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Simplicial arrangements with few double points

Dmitri Panov, Guillaume Tahar

TL;DR

The paper investigates simplicial line arrangements in $\mathbb{RP}^{2}$ with few double points, aiming to prove Grünbaum's asymptotic classification under a linear bound on double points. It leverages Green–Tao's structure theorem to show that most lines are tangential to the dual of a cubic curve and proves that the dual of an irreducible cubic cannot be tangent to most lines in a simplicial arrangement. A detailed case analysis over the possible cubic types yields quantitative bounds linking the number of lines $n$ to the number of tangent lines $k$, and a geometric rigidity analysis demonstrates that arrangements close to regular models must fall into the known families $\mathcal{R}(0)$, $\mathcal{R}(1)$, or $\mathcal{R}(2)$. Consequently, under the linear double-point bound, Grünbaum’s asymptotic classification holds, and the regular models exhibit projective rigidity. The results significantly constrain the landscape of simplicial arrangements and connect combinatorial classifications to projective-geometric rigidity phenomena.

Abstract

In their solution to the orchard-planting problem, Green and Tao established a structure theorem which proves that in a line arrangement in the real projective plane with few double points, most lines are tangent to the dual curve of a cubic curve. We provide geometric arguments to prove that in the case of a simplicial arrangement, the aforementioned cubic curve cannot be irreducible. It follows that Grünbaum's conjectural asymptotic classification of simplicial arrangements holds under the additional hypothesis of a linear bound on the number of double points.

Simplicial arrangements with few double points

TL;DR

The paper investigates simplicial line arrangements in with few double points, aiming to prove Grünbaum's asymptotic classification under a linear bound on double points. It leverages Green–Tao's structure theorem to show that most lines are tangential to the dual of a cubic curve and proves that the dual of an irreducible cubic cannot be tangent to most lines in a simplicial arrangement. A detailed case analysis over the possible cubic types yields quantitative bounds linking the number of lines to the number of tangent lines , and a geometric rigidity analysis demonstrates that arrangements close to regular models must fall into the known families , , or . Consequently, under the linear double-point bound, Grünbaum’s asymptotic classification holds, and the regular models exhibit projective rigidity. The results significantly constrain the landscape of simplicial arrangements and connect combinatorial classifications to projective-geometric rigidity phenomena.

Abstract

In their solution to the orchard-planting problem, Green and Tao established a structure theorem which proves that in a line arrangement in the real projective plane with few double points, most lines are tangent to the dual curve of a cubic curve. We provide geometric arguments to prove that in the case of a simplicial arrangement, the aforementioned cubic curve cannot be irreducible. It follows that Grünbaum's conjectural asymptotic classification of simplicial arrangements holds under the additional hypothesis of a linear bound on the number of double points.
Paper Structure (14 sections, 16 theorems, 5 equations, 3 figures, 1 table)

This paper contains 14 sections, 16 theorems, 5 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalities about simplicial arrangements
  3. Conventions
  4. Exceptional properties of family $\mathcal{R}(0)$
  5. Star of a vertex
  6. Recollection on cubic curves and their duals
  7. Estimates on lines tangent to a dual cubic in a simplicial arrangement
  8. Families of lines decomposing a convex polygon into triangles
  9. Proof of Proposition \ref{['prop:irreducible']}
  10. Proof of the main theorem
  11. Geometry of regular simplicial arrangements
  12. Arrangements differing from a regular arrangement by a small proportion of lines
  13. Proof of the main theorem
  14. Projective rigidity of regular simplicial arrangements

Key Result

Theorem 1.5

For any constant $K > 0$, there are only finitely many sporadic simplicial arrangements (combinatorial classes not belonging to $\mathcal{R}(0)$, $\mathcal{R}(1)$ or $\mathcal{R}(2)$) that have at most $Kn$ double points, where $n$ is the number of lines of the arrangement.

Figures (3)

  • Figure 1: Regular simplicial arrangement obtained from a regular $12$-gon.
  • Figure 2: The connected dual cubic and the lines of $\mathcal{A}$ tangent to it
  • Figure 3: The disconnected dual cubic and the lines of $\mathcal{A}$ tangent to it

Theorems & Definitions (41)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6: Green-Tao full structure theorem
  • Proposition 1.7
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 31 more