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Double star arrangement and the pointed multinet

Yongqiang Liu, Wentao Xie

TL;DR

The paper investigates whether translated components of the degree-one cohomology jump loci $\\mathcal{V}^1$ for complex hyperplane arrangement complements are determined by combinatorics. It reviews the structure of $\\mathcal{V}^1$ via orbifold fibrations and multinet/pointed multinet constructions, and shows that translated components can arise beyond pointed multinets. The main result is a counterexample provided by the double star arrangement, which yields a small orbifold fibration with a multiple fiber not formed by lines, producing a translated 1-dimensional component not attributable to pointed multinets. This demonstrates limitations in obtaining translated components purely from combinatorial data and connects to 2-torsion phenomena in the Aomoto complex.

Abstract

Let $\mathcal{A}$ be a hyperplane arrangement in a complex projective space. It is an open question if the degree one cohomology jump loci (with complex coefficients) are determined by the combinatorics of $\mathcal{A}$. By the work of Falk and Yuzvinsky \cite{FY}, all the irreducible components passing through the origin are determined by the multinet structure, which are combinatorially determined. Denham and Suciu introduced the pointed multinet structure to obtain examples of arrangements with translated positive-dimensional components in the degree one cohomology jump loci \cite{DS}. Suciu asked the question if all translated positive-dimensional components appear in this manner \cite{Suc14}. In this paper, we show that the double star arrangement introduced by Ishibashi, Sugawara and Yoshinaga \cite[Example 3.2]{ISY22} gives a negative answer to this question.

Double star arrangement and the pointed multinet

TL;DR

The paper investigates whether translated components of the degree-one cohomology jump loci for complex hyperplane arrangement complements are determined by combinatorics. It reviews the structure of via orbifold fibrations and multinet/pointed multinet constructions, and shows that translated components can arise beyond pointed multinets. The main result is a counterexample provided by the double star arrangement, which yields a small orbifold fibration with a multiple fiber not formed by lines, producing a translated 1-dimensional component not attributable to pointed multinets. This demonstrates limitations in obtaining translated components purely from combinatorial data and connects to 2-torsion phenomena in the Aomoto complex.

Abstract

Let be a hyperplane arrangement in a complex projective space. It is an open question if the degree one cohomology jump loci (with complex coefficients) are determined by the combinatorics of . By the work of Falk and Yuzvinsky \cite{FY}, all the irreducible components passing through the origin are determined by the multinet structure, which are combinatorially determined. Denham and Suciu introduced the pointed multinet structure to obtain examples of arrangements with translated positive-dimensional components in the degree one cohomology jump loci \cite{DS}. Suciu asked the question if all translated positive-dimensional components appear in this manner \cite{Suc14}. In this paper, we show that the double star arrangement introduced by Ishibashi, Sugawara and Yoshinaga \cite[Example 3.2]{ISY22} gives a negative answer to this question.
Paper Structure (5 sections, 5 theorems, 15 equations, 2 figures)

This paper contains 5 sections, 5 theorems, 15 equations, 2 figures.

Table of Contents

  1. Background
  2. Cohomology jump loci
  3. Multinet
  4. Pointed multinet
  5. The double star arrangement

Key Result

Theorem 1.2

Ara97ACM13 Let $\mathcal{A}$ be an essential line arrangement in $\mathbb{CP}^2$. Then we have where the unions are over the equivalence classes of orbifold fibration $f \colon X(\mathcal{A})\to \Sigma_k$ of the types indicated, and $Z$ is a finite union of isolated torsion points. In particular, $\mathcal{V}^1(\mathcal{A})$ has translated positive-dimensional components if and only if at lea

Figures (2)

  • Figure 1: $(3,4)$-multinet on $B_3$-arrangement
  • Figure 2: The double star arrangement

Theorems & Definitions (12)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.7
  • Proposition 1.8
  • Example 1.9
  • Theorem 2.1
  • ...and 2 more