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Zariski pairs of conic-line arrangements of degrees 7 and 8 via fundamental groups

Meirav Amram, Robert Shwartz, Uriel Sinichkin, Sheng-Li Tan, Hiro-o Tokunaga

TL;DR

This work addresses the problem of distinguishing Zariski pairs among conic-line plane arrangements by comparing fundamental groups of their complements. It develops a Coxeter-group–based framework: compute $\pi_1$ via the Zariski-van Kampen method, form square quotients by modding out squares of component generators, and analyze these quotients using affine and finite Coxeter structures. The key contribution is a new Zariski pair of degree $8$ with non-isomorphic square-quotient groups, together with alternative proofs for Tokunaga’s degree $7$ Zariski pairs, illustrating a systematic approach to separating combinatorially similar yet topologically distinct arrangements. This method advances understanding of the conic-line arrangement moduli space and offers a practical tool for proving Zariski non-equivalence in low degrees.

Abstract

We find a new Zariski pair with non-isomorphic fundamental groups that consists of degree $ 8 $ conic-line arrangements. Each arrangement has three conics and two lines. We use the Zariski-van Kampen Theorem and some known Coxeter groups to determine the fundamental groups. Two examples of degree $7$ Zariski pairs that were introduced in 2014 by the last named author, are given as well. They consist of a pair of conic-line arrangements with three conics in each (and thus, each has a single line) and a pair with two conics in each (and thus, each has three lines). We were able to provide alternative proof of the fact those are indeed Zariski pairs by our methods.

Zariski pairs of conic-line arrangements of degrees 7 and 8 via fundamental groups

TL;DR

This work addresses the problem of distinguishing Zariski pairs among conic-line plane arrangements by comparing fundamental groups of their complements. It develops a Coxeter-group–based framework: compute via the Zariski-van Kampen method, form square quotients by modding out squares of component generators, and analyze these quotients using affine and finite Coxeter structures. The key contribution is a new Zariski pair of degree with non-isomorphic square-quotient groups, together with alternative proofs for Tokunaga’s degree Zariski pairs, illustrating a systematic approach to separating combinatorially similar yet topologically distinct arrangements. This method advances understanding of the conic-line arrangement moduli space and offers a practical tool for proving Zariski non-equivalence in low degrees.

Abstract

We find a new Zariski pair with non-isomorphic fundamental groups that consists of degree conic-line arrangements. Each arrangement has three conics and two lines. We use the Zariski-van Kampen Theorem and some known Coxeter groups to determine the fundamental groups. Two examples of degree Zariski pairs that were introduced in 2014 by the last named author, are given as well. They consist of a pair of conic-line arrangements with three conics in each (and thus, each has a single line) and a pair with two conics in each (and thus, each has three lines). We were able to provide alternative proof of the fact those are indeed Zariski pairs by our methods.

Paper Structure

This paper contains 137 sections, 18 theorems, 526 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments:
  3. Description of the new Zariski pair
  4. Presentations of the fundamental groups
  5. Square quotients and Coxeter groups
  6. Zariski pairs
  7. Explanation about the calculations
  8. New Zariski pair with non-isomorphic fundamental groups
  9. Affine coordinates
  10. Computation of ${\pi_1(\mathbb{C} \mathbb{P}^2 - \mathcal{B}_1, *)}$
  11. Monodromy table
  12. Vertex Number 1
  13. Vertex Number 2
  14. Vertex Number 3
  15. Vertex Number 4
  16. ...and 122 more sections

Key Result

Lemma 3.1

Group ${\pi_1(\mathbb{C} \mathbb{P}^2 - \mathcal{B}_1, *)}$ is generated by $\Gamma_{C_1}, \Gamma_{C_2}, \Gamma_{C_3} , \Gamma_{L_2}$ and has the following relations:

Figures (5)

  • Figure 1: The arrangements $\mathcal{B}_1,\mathcal{B}_2$.
  • Figure 2: The arrangements $\mathcal{B}_3,\mathcal{B}_4$.
  • Figure 3: The arrangements $\mathcal{B}_5,\mathcal{B}_6$.
  • Figure 4: The arrangements $\mathcal{B}_3, \mathcal{B}_4$.
  • Figure 5: The arrangements $\mathcal{B}_5, \mathcal{B}_6$.

Theorems & Definitions (38)

  • Definition 1.1
  • Remark 1
  • Definition 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Remark 2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 28 more