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Moduli spaces of arrangements of 12 projective lines with a sextic point

Meirav Amram, Eran Lieberman, Sheng-Li Tan, Mina Teicher, Xiao-Hang Wu

Abstract

*This paper is from 2018* In this paper, we try to classify moduli spaces of arrangements of $12$ lines with sextic points. We show that moduli spaces of arrangements of $12$ lines with sextic points can consist of more than two connected components. We also present defining equations of the arrangements whose moduli spaces are not irreducible taking quotients by the complex conjugation by supply some potential Zariski pairs. Through complex conjugation we take quotients and supply some potential Zariski pairs.

Moduli spaces of arrangements of 12 projective lines with a sextic point

Abstract

*This paper is from 2018* In this paper, we try to classify moduli spaces of arrangements of lines with sextic points. We show that moduli spaces of arrangements of lines with sextic points can consist of more than two connected components. We also present defining equations of the arrangements whose moduli spaces are not irreducible taking quotients by the complex conjugation by supply some potential Zariski pairs. Through complex conjugation we take quotients and supply some potential Zariski pairs.
Paper Structure (7 sections, 15 theorems, 17 equations, 3 figures)

This paper contains 7 sections, 15 theorems, 17 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Arrangements of $12$ lines with multiple points of multiplicity at least $7$
  4. Arrangements of 12 lines with a sextic point and one quadruple points
  5. Arrangements of $12$ lines with a sextic point and no quadruple point
  6. One disjoint triple point apart from the pencil of the sextic point
  7. All triple points are in the pencil of the sextic point

Key Result

Theorem 2.1

Ryb Let $\mathcal{A}$ be an arrangement of $C_3$ of simple type. Then the moduli space $\mathcal{M}_{\mathcal{A}}$ is irreducible.

Figures (3)

  • Figure 42:
  • Figure 43:
  • Figure 44:

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.1
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 2.2
  • Theorem 3.1
  • ...and 19 more