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On the Alexander polynomials of conic-line arrangements

Alexandru Dimca, Piotr Pokora, Gabriel Sticlaru

TL;DR

This work studies the Alexander polynomials $\Delta^1_C(t)$ of conic-line arrangements in $\mathbb{P}^2$ arising from Halphen pencils, focusing on Hughes–type and Hesse configurations. It develops general tools (notably Theorems PEN, PEN2, and PEN3) linking pencil geometry to roots of $\Delta^1_C(t)$, and applies them to conic pencils to produce explicit, nontrivial examples, including roots of unity of order $7$. The centerpiece is the 1-parameter Hesse family $\mathcal{C}_{12}(\lambda)$ of 12 conics, which is free with exponents $(7,16)$ and has a computable, factorized Alexander polynomial, along with a detailed study of degenerations $\mathcal{A}(\lambda)$ and $\mathcal{B}(\lambda)$ that exhibit freeness and specific roots of unity (orders $6$ and $7$). These results broaden the understanding of how freeness and pencil structure influence Alexander polynomials for conic-line arrangements, offering new phenomena not typically seen in line arrangements.

Abstract

In the present paper we compute Alexander polynomials for certain classes of conic-line arrangements in the complex projective plane which are related to pencils. We prove two general results for curve arrangements coming from Halphen pencils of index $k\geq 2$. Then we apply them to the Hesse arrangement of conics and to some of its degenerations. The results are completed by computations using computer algebra. In particular, we construct conic-line arrangements which are non-reduced pencil-type arrangements and have as roots of their Alexander polynomials roots of unity of order 7. Such roots are not known and are conjectured not to exist in the class of line arrangements.

On the Alexander polynomials of conic-line arrangements

TL;DR

This work studies the Alexander polynomials of conic-line arrangements in arising from Halphen pencils, focusing on Hughes–type and Hesse configurations. It develops general tools (notably Theorems PEN, PEN2, and PEN3) linking pencil geometry to roots of , and applies them to conic pencils to produce explicit, nontrivial examples, including roots of unity of order . The centerpiece is the 1-parameter Hesse family of 12 conics, which is free with exponents and has a computable, factorized Alexander polynomial, along with a detailed study of degenerations and that exhibit freeness and specific roots of unity (orders and ). These results broaden the understanding of how freeness and pencil structure influence Alexander polynomials for conic-line arrangements, offering new phenomena not typically seen in line arrangements.

Abstract

In the present paper we compute Alexander polynomials for certain classes of conic-line arrangements in the complex projective plane which are related to pencils. We prove two general results for curve arrangements coming from Halphen pencils of index . Then we apply them to the Hesse arrangement of conics and to some of its degenerations. The results are completed by computations using computer algebra. In particular, we construct conic-line arrangements which are non-reduced pencil-type arrangements and have as roots of their Alexander polynomials roots of unity of order 7. Such roots are not known and are conjectured not to exist in the class of line arrangements.
Paper Structure (6 sections, 16 theorems, 95 equations)

This paper contains 6 sections, 16 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Some general properties of Alexander polynomials of curve arrangements
  3. Pencils of curves and Alexander polynomials
  4. Free curves and pencils of conics
  5. The family of the Hesse arrangements of conics
  6. The degenerations of the family of Hesse arrangements of conics

Key Result

Theorem 2.1

Let $C=\cup_{i=1}^sC_i$ be a curve arrangement in $\mathbb{P}^2$ such that the curve $C_1$ is irreducible. For any singular point $p$ of $C$ situated on $C_1$, let $\Delta^1_{(C,p)}(t)$ be the Alexander polynomial of the singularity $(C,p)$. Then the following conditions hold.

Theorems & Definitions (30)

  • Conjecture 1.1
  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Remark 3.5
  • Definition 4.1
  • ...and 20 more