Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Construction of free arrangements using point-line operators

Piotr Pokora, Xavier Roulleau

TL;DR

This work develops a method to generate free plane curve arrangements in the complex projective plane using the point-line operator $\Lambda_{\mathfrak{n},\mathfrak{m}}$, constructed from a duality operation and applied to line configurations. By analyzing the resulting Milnor algebras and Alexander polynomials, the authors produce several new rigid, free arrangements, including a 57-line arrangement $H_{57}$ with exponents $(25,31)$ and trivial monodromy, a 33-line arrangement $O_{33}$ with exponents $(15,17)$ and nontrivial $\triangle(t)$, and a pencil-type conic-line arrangement $CL$ with exponents $(4,13)$ and nontrivial $\triangle(t)$. They also obtain further free arrangements $O_{61}$ and $O_{49}$ and demonstrate rigidity and definability over explicit number fields. The paper further connects these constructions to unexpected curves arising from dual point configurations and to failures of the Strong Lefschetz Property in associated Artinian algebras, highlighting deep links between combinatorics, geometry, and algebraic properties of plane curves.

Abstract

We construct new examples of free curve arrangements in the complex projective plane using point-line operators recently defined by the second author. In particular, we construct a new example of a conic-line arrangement with ordinary quasi-homogeneous singularities that has non-trivial monodromy.

Construction of free arrangements using point-line operators

TL;DR

This work develops a method to generate free plane curve arrangements in the complex projective plane using the point-line operator , constructed from a duality operation and applied to line configurations. By analyzing the resulting Milnor algebras and Alexander polynomials, the authors produce several new rigid, free arrangements, including a 57-line arrangement with exponents and trivial monodromy, a 33-line arrangement with exponents and nontrivial , and a pencil-type conic-line arrangement with exponents and nontrivial . They also obtain further free arrangements and and demonstrate rigidity and definability over explicit number fields. The paper further connects these constructions to unexpected curves arising from dual point configurations and to failures of the Strong Lefschetz Property in associated Artinian algebras, highlighting deep links between combinatorics, geometry, and algebraic properties of plane curves.

Abstract

We construct new examples of free curve arrangements in the complex projective plane using point-line operators recently defined by the second author. In particular, we construct a new example of a conic-line arrangement with ordinary quasi-homogeneous singularities that has non-trivial monodromy.
Paper Structure (3 sections, 4 theorems, 44 equations)

This paper contains 3 sections, 4 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Constructions of curve arrangements
  3. Unexpected curves and the Strong Lefschetz Property

Key Result

Theorem 3.2

The set of points $Z$ admits an unexpected curve if and only if where $d_{1}$ denotes the first exponent in the minimal resolution of the Milnor algebra associated with $\mathcal{A}_{Z}$. If these conditions are fulfilled, then $Z$ admits an unexpected curve of degree $j$ if and only if

Theorems & Definitions (20)

  • Example 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • proof
  • Remark 2.4
  • proof
  • Remark 2.5
  • ...and 10 more