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<p>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.</p>

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Simplicial Arrangements with Few Double Points

  • 期刊论文

  • 发表日期：

    2025年11月4日

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  • Dmitri Panov,
  • Guillaume Tahar

摘要

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.