Simplicial arrangements with few double points

Dmitri Panov; Guillaume Tahar

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arXiv · 2409.01892

Simplicial arrangements with few double points

Dmitri Panov·Guillaume Tahar

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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 Gr\"{u}nbaum's conjectural asymptotic classification of simplicial arrangements holds under the additional hypothesis of a linear bound on the number of double points.

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Dmitri Panov, Guillaume Tahar. 2024-09-03. Simplicial arrangements with few double points. https://arxiv.org/abs/2409.01892

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