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On the boundedness of some real line arrangements of type at most one

Marek Janasz

TL;DR

This work investigates the boundedness of real line arrangements of type at most one under explicit multiplicity constraints on intersection points. It combines real-geometric inequalities (Melchior and Shnurnikov) with algebraic invariants from Jacobian syzygies and the Milnor algebra to derive sharp upper bounds on the number of lines. The main results show that real free line arrangements with multiplicities bounded by five have at most $522$ lines, while real plus-one generated arrangements with multiplicities bounded by four have at most $47$ lines, implying finitely many combinatorial types in each case. These findings extend boundedness phenomena beyond previously known low-multiplicity freeness results and highlight the rigidity of plus-one generated structures in the real setting, contributing to finite classification under these constraints.

Abstract

In this note, we show that real line arrangements of type at most one, admitting only intersection points of multiplicity at most five, satisfy certain boundedness properties. In particular, we prove that a free real arrangement of $d$ lines with intersection multiplicities bounded by $5$ can have at most $522$ lines and consequently there exist only finitely many combinatorial types of such arrangements.

On the boundedness of some real line arrangements of type at most one

TL;DR

This work investigates the boundedness of real line arrangements of type at most one under explicit multiplicity constraints on intersection points. It combines real-geometric inequalities (Melchior and Shnurnikov) with algebraic invariants from Jacobian syzygies and the Milnor algebra to derive sharp upper bounds on the number of lines. The main results show that real free line arrangements with multiplicities bounded by five have at most lines, while real plus-one generated arrangements with multiplicities bounded by four have at most lines, implying finitely many combinatorial types in each case. These findings extend boundedness phenomena beyond previously known low-multiplicity freeness results and highlight the rigidity of plus-one generated structures in the real setting, contributing to finite classification under these constraints.

Abstract

In this note, we show that real line arrangements of type at most one, admitting only intersection points of multiplicity at most five, satisfy certain boundedness properties. In particular, we prove that a free real arrangement of lines with intersection multiplicities bounded by can have at most lines and consequently there exist only finitely many combinatorial types of such arrangements.
Paper Structure (6 sections, 7 theorems, 43 equations)

This paper contains 6 sections, 7 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The boundedness of real free line arrangements with points of multiplicity at most five
  4. The case $d\ge k+3$.
  5. General bound.
  6. The boundedness of real plus-one generated line arrangements with points of multiplicity at most four

Key Result

Theorem 2.5

Let $C = \{f=0\}\subset \mathbb{P}^{2}_{\mathbb{C}}$ be a reduced curve of degree $d$ and let $d_1 = {\rm mdr}(f)$. Let us denote the total Tjurina number of $C$ by $\tau(C)$. Then the following two cases hold.

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2: her
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: du Plessis -- Wall, duP
  • Proposition 2.6: Dimca -- Sticlaru, DS20
  • Theorem 3.1: Shnurnikov
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 5 more