) to describe these strata. The results yield a stratification of the arrangement realization spaces and practical tools for identifying non-very generic cases, with complete lists and equations for small-line configurations (notably eight lines) to guide further analysis. These findings deepen the understanding of the sensitivity of discriminantal combinatorics to geometric data and connect to broader matroid and geometric-lattice theory contexts.

Abstract

Discriminantal arrangements are hyperplane arrangements, which are generalized braid ones. They are constructed from given hyperplane arrangements, but their combinatorics are not invariant under combinatorial equivalence. However, it is known that the combinatorics of the discriminantal arrangement are constant on a Zariski open set of the space of hyperplane arrangements. In the present paper, we introduce non-very generic varieties in the space of hyperplane arrangements to classify discriminantal arrangements and show that the Zariski open set is the complement of non-very generic varieties. We study their basic properties and construction and provide examples, including infinite families of non-very generic varieties. In particular, the construction we call degeneration is a powerful tool for constructing non-very generic varieties. As an application, we provide lists of non-very generic varieties for spaces of small line arrangements.
Paper Structure (8 sections, 14 theorems, 38 equations, 4 figures)

This paper contains 8 sections, 14 theorems, 38 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Discriminantal arrangements
  3. Notations and basics on hyperplane arrangements
  4. The space $\mathbb{S}(\mathcal{A})$ of parallel translations of $\mathcal{A}$ and its subspaces
  5. Non-very generic arrangements
  6. Generalized Crapo configuration; wheels
  7. Deletion, contraction, degeneration
  8. Applications

Key Result

Theorem 3.2

Let $n\geq k+1\geq 2$. There exists a rank $k$ real arrangement $\mathcal{A}$ consisting of $n$ hyperplanes, which is very generic and $\mathcal{L}(\mathcal{B}(\mathcal{A}))$ isomorphic to $P(n,k)$. The isomorphism is given by the canonical presentation $\mathbb{T}:\mathcal{L}(\mathcal{B}(\mathcal{A

Figures (4)

  • Figure 1:
  • Figure 4:
  • Figure 6: representatives of the minimal non-very generic intersection with $6$ or $7$ lines
  • Figure 7: a representative of $X$ in Example \ref{['exam:K33']}

Theorems & Definitions (33)

  • Example 2.1: Crapo Cr84
  • Example 2.2: Settepanella, Yamagata SeYaArxiv22
  • Definition 3.1: Bayer--Brandt BB97, Athanasiadis Ath99
  • Theorem 3.2: Bayer--Brandt BB97, Athanasiadis Ath99
  • Definition 3.3
  • Proposition 3.4
  • Definition 3.5
  • Proposition 3.6
  • proof
  • Example 3.7
  • ...and 23 more